Find the Function Rule table[[x,y],[0,2/3],[1,2],[2,6],[3,18],[4,54],[5,162]]

Math
xy02312263184545162
Check if the function rule is linear.
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To find if the table follows a function rule, check to see if the values follow the linear form y=ax+b.
y=ax+b
Build a set of equations from the table such that y=ax+b.
23=a(0)+b2=a(1)+b6=a(2)+b18=a(3)+b54=a(4)+b162=a(5)+b
Calculate the values of a and b.
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Simplify each equation.
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Simplify a(0)+b.
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Multiply a by 0.
23=0+b
2=a(1)+b
6=a(2)+b
18=a(3)+b
54=a(4)+b
162=a(5)+b
Add 0 and b.
23=b
2=a(1)+b
6=a(2)+b
18=a(3)+b
54=a(4)+b
162=a(5)+b
23=b
2=a(1)+b
6=a(2)+b
18=a(3)+b
54=a(4)+b
162=a(5)+b
Multiply a by 1.
23=b
2=a+b
6=a(2)+b
18=a(3)+b
54=a(4)+b
162=a(5)+b
Move 2 to the left of a.
23=b
2=a+b
6=2a+b
18=a(3)+b
54=a(4)+b
162=a(5)+b
Move 3 to the left of a.
23=b
2=a+b
6=2a+b
18=3a+b
54=a(4)+b
162=a(5)+b
Move 4 to the left of a.
23=b
2=a+b
6=2a+b
18=3a+b
54=4a+b
162=a(5)+b
Move 5 to the left of a.
23=b
2=a+b
6=2a+b
18=3a+b
54=4a+b
162=5a+b
23=b
2=a+b
6=2a+b
18=3a+b
54=4a+b
162=5a+b
Rewrite the equation as b=23.
b=23
2=a+b
6=2a+b
18=3a+b
54=4a+b
162=5a+b
Replace all occurrences of b with 23 in each equation.
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Replace all occurrences of b in 2=a+b with 23.
b=23
2=a+23
6=2a+b
18=3a+b
54=4a+b
162=5a+b
Replace all occurrences of b in 6=2a+b with 23.
b=23
2=a+23
6=2a+23
18=3a+b
54=4a+b
162=5a+b
Replace all occurrences of b in 18=3a+b with 23.
b=23
2=a+23
6=2a+23
18=3a+23
54=4a+b
162=5a+b
Replace all occurrences of b in 54=4a+b with 23.
b=23
2=a+23
6=2a+23
18=3a+23
54=4a+23
162=5a+b
Replace all occurrences of b in 162=5a+b with 23.
b=23
2=a+23
6=2a+23
18=3a+23
54=4a+23
162=5a+23
b=23
2=a+23
6=2a+23
18=3a+23
54=4a+23
162=5a+23
Simplify.
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Remove parentheses.
b=23
2=a+23
6=2a+23
18=3a+23
54=4a+23
162=5a+23
Remove parentheses.
b=23
2=a+23
6=2a+23
18=3a+23
54=4a+23
162=5a+23
Remove parentheses.
b=23
2=a+23
6=2a+23
18=3a+23
54=4a+23
162=5a+23
Remove parentheses.
b=23
2=a+23
6=2a+23
18=3a+23
54=4a+23
162=5a+23
Remove parentheses.
b=23
2=a+23
6=2a+23
18=3a+23
54=4a+23
162=5a+23
b=23
2=a+23
6=2a+23
18=3a+23
54=4a+23
162=5a+23
Solve for a in the second equation.
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Rewrite the equation as a+23=2.
b=23
a+23=2
6=2a+23
18=3a+23
54=4a+23
162=5a+23
Move all terms not containing a to the right side of the equation.
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Subtract 23 from both sides of the equation.
b=23
a=2-23
6=2a+23
18=3a+23
54=4a+23
162=5a+23
To write 2 as a fraction with a common denominator, multiply by 33.
b=23
a=2⋅33-23
6=2a+23
18=3a+23
54=4a+23
162=5a+23
Combine 2 and 33.
b=23
a=2⋅33-23
6=2a+23
18=3a+23
54=4a+23
162=5a+23
Combine the numerators over the common denominator.
b=23
a=2⋅3-23
6=2a+23
18=3a+23
54=4a+23
162=5a+23
Simplify the numerator.
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Multiply 2 by 3.
b=23
a=6-23
6=2a+23
18=3a+23
54=4a+23
162=5a+23
Subtract 2 from 6.
b=23
a=43
6=2a+23
18=3a+23
54=4a+23
162=5a+23
b=23
a=43
6=2a+23
18=3a+23
54=4a+23
162=5a+23
b=23
a=43
6=2a+23
18=3a+23
54=4a+23
162=5a+23
b=23
a=43
6=2a+23
18=3a+23
54=4a+23
162=5a+23
Replace all occurrences of a with 43 in each equation.
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Replace all occurrences of a in 6=2a+23 with 43.
b=23
a=43
6=2(43)+23
18=3a+23
54=4a+23
162=5a+23
Replace all occurrences of a in 18=3a+23 with 43.
b=23
a=43
6=2(43)+23
18=3(43)+23
54=4a+23
162=5a+23
Replace all occurrences of a in 54=4a+23 with 43.
b=23
a=43
6=2(43)+23
18=3(43)+23
54=4(43)+23
162=5a+23
Replace all occurrences of a in 162=5a+23 with 43.
b=23
a=43
6=2(43)+23
18=3(43)+23
54=4(43)+23
162=5(43)+23
b=23
a=43
6=2(43)+23
18=3(43)+23
54=4(43)+23
162=5(43)+23
Simplify.
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Simplify 2(43)+23.
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Multiply 2(43).
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Combine 2 and 43.
b=23
a=43
6=2⋅43+23
18=3(43)+23
54=4(43)+23
162=5(43)+23
Multiply 2 by 4.
b=23
a=43
6=83+23
18=3(43)+23
54=4(43)+23
162=5(43)+23
b=23
a=43
6=83+23
18=3(43)+23
54=4(43)+23
162=5(43)+23
Combine the numerators over the common denominator.
b=23
a=43
6=8+23
18=3(43)+23
54=4(43)+23
162=5(43)+23
Add 8 and 2.
b=23
a=43
6=103
18=3(43)+23
54=4(43)+23
162=5(43)+23
b=23
a=43
6=103
18=3(43)+23
54=4(43)+23
162=5(43)+23
Simplify 3(43)+23.
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Cancel the common factor of 3.
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Cancel the common factor.
b=23
a=43
6=103
18=3(43)+23
54=4(43)+23
162=5(43)+23
Rewrite the expression.
b=23
a=43
6=103
18=4+23
54=4(43)+23
162=5(43)+23
b=23
a=43
6=103
18=4+23
54=4(43)+23
162=5(43)+23
To write 4 as a fraction with a common denominator, multiply by 33.
b=23
a=43
6=103
18=4⋅33+23
54=4(43)+23
162=5(43)+23
Combine 4 and 33.
b=23
a=43
6=103
18=4⋅33+23
54=4(43)+23
162=5(43)+23
Combine the numerators over the common denominator.
b=23
a=43
6=103
18=4⋅3+23
54=4(43)+23
162=5(43)+23
Simplify the numerator.
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Multiply 4 by 3.
b=23
a=43
6=103
18=12+23
54=4(43)+23
162=5(43)+23
Add 12 and 2.
b=23
a=43
6=103
18=143
54=4(43)+23
162=5(43)+23
b=23
a=43
6=103
18=143
54=4(43)+23
162=5(43)+23
b=23
a=43
6=103
18=143
54=4(43)+23
162=5(43)+23
Simplify 4(43)+23.
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Multiply 4(43).
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Combine 4 and 43.
b=23
a=43
6=103
18=143
54=4⋅43+23
162=5(43)+23
Multiply 4 by 4.
b=23
a=43
6=103
18=143
54=163+23
162=5(43)+23
b=23
a=43
6=103
18=143
54=163+23
162=5(43)+23
Combine the numerators over the common denominator.
b=23
a=43
6=103
18=143
54=16+23
162=5(43)+23
Add 16 and 2.
b=23
a=43
6=103
18=143
54=183
162=5(43)+23
Divide 18 by 3.
b=23
a=43
6=103
18=143
54=6
162=5(43)+23
b=23
a=43
6=103
18=143
54=6
162=5(43)+23
Simplify 5(43)+23.
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Multiply 5(43).
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Combine 5 and 43.
b=23
a=43
6=103
18=143
54=6
162=5⋅43+23
Multiply 5 by 4.
b=23
a=43
6=103
18=143
54=6
162=203+23
b=23
a=43
6=103
18=143
54=6
162=203+23
Combine the numerators over the common denominator.
b=23
a=43
6=103
18=143
54=6
162=20+23
Add 20 and 2.
b=23
a=43
6=103
18=143
54=6
162=223
b=23
a=43
6=103
18=143
54=6
162=223
b=23
a=43
6=103
18=143
54=6
162=223
Since 6≠103, there are no solutions.
b=23
a=43
No solution
18=143
54=6
162=223
Since 18≠143, there are no solutions.
b=23
a=43
No solution
No solution
54=6
162=223
Since 54≠6, there are no solutions.
b=23
a=43
No solution
No solution
No solution
162=223
Since 162≠223, there are no solutions.
b=23
a=43
No solution
No solution
No solution
No solution
b=23
a=43
No solution
No solution
No solution
No solution
Calculate the value of y using each x value in the relation and compare this value to the given y value in the relation.
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Calculate the value of y when a=43, b=23, and x=0.
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Multiply 43 by 0.
y=0+23
Add 0 and 23.
y=23
y=23
If the table has a linear function rule, y=y for the corresponding x value, x=0. This check passes since y=23 and y=23.
23=23
Calculate the value of y when a=43, b=23, and x=1.
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Multiply 43 by 1.
y=43+23
Combine the numerators over the common denominator.
y=4+23
Add 4 and 2.
y=63
Divide 6 by 3.
y=2
y=2
If the table has a linear function rule, y=y for the corresponding x value, x=1. This check passes since y=2 and y=2.
2=2
Calculate the value of y when a=43, b=23, and x=2.
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Multiply (43)(2).
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Combine 43 and 2.
y=4⋅23+23
Multiply 4 by 2.
y=83+23
y=83+23
Combine the numerators over the common denominator.
y=8+23
Add 8 and 2.
y=103
y=103
If the table has a linear function rule, y=y for the corresponding x value, x=2. This check does not pass, since y=103 and y=6. The function rule can’t be linear.
103≠6
Since y≠y for the corresponding x values, the function is not linear.
The function is not linear
The function is not linear
The function is not linear
Check if the function rule is quadratic.
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To find if the table follows a function rule, check whether the function rule could follow the form y=ax2+bx+c.
y=ax2+bx+c
Build a set of 3 equations from the table such that y=ax2+bx+c.
Calculate the values of a, b, and c.
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Simplify each equation.
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Simplify a(0)2+b(0)+c.
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Simplify each term.
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Raising 0 to any positive power yields 0.
23=a⋅0+b(0)+c
2=a(1)2+b(1)+c
6=a(2)2+b(2)+c
18=a(3)2+b(3)+c
54=a(4)2+b(4)+c
162=a(5)2+b(5)+c
Multiply a by 0.
23=0+b(0)+c
2=a(1)2+b(1)+c
6=a(2)2+b(2)+c
18=a(3)2+b(3)+c
54=a(4)2+b(4)+c
162=a(5)2+b(5)+c
Multiply b by 0.
23=0+0+c
2=a(1)2+b(1)+c
6=a(2)2+b(2)+c
18=a(3)2+b(3)+c
54=a(4)2+b(4)+c
162=a(5)2+b(5)+c
23=0+0+c
2=a(1)2+b(1)+c
6=a(2)2+b(2)+c
18=a(3)2+b(3)+c
54=a(4)2+b(4)+c
162=a(5)2+b(5)+c
Combine the opposite terms in 0+0+c.
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Add 0 and 0.
23=0+c
2=a(1)2+b(1)+c
6=a(2)2+b(2)+c
18=a(3)2+b(3)+c
54=a(4)2+b(4)+c
162=a(5)2+b(5)+c
Add 0 and c.
23=c
2=a(1)2+b(1)+c
6=a(2)2+b(2)+c
18=a(3)2+b(3)+c
54=a(4)2+b(4)+c
162=a(5)2+b(5)+c
23=c
2=a(1)2+b(1)+c
6=a(2)2+b(2)+c
18=a(3)2+b(3)+c
54=a(4)2+b(4)+c
162=a(5)2+b(5)+c
23=c
2=a(1)2+b(1)+c
6=a(2)2+b(2)+c
18=a(3)2+b(3)+c
54=a(4)2+b(4)+c
162=a(5)2+b(5)+c
Simplify each term.
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One to any power is one.
23=c
2=a⋅1+b(1)+c
6=a(2)2+b(2)+c
18=a(3)2+b(3)+c
54=a(4)2+b(4)+c
162=a(5)2+b(5)+c
Multiply a by 1.
23=c
2=a+b(1)+c
6=a(2)2+b(2)+c
18=a(3)2+b(3)+c
54=a(4)2+b(4)+c
162=a(5)2+b(5)+c
Multiply b by 1.
23=c
2=a+b+c
6=a(2)2+b(2)+c
18=a(3)2+b(3)+c
54=a(4)2+b(4)+c
162=a(5)2+b(5)+c
23=c
2=a+b+c
6=a(2)2+b(2)+c
18=a(3)2+b(3)+c
54=a(4)2+b(4)+c
162=a(5)2+b(5)+c
Simplify each term.
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Raise 2 to the power of 2.
23=c
2=a+b+c
6=a⋅4+b(2)+c
18=a(3)2+b(3)+c
54=a(4)2+b(4)+c
162=a(5)2+b(5)+c
Move 4 to the left of a.
23=c
2=a+b+c
6=4⋅a+b(2)+c
18=a(3)2+b(3)+c
54=a(4)2+b(4)+c
162=a(5)2+b(5)+c
Move 2 to the left of b.
23=c
2=a+b+c
6=4a+2b+c
18=a(3)2+b(3)+c
54=a(4)2+b(4)+c
162=a(5)2+b(5)+c
23=c
2=a+b+c
6=4a+2b+c
18=a(3)2+b(3)+c
54=a(4)2+b(4)+c
162=a(5)2+b(5)+c
Simplify each term.
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Raise 3 to the power of 2.
23=c
2=a+b+c
6=4a+2b+c
18=a⋅9+b(3)+c
54=a(4)2+b(4)+c
162=a(5)2+b(5)+c
Move 9 to the left of a.
23=c
2=a+b+c
6=4a+2b+c
18=9⋅a+b(3)+c
54=a(4)2+b(4)+c
162=a(5)2+b(5)+c
Move 3 to the left of b.
23=c
2=a+b+c
6=4a+2b+c
18=9a+3b+c
54=a(4)2+b(4)+c
162=a(5)2+b(5)+c
23=c
2=a+b+c
6=4a+2b+c
18=9a+3b+c
54=a(4)2+b(4)+c
162=a(5)2+b(5)+c
Simplify each term.
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Raise 4 to the power of 2.
23=c
2=a+b+c
6=4a+2b+c
18=9a+3b+c
54=a⋅16+b(4)+c
162=a(5)2+b(5)+c
Move 16 to the left of a.
23=c
2=a+b+c
6=4a+2b+c
18=9a+3b+c
54=16⋅a+b(4)+c
162=a(5)2+b(5)+c
Move 4 to the left of b.
23=c
2=a+b+c
6=4a+2b+c
18=9a+3b+c
54=16a+4b+c
162=a(5)2+b(5)+c
23=c
2=a+b+c
6=4a+2b+c
18=9a+3b+c
54=16a+4b+c
162=a(5)2+b(5)+c
Simplify each term.
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Raise 5 to the power of 2.
23=c
2=a+b+c
6=4a+2b+c
18=9a+3b+c
54=16a+4b+c
162=a⋅25+b(5)+c
Move 25 to the left of a.
23=c
2=a+b+c
6=4a+2b+c
18=9a+3b+c
54=16a+4b+c
162=25⋅a+b(5)+c
Move 5 to the left of b.
23=c
2=a+b+c
6=4a+2b+c
18=9a+3b+c
54=16a+4b+c
162=25a+5b+c
23=c
2=a+b+c
6=4a+2b+c
18=9a+3b+c
54=16a+4b+c
162=25a+5b+c
23=c
2=a+b+c
6=4a+2b+c
18=9a+3b+c
54=16a+4b+c
162=25a+5b+c
Rewrite the equation as c=23.
c=23
2=a+b+c
6=4a+2b+c
18=9a+3b+c
54=16a+4b+c
162=25a+5b+c
Replace all occurrences of c with 23 in each equation.
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Replace all occurrences of c in 2=a+b+c with 23.
c=23
2=a+b+23
6=4a+2b+c
18=9a+3b+c
54=16a+4b+c
162=25a+5b+c
Replace all occurrences of c in 6=4a+2b+c with 23.
c=23
2=a+b+23
6=4a+2b+23
18=9a+3b+c
54=16a+4b+c
162=25a+5b+c
Replace all occurrences of c in 18=9a+3b+c with 23.
c=23
2=a+b+23
6=4a+2b+23
18=9a+3b+23
54=16a+4b+c
162=25a+5b+c
Replace all occurrences of c in 54=16a+4b+c with 23.
c=23
2=a+b+23
6=4a+2b+23
18=9a+3b+23
54=16a+4b+23
162=25a+5b+c
Replace all occurrences of c in 162=25a+5b+c with 23.
c=23
2=a+b+23
6=4a+2b+23
18=9a+3b+23
54=16a+4b+23
162=25a+5b+23
c=23
2=a+b+23
6=4a+2b+23
18=9a+3b+23
54=16a+4b+23
162=25a+5b+23
Simplify.
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Remove parentheses.
c=23
2=a+b+23
6=4a+2b+23
18=9a+3b+23
54=16a+4b+23
162=25a+5b+23
Remove parentheses.
c=23
2=a+b+23
6=4a+2b+23
18=9a+3b+23
54=16a+4b+23
162=25a+5b+23
Remove parentheses.
c=23
2=a+b+23
6=4a+2b+23
18=9a+3b+23
54=16a+4b+23
162=25a+5b+23
Remove parentheses.
c=23
2=a+b+23
6=4a+2b+23
18=9a+3b+23
54=16a+4b+23
162=25a+5b+23
Remove parentheses.
c=23
2=a+b+23
6=4a+2b+23
18=9a+3b+23
54=16a+4b+23
162=25a+5b+23
c=23
2=a+b+23
6=4a+2b+23
18=9a+3b+23
54=16a+4b+23
162=25a+5b+23
Solve for a in the second equation.
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Rewrite the equation as a+b+23=2.
c=23
a+b+23=2
6=4a+2b+23
18=9a+3b+23
54=16a+4b+23
162=25a+5b+23
Move all terms not containing a to the right side of the equation.
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Subtract b from both sides of the equation.
c=23
a+23=2-b
6=4a+2b+23
18=9a+3b+23
54=16a+4b+23
162=25a+5b+23
Subtract 23 from both sides of the equation.
c=23
a=2-b-23
6=4a+2b+23
18=9a+3b+23
54=16a+4b+23
162=25a+5b+23
To write 2 as a fraction with a common denominator, multiply by 33.
c=23
a=-b+2⋅33-23
6=4a+2b+23
18=9a+3b+23
54=16a+4b+23
162=25a+5b+23
Combine 2 and 33.
c=23
a=-b+2⋅33-23
6=4a+2b+23
18=9a+3b+23
54=16a+4b+23
162=25a+5b+23
Combine the numerators over the common denominator.
c=23
a=-b+2⋅3-23
6=4a+2b+23
18=9a+3b+23
54=16a+4b+23
162=25a+5b+23
Simplify the numerator.
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Multiply 2 by 3.
c=23
a=-b+6-23
6=4a+2b+23
18=9a+3b+23
54=16a+4b+23
162=25a+5b+23
Subtract 2 from 6.
c=23
a=-b+43
6=4a+2b+23
18=9a+3b+23
54=16a+4b+23
162=25a+5b+23
c=23
a=-b+43
6=4a+2b+23
18=9a+3b+23
54=16a+4b+23
162=25a+5b+23
c=23
a=-b+43
6=4a+2b+23
18=9a+3b+23
54=16a+4b+23
162=25a+5b+23
c=23
a=-b+43
6=4a+2b+23
18=9a+3b+23
54=16a+4b+23
162=25a+5b+23
Replace all occurrences of a with -b+43 in each equation.
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Replace all occurrences of a in 6=4a+2b+23 with -b+43.
c=23
a=-b+43
6=4(-b+43)+2b+23
18=9a+3b+23
54=16a+4b+23
162=25a+5b+23
Replace all occurrences of a in 18=9a+3b+23 with -b+43.
c=23
a=-b+43
6=4(-b+43)+2b+23
18=9(-b+43)+3b+23
54=16a+4b+23
162=25a+5b+23
Replace all occurrences of a in 54=16a+4b+23 with -b+43.
c=23
a=-b+43
6=4(-b+43)+2b+23
18=9(-b+43)+3b+23
54=16(-b+43)+4b+23
162=25a+5b+23
Replace all occurrences of a in 162=25a+5b+23 with -b+43.
c=23
a=-b+43
6=4(-b+43)+2b+23
18=9(-b+43)+3b+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
c=23
a=-b+43
6=4(-b+43)+2b+23
18=9(-b+43)+3b+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
Simplify each equation.
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Simplify 4(-b+43)+2b+23.
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Simplify each term.
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Apply the distributive property.
c=23
a=-b+43
6=4(-b)+4(43)+2b+23
18=9(-b+43)+3b+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
Multiply -1 by 4.
c=23
a=-b+43
6=-4b+4(43)+2b+23
18=9(-b+43)+3b+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
Multiply 4(43).
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Combine 4 and 43.
c=23
a=-b+43
6=-4b+4⋅43+2b+23
18=9(-b+43)+3b+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
Multiply 4 by 4.
c=23
a=-b+43
6=-4b+163+2b+23
18=9(-b+43)+3b+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
c=23
a=-b+43
6=-4b+163+2b+23
18=9(-b+43)+3b+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
c=23
a=-b+43
6=-4b+163+2b+23
18=9(-b+43)+3b+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
Simplify terms.
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Combine fractions with similar denominators.
c=23
a=-b+43
6=-4b+2b+16+23
18=9(-b+43)+3b+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
Simplify the expression.
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Add 16 and 2.
c=23
a=-b+43
6=-4b+2b+183
18=9(-b+43)+3b+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
Divide 18 by 3.
c=23
a=-b+43
6=-4b+2b+6
18=9(-b+43)+3b+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
c=23
a=-b+43
6=-4b+2b+6
18=9(-b+43)+3b+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
Add -4b and 2b.
c=23
a=-b+43
6=-2b+6
18=9(-b+43)+3b+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
c=23
a=-b+43
6=-2b+6
18=9(-b+43)+3b+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
c=23
a=-b+43
6=-2b+6
18=9(-b+43)+3b+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
Simplify 9(-b+43)+3b+23.
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Simplify each term.
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Apply the distributive property.
c=23
a=-b+43
6=-2b+6
18=9(-b)+9(43)+3b+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
Multiply -1 by 9.
c=23
a=-b+43
6=-2b+6
18=-9b+9(43)+3b+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
Cancel the common factor of 3.
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Factor 3 out of 9.
c=23
a=-b+43
6=-2b+6
18=-9b+3(3)(43)+3b+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
Cancel the common factor.
c=23
a=-b+43
6=-2b+6
18=-9b+3⋅(3(43))+3b+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
Rewrite the expression.
c=23
a=-b+43
6=-2b+6
18=-9b+3⋅4+3b+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
c=23
a=-b+43
6=-2b+6
18=-9b+3⋅4+3b+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
Multiply 3 by 4.
c=23
a=-b+43
6=-2b+6
18=-9b+12+3b+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
c=23
a=-b+43
6=-2b+6
18=-9b+12+3b+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
Add -9b and 3b.
c=23
a=-b+43
6=-2b+6
18=-6b+12+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
To write 12 as a fraction with a common denominator, multiply by 33.
c=23
a=-b+43
6=-2b+6
18=-6b+12⋅33+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
Combine 12 and 33.
c=23
a=-b+43
6=-2b+6
18=-6b+12⋅33+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
Combine the numerators over the common denominator.
c=23
a=-b+43
6=-2b+6
18=-6b+12⋅3+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
Simplify the numerator.
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Multiply 12 by 3.
c=23
a=-b+43
6=-2b+6
18=-6b+36+23
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
Add 36 and 2.
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=16(-b+43)+4b+23
162=25(-b+43)+5b+23
Simplify 16(-b+43)+4b+23.
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Simplify each term.
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Apply the distributive property.
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=16(-b)+16(43)+4b+23
162=25(-b+43)+5b+23
Multiply -1 by 16.
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-16b+16(43)+4b+23
162=25(-b+43)+5b+23
Multiply 16(43).
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Combine 16 and 43.
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-16b+16⋅43+4b+23
162=25(-b+43)+5b+23
Multiply 16 by 4.
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-16b+643+4b+23
162=25(-b+43)+5b+23
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-16b+643+4b+23
162=25(-b+43)+5b+23
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-16b+643+4b+23
162=25(-b+43)+5b+23
Simplify terms.
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Combine fractions with similar denominators.
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-16b+4b+64+23
162=25(-b+43)+5b+23
Simplify the expression.
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Add 64 and 2.
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-16b+4b+663
162=25(-b+43)+5b+23
Divide 66 by 3.
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-16b+4b+22
162=25(-b+43)+5b+23
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-16b+4b+22
162=25(-b+43)+5b+23
Add -16b and 4b.
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-12b+22
162=25(-b+43)+5b+23
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-12b+22
162=25(-b+43)+5b+23
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-12b+22
162=25(-b+43)+5b+23
Simplify 25(-b+43)+5b+23.
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Simplify each term.
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Apply the distributive property.
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-12b+22
162=25(-b)+25(43)+5b+23
Multiply -1 by 25.
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-12b+22
162=-25b+25(43)+5b+23
Multiply 25(43).
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Combine 25 and 43.
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-12b+22
162=-25b+25⋅43+5b+23
Multiply 25 by 4.
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-12b+22
162=-25b+1003+5b+23
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-12b+22
162=-25b+1003+5b+23
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-12b+22
162=-25b+1003+5b+23
Simplify terms.
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Combine fractions with similar denominators.
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-12b+22
162=-25b+5b+100+23
Simplify the expression.
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Add 100 and 2.
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-12b+22
162=-25b+5b+1023
Divide 102 by 3.
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-12b+22
162=-25b+5b+34
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-12b+22
162=-25b+5b+34
Add -25b and 5b.
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-12b+22
162=-20b+34
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-12b+22
162=-20b+34
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-12b+22
162=-20b+34
c=23
a=-b+43
6=-2b+6
18=-6b+383
54=-12b+22
162=-20b+34
Solve for b in the third equation.
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Rewrite the equation as -2b+6=6.
c=23
a=-b+43
-2b+6=6
18=-6b+383
54=-12b+22
162=-20b+34
Move all terms not containing b to the right side of the equation.
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Subtract 6 from both sides of the equation.
c=23
a=-b+43
-2b=6-6
18=-6b+383
54=-12b+22
162=-20b+34
Subtract 6 from 6.
c=23
a=-b+43
-2b=0
18=-6b+383
54=-12b+22
162=-20b+34
c=23
a=-b+43
-2b=0
18=-6b+383
54=-12b+22
162=-20b+34
Divide each term by -2 and simplify.
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Divide each term in -2b=0 by -2.
c=23
a=-b+43
-2b-2=0-2
18=-6b+383
54=-12b+22
162=-20b+34
Cancel the common factor of -2.
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Cancel the common factor.
c=23
a=-b+43
-2b-2=0-2
18=-6b+383
54=-12b+22
162=-20b+34
Divide b by 1.
c=23
a=-b+43
b=0-2
18=-6b+383
54=-12b+22
162=-20b+34
c=23
a=-b+43
b=0-2
18=-6b+383
54=-12b+22
162=-20b+34
Divide 0 by -2.
c=23
a=-b+43
b=0
18=-6b+383
54=-12b+22
162=-20b+34
c=23
a=-b+43
b=0
18=-6b+383
54=-12b+22
162=-20b+34
c=23
a=-b+43
b=0
18=-6b+383
54=-12b+22
162=-20b+34
Replace all occurrences of b with 0 in each equation.
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Replace all occurrences of b in a=-b+43 with 0.
c=23
a=-(0)+43
b=0
18=-6b+383
54=-12b+22
162=-20b+34
Replace all occurrences of b in 18=-6b+383 with 0.
c=23
a=-(0)+43
b=0
18=-6⋅0+383
54=-12b+22
162=-20b+34
Replace all occurrences of b in 54=-12b+22 with 0.
c=23
a=-(0)+43
b=0
18=-6⋅0+383
54=-12⋅0+22
162=-20b+34
Replace all occurrences of b in 162=-20b+34 with 0.
c=23
a=-(0)+43
b=0
18=-6⋅0+383
54=-12⋅0+22
162=-20⋅0+34
c=23
a=-(0)+43
b=0
18=-6⋅0+383
54=-12⋅0+22
162=-20⋅0+34
Simplify.
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Simplify -(0)+43.
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Multiply -1 by 0.
c=23
a=0+43
b=0
18=-6⋅0+383
54=-12⋅0+22
162=-20⋅0+34
Add 0 and 43.
c=23
a=43
b=0
18=-6⋅0+383
54=-12⋅0+22
162=-20⋅0+34
c=23
a=43
b=0
18=-6⋅0+383
54=-12⋅0+22
162=-20⋅0+34
Simplify -6⋅0+383.
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Multiply -6 by 0.
c=23
a=43
b=0
18=0+383
54=-12⋅0+22
162=-20⋅0+34
Add 0 and 383.
c=23
a=43
b=0
18=383
54=-12⋅0+22
162=-20⋅0+34
c=23
a=43
b=0
18=383
54=-12⋅0+22
162=-20⋅0+34
Simplify -12⋅0+22.
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Multiply -12 by 0.
c=23
a=43
b=0
18=383
54=0+22
162=-20⋅0+34
Add 0 and 22.
c=23
a=43
b=0
18=383
54=22
162=-20⋅0+34
c=23
a=43
b=0
18=383
54=22
162=-20⋅0+34
Simplify -20⋅0+34.
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Multiply -20 by 0.
c=23
a=43
b=0
18=383
54=22
162=0+34
Add 0 and 34.
c=23
a=43
b=0
18=383
54=22
162=34
c=23
a=43
b=0
18=383
54=22
162=34
c=23
a=43
b=0
18=383
54=22
162=34
Since 18≠383, there are no solutions.
c=23
a=43
b=0
No solution
54=22
162=34
Since 54≠22, there are no solutions.
c=23
a=43
b=0
No solution
No solution
162=34
Since 162≠34, there are no solutions.
c=23
a=43
b=0
No solution
No solution
No solution
c=23
a=43
b=0
No solution
No solution
No solution
Calculate the value of y using each x value in the table and compare this value to the given y value in the table.
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Calculate the value of y such that y=ax2+b when a=43, b=0, c=23, and x=0.
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Simplify each term.
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Raising 0 to any positive power yields 0.
y=43⋅0+(0)⋅(0)+23
Multiply 43 by 0.
y=0+(0)⋅(0)+23
Multiply 0 by 0.
y=0+0+23
y=0+0+23
Simplify by adding zeros.
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Add 0 and 0.
y=0+23
Add 0 and 23.
y=23
y=23
y=23
If the table has a quadratic function rule, y=y for the corresponding x value, x=0. This check passes since y=23 and y=23.
23=23
Calculate the value of y such that y=ax2+b when a=43, b=0, c=23, and x=1.
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Simplify each term.
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One to any power is one.
y=43⋅1+(0)⋅(1)+23
Multiply 43 by 1.
y=43+(0)⋅(1)+23
Multiply 0 by 1.
y=43+0+23
y=43+0+23
Combine fractions.
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Combine fractions with similar denominators.
y=4+23
Simplify the expression.
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Add 4 and 2.
y=63
Divide 6 by 3.
y=2
y=2
y=2
y=2
If the table has a quadratic function rule, y=y for the corresponding x value, x=1. This check passes since y=2 and y=2.
2=2
Calculate the value of y such that y=ax2+b when a=43, b=0, c=23, and x=2.
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Simplify each term.
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Raise 2 to the power of 2.
y=43⋅4+(0)⋅(2)+23
Multiply 43⋅4.
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Combine 43 and 4.
y=4⋅43+(0)⋅(2)+23
Multiply 4 by 4.
y=163+(0)⋅(2)+23
y=163+(0)⋅(2)+23
Multiply 0 by 2.
y=163+0+23
y=163+0+23
Combine fractions.
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Combine fractions with similar denominators.
y=16+23
Simplify the expression.
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Add 16 and 2.
y=183
Divide 18 by 3.
y=6
y=6
y=6
y=6
If the table has a quadratic function rule, y=y for the corresponding x value, x=2. This check passes since y=6 and y=6.
6=6
Calculate the value of y such that y=ax2+b when a=43, b=0, c=23, and x=3.
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Simplify each term.
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Raise 3 to the power of 2.
y=43⋅9+(0)⋅(3)+23
Cancel the common factor of 3.
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Factor 3 out of 9.
y=43⋅(3(3))+(0)⋅(3)+23
Cancel the common factor.
y=43⋅(3⋅3)+(0)⋅(3)+23
Rewrite the expression.
y=4⋅3+(0)⋅(3)+23
y=4⋅3+(0)⋅(3)+23
Multiply 4 by 3.
y=12+(0)⋅(3)+23
Multiply 0 by 3.
y=12+0+23
y=12+0+23
Add 12 and 0.
y=12+23
To write 12 as a fraction with a common denominator, multiply by 33.
y=12⋅33+23
Combine 12 and 33.
y=12⋅33+23
Combine the numerators over the common denominator.
y=12⋅3+23
Simplify the numerator.
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Multiply 12 by 3.
y=36+23
Add 36 and 2.
y=383
y=383
y=383
If the table has a quadratic function rule, y=y for the corresponding x value, x=3. This check does not pass, since y=383 and y=18. The function rule can’t be quadratic.
383≠18
Since y≠y for the corresponding x values, the function is not quadratic.
The function is not quadratic
The function is not quadratic
The function is not quadratic
Check if the function rule is cubic.
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To find if the table follows a function rule, check whether the function rule could follow the form y=ax3+bx2+cx+d.
y=ax3+bx2+cx+d
Build a set of 4 equations from the table such that y=ax3+bx2+cx+d.
Calculate the values of a, b, c, and d.
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Simplify each equation.
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Simplify a(0)3+b(0)2+c(0)+d.
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Simplify each term.
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Raising 0 to any positive power yields 0.
23=a⋅0+b(0)2+c(0)+d
2=a(1)3+b(0)2+c(0)+d
6=a(2)3+b(0)2+c(0)+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Multiply a by 0.
23=0+b(0)2+c(0)+d
2=a(1)3+b(0)2+c(0)+d
6=a(2)3+b(0)2+c(0)+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Raising 0 to any positive power yields 0.
23=0+b⋅0+c(0)+d
2=a(1)3+b(0)2+c(0)+d
6=a(2)3+b(0)2+c(0)+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Multiply b by 0.
23=0+0+c(0)+d
2=a(1)3+b(0)2+c(0)+d
6=a(2)3+b(0)2+c(0)+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Multiply c by 0.
23=0+0+0+d
2=a(1)3+b(0)2+c(0)+d
6=a(2)3+b(0)2+c(0)+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
23=0+0+0+d
2=a(1)3+b(0)2+c(0)+d
6=a(2)3+b(0)2+c(0)+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Combine the opposite terms in 0+0+0+d.
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Add 0 and 0.
23=0+0+d
2=a(1)3+b(0)2+c(0)+d
6=a(2)3+b(0)2+c(0)+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Add 0 and 0.
23=0+d
2=a(1)3+b(0)2+c(0)+d
6=a(2)3+b(0)2+c(0)+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Add 0 and d.
23=d
2=a(1)3+b(0)2+c(0)+d
6=a(2)3+b(0)2+c(0)+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
23=d
2=a(1)3+b(0)2+c(0)+d
6=a(2)3+b(0)2+c(0)+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
23=d
2=a(1)3+b(0)2+c(0)+d
6=a(2)3+b(0)2+c(0)+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Simplify a(1)3+b(0)2+c(0)+d.
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Simplify each term.
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One to any power is one.
23=d
2=a⋅1+b(0)2+c(0)+d
6=a(2)3+b(0)2+c(0)+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Multiply a by 1.
23=d
2=a+b(0)2+c(0)+d
6=a(2)3+b(0)2+c(0)+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Raising 0 to any positive power yields 0.
23=d
2=a+b⋅0+c(0)+d
6=a(2)3+b(0)2+c(0)+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Multiply b by 0.
23=d
2=a+0+c(0)+d
6=a(2)3+b(0)2+c(0)+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Multiply c by 0.
23=d
2=a+0+0+d
6=a(2)3+b(0)2+c(0)+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
23=d
2=a+0+0+d
6=a(2)3+b(0)2+c(0)+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Combine the opposite terms in a+0+0+d.
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Add a and 0.
23=d
2=a+0+d
6=a(2)3+b(0)2+c(0)+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Add a and 0.
23=d
2=a+d
6=a(2)3+b(0)2+c(0)+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
23=d
2=a+d
6=a(2)3+b(0)2+c(0)+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
23=d
2=a+d
6=a(2)3+b(0)2+c(0)+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Simplify a(2)3+b(0)2+c(0)+d.
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Simplify each term.
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Raise 2 to the power of 3.
23=d
2=a+d
6=a⋅8+b(0)2+c(0)+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Move 8 to the left of a.
23=d
2=a+d
6=8⋅a+b(0)2+c(0)+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Raising 0 to any positive power yields 0.
23=d
2=a+d
6=8a+b⋅0+c(0)+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Multiply b by 0.
23=d
2=a+d
6=8a+0+c(0)+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Multiply c by 0.
23=d
2=a+d
6=8a+0+0+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
23=d
2=a+d
6=8a+0+0+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Combine the opposite terms in 8a+0+0+d.
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Add 8a and 0.
23=d
2=a+d
6=8a+0+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Add 8a and 0.
23=d
2=a+d
6=8a+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
23=d
2=a+d
6=8a+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
23=d
2=a+d
6=8a+d
18=a(3)3+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Simplify a(3)3+b(0)2+c(0)+d.
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Simplify each term.
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Raise 3 to the power of 3.
23=d
2=a+d
6=8a+d
18=a⋅27+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Move 27 to the left of a.
23=d
2=a+d
6=8a+d
18=27⋅a+b(0)2+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Raising 0 to any positive power yields 0.
23=d
2=a+d
6=8a+d
18=27a+b⋅0+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Multiply b by 0.
23=d
2=a+d
6=8a+d
18=27a+0+c(0)+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Multiply c by 0.
23=d
2=a+d
6=8a+d
18=27a+0+0+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
23=d
2=a+d
6=8a+d
18=27a+0+0+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Combine the opposite terms in 27a+0+0+d.
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Add 27a and 0.
23=d
2=a+d
6=8a+d
18=27a+0+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Add 27a and 0.
23=d
2=a+d
6=8a+d
18=27a+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
23=d
2=a+d
6=8a+d
18=27a+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
23=d
2=a+d
6=8a+d
18=27a+d
54=a(4)3+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Simplify a(4)3+b(0)2+c(0)+d.
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Simplify each term.
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Raise 4 to the power of 3.
23=d
2=a+d
6=8a+d
18=27a+d
54=a⋅64+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Move 64 to the left of a.
23=d
2=a+d
6=8a+d
18=27a+d
54=64⋅a+b(0)2+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Raising 0 to any positive power yields 0.
23=d
2=a+d
6=8a+d
18=27a+d
54=64a+b⋅0+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Multiply b by 0.
23=d
2=a+d
6=8a+d
18=27a+d
54=64a+0+c(0)+d
162=a(5)3+b(0)2+c(0)+d
Multiply c by 0.
23=d
2=a+d
6=8a+d
18=27a+d
54=64a+0+0+d
162=a(5)3+b(0)2+c(0)+d
23=d
2=a+d
6=8a+d
18=27a+d
54=64a+0+0+d
162=a(5)3+b(0)2+c(0)+d
Combine the opposite terms in 64a+0+0+d.
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Add 64a and 0.
23=d
2=a+d
6=8a+d
18=27a+d
54=64a+0+d
162=a(5)3+b(0)2+c(0)+d
Add 64a and 0.
23=d
2=a+d
6=8a+d
18=27a+d
54=64a+d
162=a(5)3+b(0)2+c(0)+d
23=d
2=a+d
6=8a+d
18=27a+d
54=64a+d
162=a(5)3+b(0)2+c(0)+d
23=d
2=a+d
6=8a+d
18=27a+d
54=64a+d
162=a(5)3+b(0)2+c(0)+d
Simplify a(5)3+b(0)2+c(0)+d.
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Simplify each term.
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Raise 5 to the power of 3.
23=d
2=a+d
6=8a+d
18=27a+d
54=64a+d
162=a⋅125+b(0)2+c(0)+d
Move 125 to the left of a.
23=d
2=a+d
6=8a+d
18=27a+d
54=64a+d
162=125⋅a+b(0)2+c(0)+d
Raising 0 to any positive power yields 0.
23=d
2=a+d
6=8a+d
18=27a+d
54=64a+d
162=125a+b⋅0+c(0)+d
Multiply b by 0.
23=d
2=a+d
6=8a+d
18=27a+d
54=64a+d
162=125a+0+c(0)+d
Multiply c by 0.
23=d
2=a+d
6=8a+d
18=27a+d
54=64a+d
162=125a+0+0+d
23=d
2=a+d
6=8a+d
18=27a+d
54=64a+d
162=125a+0+0+d
Combine the opposite terms in 125a+0+0+d.
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Add 125a and 0.
23=d
2=a+d
6=8a+d
18=27a+d
54=64a+d
162=125a+0+d
Add 125a and 0.
23=d
2=a+d
6=8a+d
18=27a+d
54=64a+d
162=125a+d
23=d
2=a+d
6=8a+d
18=27a+d
54=64a+d
162=125a+d
23=d
2=a+d
6=8a+d
18=27a+d
54=64a+d
162=125a+d
23=d
2=a+d
6=8a+d
18=27a+d
54=64a+d
162=125a+d
Rewrite the equation as d=23.
d=23
2=a+d
6=8a+d
18=27a+d
54=64a+d
162=125a+d
Replace all occurrences of d with 23 in each equation.
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Replace all occurrences of d in 2=a+d with 23.
d=23
2=a+23
6=8a+d
18=27a+d
54=64a+d
162=125a+d
Replace all occurrences of d in 6=8a+d with 23.
d=23
2=a+23
6=8a+23
18=27a+d
54=64a+d
162=125a+d
Replace all occurrences of d in 18=27a+d with 23.
d=23
2=a+23
6=8a+23
18=27a+23
54=64a+d
162=125a+d
Replace all occurrences of d in 54=64a+d with 23.
d=23
2=a+23
6=8a+23
18=27a+23
54=64a+23
162=125a+d
Replace all occurrences of d in 162=125a+d with 23.
d=23
2=a+23
6=8a+23
18=27a+23
54=64a+23
162=125a+23
d=23
2=a+23
6=8a+23
18=27a+23
54=64a+23
162=125a+23
Simplify.
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Remove parentheses.
d=23
2=a+23
6=8a+23
18=27a+23
54=64a+23
162=125a+23
Remove parentheses.
d=23
2=a+23
6=8a+23
18=27a+23
54=64a+23
162=125a+23
Remove parentheses.
d=23
2=a+23
6=8a+23
18=27a+23
54=64a+23
162=125a+23
Remove parentheses.
d=23
2=a+23
6=8a+23
18=27a+23
54=64a+23
162=125a+23
Remove parentheses.
d=23
2=a+23
6=8a+23
18=27a+23
54=64a+23
162=125a+23
d=23
2=a+23
6=8a+23
18=27a+23
54=64a+23
162=125a+23
Solve for a in the second equation.
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Rewrite the equation as a+23=2.
d=23
a+23=2
6=8a+23
18=27a+23
54=64a+23
162=125a+23
Move all terms not containing a to the right side of the equation.
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Subtract 23 from both sides of the equation.
d=23
a=2-23
6=8a+23
18=27a+23
54=64a+23
162=125a+23
To write 2 as a fraction with a common denominator, multiply by 33.
d=23
a=2⋅33-23
6=8a+23
18=27a+23
54=64a+23
162=125a+23
Combine 2 and 33.
d=23
a=2⋅33-23
6=8a+23
18=27a+23
54=64a+23
162=125a+23
Combine the numerators over the common denominator.
d=23
a=2⋅3-23
6=8a+23
18=27a+23
54=64a+23
162=125a+23
Simplify the numerator.
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Multiply 2 by 3.
d=23
a=6-23
6=8a+23
18=27a+23
54=64a+23
162=125a+23
Subtract 2 from 6.
d=23
a=43
6=8a+23
18=27a+23
54=64a+23
162=125a+23
d=23
a=43
6=8a+23
18=27a+23
54=64a+23
162=125a+23
d=23
a=43
6=8a+23
18=27a+23
54=64a+23
162=125a+23
d=23
a=43
6=8a+23
18=27a+23
54=64a+23
162=125a+23
Replace all occurrences of a with 43 in each equation.
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Replace all occurrences of a in 6=8a+23 with 43.
d=23
a=43
6=8(43)+23
18=27a+23
54=64a+23
162=125a+23
Replace all occurrences of a in 18=27a+23 with 43.
d=23
a=43
6=8(43)+23
18=27(43)+23
54=64a+23
162=125a+23
Replace all occurrences of a in 54=64a+23 with 43.
d=23
a=43
6=8(43)+23
18=27(43)+23
54=64(43)+23
162=125a+23
Replace all occurrences of a in 162=125a+23 with 43.
d=23
a=43
6=8(43)+23
18=27(43)+23
54=64(43)+23
162=125(43)+23
d=23
a=43
6=8(43)+23
18=27(43)+23
54=64(43)+23
162=125(43)+23
Simplify.
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Simplify 8(43)+23.
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Multiply 8(43).
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Combine 8 and 43.
d=23
a=43
6=8⋅43+23
18=27(43)+23
54=64(43)+23
162=125(43)+23
Multiply 8 by 4.
d=23
a=43
6=323+23
18=27(43)+23
54=64(43)+23
162=125(43)+23
d=23
a=43
6=323+23
18=27(43)+23
54=64(43)+23
162=125(43)+23
Combine the numerators over the common denominator.
d=23
a=43
6=32+23
18=27(43)+23
54=64(43)+23
162=125(43)+23
Add 32 and 2.
d=23
a=43
6=343
18=27(43)+23
54=64(43)+23
162=125(43)+23
d=23
a=43
6=343
18=27(43)+23
54=64(43)+23
162=125(43)+23
Simplify 27(43)+23.
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Simplify each term.
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Cancel the common factor of 3.
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Factor 3 out of 27.
d=23
a=43
6=343
18=3(9)(43)+23
54=64(43)+23
162=125(43)+23
Cancel the common factor.
d=23
a=43
6=343
18=3⋅(9(43))+23
54=64(43)+23
162=125(43)+23
Rewrite the expression.
d=23
a=43
6=343
18=9⋅4+23
54=64(43)+23
162=125(43)+23
d=23
a=43
6=343
18=9⋅4+23
54=64(43)+23
162=125(43)+23
Multiply 9 by 4.
d=23
a=43
6=343
18=36+23
54=64(43)+23
162=125(43)+23
d=23
a=43
6=343
18=36+23
54=64(43)+23
162=125(43)+23
To write 36 as a fraction with a common denominator, multiply by 33.
d=23
a=43
6=343
18=36⋅33+23
54=64(43)+23
162=125(43)+23
Combine 36 and 33.
d=23
a=43
6=343
18=36⋅33+23
54=64(43)+23
162=125(43)+23
Combine the numerators over the common denominator.
d=23
a=43
6=343
18=36⋅3+23
54=64(43)+23
162=125(43)+23
Simplify the numerator.
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Multiply 36 by 3.
d=23
a=43
6=343
18=108+23
54=64(43)+23
162=125(43)+23
Add 108 and 2.
d=23
a=43
6=343
18=1103
54=64(43)+23
162=125(43)+23
d=23
a=43
6=343
18=1103
54=64(43)+23
162=125(43)+23
d=23
a=43
6=343
18=1103
54=64(43)+23
162=125(43)+23
Simplify 64(43)+23.
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Multiply 64(43).
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Combine 64 and 43.
d=23
a=43
6=343
18=1103
54=64⋅43+23
162=125(43)+23
Multiply 64 by 4.
d=23
a=43
6=343
18=1103
54=2563+23
162=125(43)+23
d=23
a=43
6=343
18=1103
54=2563+23
162=125(43)+23
Combine the numerators over the common denominator.
d=23
a=43
6=343
18=1103
54=256+23
162=125(43)+23
Add 256 and 2.
d=23
a=43
6=343
18=1103
54=2583
162=125(43)+23
Divide 258 by 3.
d=23
a=43
6=343
18=1103
54=86
162=125(43)+23
d=23
a=43
6=343
18=1103
54=86
162=125(43)+23
Simplify 125(43)+23.
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Multiply 125(43).
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Combine 125 and 43.
d=23
a=43
6=343
18=1103
54=86
162=125⋅43+23
Multiply 125 by 4.
d=23
a=43
6=343
18=1103
54=86
162=5003+23
d=23
a=43
6=343
18=1103
54=86
162=5003+23
Combine the numerators over the common denominator.
d=23
a=43
6=343
18=1103
54=86
162=500+23
Add 500 and 2.
d=23
a=43
6=343
18=1103
54=86
162=5023
d=23
a=43
6=343
18=1103
54=86
162=5023
d=23
a=43
6=343
18=1103
54=86
162=5023
Since 6≠343, there are no solutions.
d=23
a=43
No solution
18=1103
54=86
162=5023
Since 18≠1103, there are no solutions.
d=23
a=43
No solution
No solution
54=86
162=5023
Since 54≠86, there are no solutions.
d=23
a=43
No solution
No solution
No solution
162=5023
Since 162≠5023, there are no solutions.
d=23
a=43
No solution
No solution
No solution
No solution
d=23
a=43
No solution
No solution
No solution
No solution
Calculate the value of y using each x value in the table and compare this value to the given y value in the table.
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Calculate the value of y such that y=ax3+b when a=43, b=0, c=0, d=23, and x=0.
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Simplify each term.
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Raising 0 to any positive power yields 0.
y=43⋅0+(0)⋅(02)+(0)⋅(0)+23
Multiply 43 by 0.
y=0+(0)⋅(02)+(0)⋅(0)+23
Multiply 0 by 02 by adding the exponents.
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Multiply 0 by 02.
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Raise 0 to the power of 1.
y=0+0⋅02+(0)⋅(0)+23
Use the power rule aman=am+n to combine exponents.
y=0+01+2+(0)⋅(0)+23
y=0+01+2+(0)⋅(0)+23
Add 1 and 2.
y=0+03+(0)⋅(0)+23
y=0+03+(0)⋅(0)+23
Raising 0 to any positive power yields 0.
y=0+0+(0)⋅(0)+23
Multiply 0 by 0.
y=0+0+0+23
y=0+0+0+23
Simplify by adding zeros.
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Add 0 and 0.
y=0+0+23
Add 0 and 0.
y=0+23
Add 0 and 23.
y=23
y=23
y=23
If the table has a cubic function rule, y=y for the corresponding x value, x=0. This check passes since y=23 and y=23.
23=23
Calculate the value of y such that y=ax3+b when a=43, b=0, c=0, d=23, and x=1.
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Simplify each term.
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One to any power is one.
y=43⋅1+(0)⋅(12)+(0)⋅(1)+23
Multiply 43 by 1.
y=43+(0)⋅(12)+(0)⋅(1)+23
One to any power is one.
y=43+0⋅1+(0)⋅(1)+23
Multiply 0 by 1.
y=43+0+(0)⋅(1)+23
Multiply 0 by 1.
y=43+0+0+23
y=43+0+0+23
Combine fractions.
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Combine fractions with similar denominators.
y=4+23
Simplify the expression.
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Add 4 and 2.
y=63
Divide 6 by 3.
y=2
y=2
y=2
y=2
If the table has a cubic function rule, y=y for the corresponding x value, x=1. This check passes since y=2 and y=2.
2=2
Calculate the value of y such that y=ax3+b when a=43, b=0, c=0, d=23, and x=2.
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Simplify each term.
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Raise 2 to the power of 3.
y=43⋅8+(0)⋅(22)+(0)⋅(2)+23
Multiply 43⋅8.
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Combine 43 and 8.
y=4⋅83+(0)⋅(22)+(0)⋅(2)+23
Multiply 4 by 8.
y=323+(0)⋅(22)+(0)⋅(2)+23
y=323+(0)⋅(22)+(0)⋅(2)+23
Raise 2 to the power of 2.
y=323+0⋅4+(0)⋅(2)+23
Multiply 0 by 4.
y=323+0+(0)⋅(2)+23
Multiply 0 by 2.
y=323+0+0+23
y=323+0+0+23
Combine fractions.
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Combine fractions with similar denominators.
y=32+23
Add 32 and 2.
y=343
y=343
y=343
If the table has a cubic function rule, y=y for the corresponding x value, x=2. This check does not pass, since y=343 and y=6. The function rule can’t be cubic.
343≠6
Since y≠y for the corresponding x values, the function is not cubic.
The function is not cubic
The function is not cubic
The function is not cubic
There are no values for a, b, c, and d in the equations y=ax+b, y=ax2+bx+c, and y=ax3+bx2+cx+d that work for every pair of x and y.
The table does not have a function rule that is linear, quadratic, or cubic.
Find the Function Rule table[[x,y],[0,2/3],[1,2],[2,6],[3,18],[4,54],[5,162]]

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