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

xy02312263184545162
Check if the function rule is linear.
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.
Simplify each equation.
Simplify a(0)+b.
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
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.
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.
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.
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.
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.
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.
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.
Simplify 2(43)+23.
Multiply 2(43).
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
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.
Cancel the common factor of 3.
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.
Multiply 4 by 3.
b=23
a=43
6=103
18=12+23
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
b=23
a=43
6=103
18=143
54=4(43)+23
162=5(43)+23
Simplify 4(43)+23.
Multiply 4(43).
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
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.
Multiply 5(43).
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
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.
Calculate the value of y when a=43, b=23, and x=0.
Multiply 43 by 0.
y=0+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.
Multiply 43 by 1.
y=43+23
Combine the numerators over the common denominator.
y=4+23
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.
Multiply (43)(2).
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
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.
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.
Simplify each equation.
Simplify a(0)2+b(0)+c.
Simplify each term.
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.
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Simplify 4(-b+43)+2b+23.
Simplify each term.
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).
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.
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.
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
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.
Simplify each term.
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.
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
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.
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
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.
Simplify each term.
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).
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.
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.
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
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.
Simplify each term.
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).
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.
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.
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
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.
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.
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.
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.
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.
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.
Simplify -(0)+43.
Multiply -1 by 0.
c=23
a=0+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
c=23
a=43
b=0
18=-6⋅0+383
54=-12⋅0+22
162=-20⋅0+34
Simplify -6⋅0+383.
Multiply -6 by 0.
c=23
a=43
b=0
18=0+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
c=23
a=43
b=0
18=383
54=-12⋅0+22
162=-20⋅0+34
Simplify -12⋅0+22.
Multiply -12 by 0.
c=23
a=43
b=0
18=383
54=0+22
162=-20⋅0+34
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.
Multiply -20 by 0.
c=23
a=43
b=0
18=383
54=22
162=0+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.
Calculate the value of y such that y=ax2+b when a=43, b=0, c=23, and x=0.
Simplify each term.
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
y=0+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.
Simplify each term.
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.
Combine fractions with similar denominators.
y=4+23
Simplify the expression.
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.
Simplify each term.
Raise 2 to the power of 2.
y=43⋅4+(0)⋅(2)+23
Multiply 43⋅4.
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.
Combine fractions with similar denominators.
y=16+23
Simplify the expression.
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.
Simplify each term.
Raise 3 to the power of 2.
y=43⋅9+(0)⋅(3)+23
Cancel the common factor of 3.
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
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.
Multiply 12 by 3.
y=36+23
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.
Check if the function rule is cubic.
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.
Simplify each equation.
Simplify a(0)3+b(0)2+c(0)+d.
Simplify each term.
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.
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
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
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.
Simplify each term.
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.
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
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.
Simplify each term.
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.
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
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.
Simplify each term.
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.
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
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.
Simplify each term.
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.
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
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.
Simplify each term.
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.
23=d
2=a+d
6=8a+d
18=27a+d
54=64a+d
162=125a+0+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
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.
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.
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.
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.
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.
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.
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.
Simplify 8(43)+23.
Multiply 8(43).
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
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.
Simplify each term.
Cancel the common factor of 3.
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.
Multiply 36 by 3.
d=23
a=43
6=343
18=108+23
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
d=23
a=43
6=343
18=1103
54=64(43)+23
162=125(43)+23
Simplify 64(43)+23.
Multiply 64(43).
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
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.
Multiply 125(43).
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
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.
Calculate the value of y such that y=ax3+b when a=43, b=0, c=0, d=23, and x=0.
Simplify each term.
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.
Multiply 0 by 02.
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
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
y=0+0+23
y=0+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.
Simplify each term.
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.
Combine fractions with similar denominators.
y=4+23
Simplify the expression.
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.
Simplify each term.
Raise 2 to the power of 3.
y=43⋅8+(0)⋅(22)+(0)⋅(2)+23
Multiply 43⋅8.
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.
Combine fractions with similar denominators.
y=32+23
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]]