# Find the Function Rule table[[x,y],[0,3],[1,2],[2,4],[3,1]] xy03122431
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.
3=a(0)+b2=a(1)+b4=a(2)+b1=a(3)+b
Calculate the values of a and b.
Simplify each equation.
Simplify a(0)+b.
Multiply a by 0.
3=0+b
2=a(1)+b
4=a(2)+b
1=a(3)+b
3=b
2=a(1)+b
4=a(2)+b
1=a(3)+b
3=b
2=a(1)+b
4=a(2)+b
1=a(3)+b
Multiply a by 1.
3=b
2=a+b
4=a(2)+b
1=a(3)+b
Move 2 to the left of a.
3=b
2=a+b
4=2a+b
1=a(3)+b
Move 3 to the left of a.
3=b
2=a+b
4=2a+b
1=3a+b
3=b
2=a+b
4=2a+b
1=3a+b
Rewrite the equation as b=3.
b=3
2=a+b
4=2a+b
1=3a+b
Replace all occurrences of b with 3 in each equation.
Replace all occurrences of b in 2=a+b with 3.
b=3
2=a+3
4=2a+b
1=3a+b
Replace all occurrences of b in 4=2a+b with 3.
b=3
2=a+3
4=2a+3
1=3a+b
Replace all occurrences of b in 1=3a+b with 3.
b=3
2=a+3
4=2a+3
1=3a+3
b=3
2=a+3
4=2a+3
1=3a+3
Simplify.
Remove parentheses.
b=3
2=a+3
4=2a+3
1=3a+3
Remove parentheses.
b=3
2=a+3
4=2a+3
1=3a+3
Remove parentheses.
b=3
2=a+3
4=2a+3
1=3a+3
b=3
2=a+3
4=2a+3
1=3a+3
Solve for a in the second equation.
Rewrite the equation as a+3=2.
b=3
a+3=2
4=2a+3
1=3a+3
Move all terms not containing a to the right side of the equation.
Subtract 3 from both sides of the equation.
b=3
a=2-3
4=2a+3
1=3a+3
Subtract 3 from 2.
b=3
a=-1
4=2a+3
1=3a+3
b=3
a=-1
4=2a+3
1=3a+3
b=3
a=-1
4=2a+3
1=3a+3
Replace all occurrences of a with -1 in each equation.
Replace all occurrences of a in 4=2a+3 with -1.
b=3
a=-1
4=2(-1)+3
1=3a+3
Replace all occurrences of a in 1=3a+3 with -1.
b=3
a=-1
4=2(-1)+3
1=3(-1)+3
b=3
a=-1
4=2(-1)+3
1=3(-1)+3
Simplify.
Simplify 2(-1)+3.
Multiply 2 by -1.
b=3
a=-1
4=-2+3
1=3(-1)+3
b=3
a=-1
4=1
1=3(-1)+3
b=3
a=-1
4=1
1=3(-1)+3
Simplify 3(-1)+3.
Multiply 3 by -1.
b=3
a=-1
4=1
1=-3+3
b=3
a=-1
4=1
1=0
b=3
a=-1
4=1
1=0
b=3
a=-1
4=1
1=0
Since 4≠1, there are no solutions.
b=3
a=-1
No solution
1=0
Since 1≠0, there are no solutions.
b=3
a=-1
No solution
No solution
b=3
a=-1
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=-1, b=3, and x=0.
Multiply -1 by 0.
y=0+3
y=3
y=3
If the table has a linear function rule, y=y for the corresponding x value, x=0. This check passes since y=3 and y=3.
3=3
Calculate the value of y when a=-1, b=3, and x=1.
Multiply -1 by 1.
y=-1+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=-1, b=3, and x=2.
Multiply -1 by 2.
y=-2+3
y=1
y=1
If the table has a linear function rule, y=y for the corresponding x value, x=2. This check does not pass, since y=1 and y=4. The function rule can’t be linear.
1≠4
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.
3=a⋅0+b(0)+c
2=a(1)2+b(1)+c
4=a(2)2+b(2)+c
1=a(3)2+b(3)+c
Multiply a by 0.
3=0+b(0)+c
2=a(1)2+b(1)+c
4=a(2)2+b(2)+c
1=a(3)2+b(3)+c
Multiply b by 0.
3=0+0+c
2=a(1)2+b(1)+c
4=a(2)2+b(2)+c
1=a(3)2+b(3)+c
3=0+0+c
2=a(1)2+b(1)+c
4=a(2)2+b(2)+c
1=a(3)2+b(3)+c
Combine the opposite terms in 0+0+c.
3=0+c
2=a(1)2+b(1)+c
4=a(2)2+b(2)+c
1=a(3)2+b(3)+c
3=c
2=a(1)2+b(1)+c
4=a(2)2+b(2)+c
1=a(3)2+b(3)+c
3=c
2=a(1)2+b(1)+c
4=a(2)2+b(2)+c
1=a(3)2+b(3)+c
3=c
2=a(1)2+b(1)+c
4=a(2)2+b(2)+c
1=a(3)2+b(3)+c
Simplify each term.
One to any power is one.
3=c
2=a⋅1+b(1)+c
4=a(2)2+b(2)+c
1=a(3)2+b(3)+c
Multiply a by 1.
3=c
2=a+b(1)+c
4=a(2)2+b(2)+c
1=a(3)2+b(3)+c
Multiply b by 1.
3=c
2=a+b+c
4=a(2)2+b(2)+c
1=a(3)2+b(3)+c
3=c
2=a+b+c
4=a(2)2+b(2)+c
1=a(3)2+b(3)+c
Simplify each term.
Raise 2 to the power of 2.
3=c
2=a+b+c
4=a⋅4+b(2)+c
1=a(3)2+b(3)+c
Move 4 to the left of a.
3=c
2=a+b+c
4=4⋅a+b(2)+c
1=a(3)2+b(3)+c
Move 2 to the left of b.
3=c
2=a+b+c
4=4a+2b+c
1=a(3)2+b(3)+c
3=c
2=a+b+c
4=4a+2b+c
1=a(3)2+b(3)+c
Simplify each term.
Raise 3 to the power of 2.
3=c
2=a+b+c
4=4a+2b+c
1=a⋅9+b(3)+c
Move 9 to the left of a.
3=c
2=a+b+c
4=4a+2b+c
1=9⋅a+b(3)+c
Move 3 to the left of b.
3=c
2=a+b+c
4=4a+2b+c
1=9a+3b+c
3=c
2=a+b+c
4=4a+2b+c
1=9a+3b+c
3=c
2=a+b+c
4=4a+2b+c
1=9a+3b+c
Rewrite the equation as c=3.
c=3
2=a+b+c
4=4a+2b+c
1=9a+3b+c
Replace all occurrences of c with 3 in each equation.
Replace all occurrences of c in 2=a+b+c with 3.
c=3
2=a+b+3
4=4a+2b+c
1=9a+3b+c
Replace all occurrences of c in 4=4a+2b+c with 3.
c=3
2=a+b+3
4=4a+2b+3
1=9a+3b+c
Replace all occurrences of c in 1=9a+3b+c with 3.
c=3
2=a+b+3
4=4a+2b+3
1=9a+3b+3
c=3
2=a+b+3
4=4a+2b+3
1=9a+3b+3
Simplify.
Remove parentheses.
c=3
2=a+b+3
4=4a+2b+3
1=9a+3b+3
Remove parentheses.
c=3
2=a+b+3
4=4a+2b+3
1=9a+3b+3
Remove parentheses.
c=3
2=a+b+3
4=4a+2b+3
1=9a+3b+3
c=3
2=a+b+3
4=4a+2b+3
1=9a+3b+3
Solve for a in the second equation.
Rewrite the equation as a+b+3=2.
c=3
a+b+3=2
4=4a+2b+3
1=9a+3b+3
Move all terms not containing a to the right side of the equation.
Subtract b from both sides of the equation.
c=3
a+3=2-b
4=4a+2b+3
1=9a+3b+3
Subtract 3 from both sides of the equation.
c=3
a=2-b-3
4=4a+2b+3
1=9a+3b+3
Subtract 3 from 2.
c=3
a=-b-1
4=4a+2b+3
1=9a+3b+3
c=3
a=-b-1
4=4a+2b+3
1=9a+3b+3
c=3
a=-b-1
4=4a+2b+3
1=9a+3b+3
Replace all occurrences of a with -b-1 in each equation.
Replace all occurrences of a in 4=4a+2b+3 with -b-1.
c=3
a=-b-1
4=4(-b-1)+2b+3
1=9a+3b+3
Replace all occurrences of a in 1=9a+3b+3 with -b-1.
c=3
a=-b-1
4=4(-b-1)+2b+3
1=9(-b-1)+3b+3
c=3
a=-b-1
4=4(-b-1)+2b+3
1=9(-b-1)+3b+3
Simplify each equation.
Simplify 4(-b-1)+2b+3.
Simplify each term.
Apply the distributive property.
c=3
a=-b-1
4=4(-b)+4⋅-1+2b+3
1=9(-b-1)+3b+3
Multiply -1 by 4.
c=3
a=-b-1
4=-4b+4⋅-1+2b+3
1=9(-b-1)+3b+3
Multiply 4 by -1.
c=3
a=-b-1
4=-4b-4+2b+3
1=9(-b-1)+3b+3
c=3
a=-b-1
4=-4b-4+2b+3
1=9(-b-1)+3b+3
c=3
a=-b-1
4=-2b-4+3
1=9(-b-1)+3b+3
c=3
a=-b-1
4=-2b-1
1=9(-b-1)+3b+3
c=3
a=-b-1
4=-2b-1
1=9(-b-1)+3b+3
c=3
a=-b-1
4=-2b-1
1=9(-b-1)+3b+3
Simplify 9(-b-1)+3b+3.
Simplify each term.
Apply the distributive property.
c=3
a=-b-1
4=-2b-1
1=9(-b)+9⋅-1+3b+3
Multiply -1 by 9.
c=3
a=-b-1
4=-2b-1
1=-9b+9⋅-1+3b+3
Multiply 9 by -1.
c=3
a=-b-1
4=-2b-1
1=-9b-9+3b+3
c=3
a=-b-1
4=-2b-1
1=-9b-9+3b+3
c=3
a=-b-1
4=-2b-1
1=-6b-9+3
c=3
a=-b-1
4=-2b-1
1=-6b-6
c=3
a=-b-1
4=-2b-1
1=-6b-6
c=3
a=-b-1
4=-2b-1
1=-6b-6
c=3
a=-b-1
4=-2b-1
1=-6b-6
Solve for b in the third equation.
Rewrite the equation as -2b-1=4.
c=3
a=-b-1
-2b-1=4
1=-6b-6
Move all terms not containing b to the right side of the equation.
Add 1 to both sides of the equation.
c=3
a=-b-1
-2b=4+1
1=-6b-6
c=3
a=-b-1
-2b=5
1=-6b-6
c=3
a=-b-1
-2b=5
1=-6b-6
Divide each term by -2 and simplify.
Divide each term in -2b=5 by -2.
c=3
a=-b-1
-2b-2=5-2
1=-6b-6
Cancel the common factor of -2.
Cancel the common factor.
c=3
a=-b-1
-2b-2=5-2
1=-6b-6
Divide b by 1.
c=3
a=-b-1
b=5-2
1=-6b-6
c=3
a=-b-1
b=5-2
1=-6b-6
Move the negative in front of the fraction.
c=3
a=-b-1
b=-52
1=-6b-6
c=3
a=-b-1
b=-52
1=-6b-6
c=3
a=-b-1
b=-52
1=-6b-6
Replace all occurrences of b with -52 in each equation.
Replace all occurrences of b in a=-b-1 with -52.
c=3
a=-(-52)-1
b=-52
1=-6b-6
Replace all occurrences of b in 1=-6b-6 with -52.
c=3
a=-(-52)-1
b=-52
1=-6(-52)-6
c=3
a=-(-52)-1
b=-52
1=-6(-52)-6
Simplify.
Simplify -(-52)-1.
Multiply -(-52).
Multiply -1 by -1.
c=3
a=1(52)-1
b=-52
1=-6(-52)-6
Multiply 52 by 1.
c=3
a=52-1
b=-52
1=-6(-52)-6
c=3
a=52-1
b=-52
1=-6(-52)-6
To write -1 as a fraction with a common denominator, multiply by 22.
c=3
a=52-1⋅22
b=-52
1=-6(-52)-6
Combine -1 and 22.
c=3
a=52+-1⋅22
b=-52
1=-6(-52)-6
Combine the numerators over the common denominator.
c=3
a=5-1⋅22
b=-52
1=-6(-52)-6
Simplify the numerator.
Multiply -1 by 2.
c=3
a=5-22
b=-52
1=-6(-52)-6
Subtract 2 from 5.
c=3
a=32
b=-52
1=-6(-52)-6
c=3
a=32
b=-52
1=-6(-52)-6
c=3
a=32
b=-52
1=-6(-52)-6
Simplify -6(-52)-6.
Simplify each term.
Cancel the common factor of 2.
Move the leading negative in -52 into the numerator.
c=3
a=32
b=-52
1=-6(-52)-6
Factor 2 out of -6.
c=3
a=32
b=-52
1=2(-3)(-52)-6
Cancel the common factor.
c=3
a=32
b=-52
1=2⋅(-3(-52))-6
Rewrite the expression.
c=3
a=32
b=-52
1=-3⋅-5-6
c=3
a=32
b=-52
1=-3⋅-5-6
Multiply -3 by -5.
c=3
a=32
b=-52
1=15-6
c=3
a=32
b=-52
1=15-6
Subtract 6 from 15.
c=3
a=32
b=-52
1=9
c=3
a=32
b=-52
1=9
c=3
a=32
b=-52
1=9
Since 1≠9, there are no solutions.
c=3
a=32
b=-52
No solution
c=3
a=32
b=-52
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=32, b=-52, c=3, and x=0.
Simplify each term.
Raising 0 to any positive power yields 0.
y=32⋅0+(-52)⋅(0)+3
Multiply 32 by 0.
y=0+(-52)⋅(0)+3
Multiply (-52)(0).
Multiply 0 by -1.
y=0+0(52)+3
Multiply 0 by 52.
y=0+0+3
y=0+0+3
y=0+0+3
y=0+3
y=3
y=3
y=3
If the table has a quadratic function rule, y=y for the corresponding x value, x=0. This check passes since y=3 and y=3.
3=3
Calculate the value of y such that y=ax2+b when a=32, b=-52, c=3, and x=1.
Simplify each term.
One to any power is one.
y=32⋅1+(-52)⋅(1)+3
Multiply 32 by 1.
y=32+(-52)⋅(1)+3
Multiply -1 by 1.
y=32-52+3
y=32-52+3
Combine fractions.
Combine fractions with similar denominators.
y=3+3-52
Simplify the expression.
Subtract 5 from 3.
y=3+-22
Divide -2 by 2.
y=3-1
Subtract 1 from 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=32, b=-52, c=3, and x=2.
Simplify each term.
Raise 2 to the power of 2.
y=32⋅4+(-52)⋅(2)+3
Cancel the common factor of 2.
Factor 2 out of 4.
y=32⋅(2(2))+(-52)⋅(2)+3
Cancel the common factor.
y=32⋅(2⋅2)+(-52)⋅(2)+3
Rewrite the expression.
y=3⋅2+(-52)⋅(2)+3
y=3⋅2+(-52)⋅(2)+3
Multiply 3 by 2.
y=6+(-52)⋅(2)+3
Cancel the common factor of 2.
Move the leading negative in -52 into the numerator.
y=6+-52⋅2+3
Cancel the common factor.
y=6+-52⋅2+3
Rewrite the expression.
y=6-5+3
y=6-5+3
y=6-5+3
Subtract 5 from 6.
y=1+3
y=4
y=4
y=4
If the table has a quadratic function rule, y=y for the corresponding x value, x=2. This check passes since y=4 and y=4.
4=4
Calculate the value of y such that y=ax2+b when a=32, b=-52, c=3, and x=3.
Simplify each term.
Raise 3 to the power of 2.
y=32⋅9+(-52)⋅(3)+3
Multiply 32⋅9.
Combine 32 and 9.
y=3⋅92+(-52)⋅(3)+3
Multiply 3 by 9.
y=272+(-52)⋅(3)+3
y=272+(-52)⋅(3)+3
Multiply (-52)(3).
Multiply 3 by -1.
y=272-3(52)+3
Combine -3 and 52.
y=272+-3⋅52+3
Multiply -3 by 5.
y=272+-152+3
y=272+-152+3
Move the negative in front of the fraction.
y=272-152+3
y=272-152+3
Combine fractions.
Combine fractions with similar denominators.
y=3+27-152
Simplify the expression.
Subtract 15 from 27.
y=3+122
Divide 12 by 2.
y=3+6
y=9
y=9
y=9
y=9
If the table has a quadratic function rule, y=y for the corresponding x value, x=3. This check does not pass, since y=9 and y=1. The function rule can’t be quadratic.
9≠1
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.
3=a⋅0+b(0)2+c(0)+d
2=a(1)3+b(0)2+c(0)+d
4=a(2)3+b(0)2+c(0)+d
1=a(3)3+b(0)2+c(0)+d
Multiply a by 0.
3=0+b(0)2+c(0)+d
2=a(1)3+b(0)2+c(0)+d
4=a(2)3+b(0)2+c(0)+d
1=a(3)3+b(0)2+c(0)+d
Raising 0 to any positive power yields 0.
3=0+b⋅0+c(0)+d
2=a(1)3+b(0)2+c(0)+d
4=a(2)3+b(0)2+c(0)+d
1=a(3)3+b(0)2+c(0)+d
Multiply b by 0.
3=0+0+c(0)+d
2=a(1)3+b(0)2+c(0)+d
4=a(2)3+b(0)2+c(0)+d
1=a(3)3+b(0)2+c(0)+d
Multiply c by 0.
3=0+0+0+d
2=a(1)3+b(0)2+c(0)+d
4=a(2)3+b(0)2+c(0)+d
1=a(3)3+b(0)2+c(0)+d
3=0+0+0+d
2=a(1)3+b(0)2+c(0)+d
4=a(2)3+b(0)2+c(0)+d
1=a(3)3+b(0)2+c(0)+d
Combine the opposite terms in 0+0+0+d.
3=0+0+d
2=a(1)3+b(0)2+c(0)+d
4=a(2)3+b(0)2+c(0)+d
1=a(3)3+b(0)2+c(0)+d
3=0+d
2=a(1)3+b(0)2+c(0)+d
4=a(2)3+b(0)2+c(0)+d
1=a(3)3+b(0)2+c(0)+d
3=d
2=a(1)3+b(0)2+c(0)+d
4=a(2)3+b(0)2+c(0)+d
1=a(3)3+b(0)2+c(0)+d
3=d
2=a(1)3+b(0)2+c(0)+d
4=a(2)3+b(0)2+c(0)+d
1=a(3)3+b(0)2+c(0)+d
3=d
2=a(1)3+b(0)2+c(0)+d
4=a(2)3+b(0)2+c(0)+d
1=a(3)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.
3=d
2=a⋅1+b(0)2+c(0)+d
4=a(2)3+b(0)2+c(0)+d
1=a(3)3+b(0)2+c(0)+d
Multiply a by 1.
3=d
2=a+b(0)2+c(0)+d
4=a(2)3+b(0)2+c(0)+d
1=a(3)3+b(0)2+c(0)+d
Raising 0 to any positive power yields 0.
3=d
2=a+b⋅0+c(0)+d
4=a(2)3+b(0)2+c(0)+d
1=a(3)3+b(0)2+c(0)+d
Multiply b by 0.
3=d
2=a+0+c(0)+d
4=a(2)3+b(0)2+c(0)+d
1=a(3)3+b(0)2+c(0)+d
Multiply c by 0.
3=d
2=a+0+0+d
4=a(2)3+b(0)2+c(0)+d
1=a(3)3+b(0)2+c(0)+d
3=d
2=a+0+0+d
4=a(2)3+b(0)2+c(0)+d
1=a(3)3+b(0)2+c(0)+d
Combine the opposite terms in a+0+0+d.
3=d
2=a+0+d
4=a(2)3+b(0)2+c(0)+d
1=a(3)3+b(0)2+c(0)+d
3=d
2=a+d
4=a(2)3+b(0)2+c(0)+d
1=a(3)3+b(0)2+c(0)+d
3=d
2=a+d
4=a(2)3+b(0)2+c(0)+d
1=a(3)3+b(0)2+c(0)+d
3=d
2=a+d
4=a(2)3+b(0)2+c(0)+d
1=a(3)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.
3=d
2=a+d
4=a⋅8+b(0)2+c(0)+d
1=a(3)3+b(0)2+c(0)+d
Move 8 to the left of a.
3=d
2=a+d
4=8⋅a+b(0)2+c(0)+d
1=a(3)3+b(0)2+c(0)+d
Raising 0 to any positive power yields 0.
3=d
2=a+d
4=8a+b⋅0+c(0)+d
1=a(3)3+b(0)2+c(0)+d
Multiply b by 0.
3=d
2=a+d
4=8a+0+c(0)+d
1=a(3)3+b(0)2+c(0)+d
Multiply c by 0.
3=d
2=a+d
4=8a+0+0+d
1=a(3)3+b(0)2+c(0)+d
3=d
2=a+d
4=8a+0+0+d
1=a(3)3+b(0)2+c(0)+d
Combine the opposite terms in 8a+0+0+d.
3=d
2=a+d
4=8a+0+d
1=a(3)3+b(0)2+c(0)+d
3=d
2=a+d
4=8a+d
1=a(3)3+b(0)2+c(0)+d
3=d
2=a+d
4=8a+d
1=a(3)3+b(0)2+c(0)+d
3=d
2=a+d
4=8a+d
1=a(3)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.
3=d
2=a+d
4=8a+d
1=a⋅27+b(0)2+c(0)+d
Move 27 to the left of a.
3=d
2=a+d
4=8a+d
1=27⋅a+b(0)2+c(0)+d
Raising 0 to any positive power yields 0.
3=d
2=a+d
4=8a+d
1=27a+b⋅0+c(0)+d
Multiply b by 0.
3=d
2=a+d
4=8a+d
1=27a+0+c(0)+d
Multiply c by 0.
3=d
2=a+d
4=8a+d
1=27a+0+0+d
3=d
2=a+d
4=8a+d
1=27a+0+0+d
Combine the opposite terms in 27a+0+0+d.
3=d
2=a+d
4=8a+d
1=27a+0+d
3=d
2=a+d
4=8a+d
1=27a+d
3=d
2=a+d
4=8a+d
1=27a+d
3=d
2=a+d
4=8a+d
1=27a+d
3=d
2=a+d
4=8a+d
1=27a+d
Rewrite the equation as d=3.
d=3
2=a+d
4=8a+d
1=27a+d
Replace all occurrences of d with 3 in each equation.
Replace all occurrences of d in 2=a+d with 3.
d=3
2=a+3
4=8a+d
1=27a+d
Replace all occurrences of d in 4=8a+d with 3.
d=3
2=a+3
4=8a+3
1=27a+d
Replace all occurrences of d in 1=27a+d with 3.
d=3
2=a+3
4=8a+3
1=27a+3
d=3
2=a+3
4=8a+3
1=27a+3
Simplify.
Remove parentheses.
d=3
2=a+3
4=8a+3
1=27a+3
Remove parentheses.
d=3
2=a+3
4=8a+3
1=27a+3
Remove parentheses.
d=3
2=a+3
4=8a+3
1=27a+3
d=3
2=a+3
4=8a+3
1=27a+3
Solve for a in the second equation.
Rewrite the equation as a+3=2.
d=3
a+3=2
4=8a+3
1=27a+3
Move all terms not containing a to the right side of the equation.
Subtract 3 from both sides of the equation.
d=3
a=2-3
4=8a+3
1=27a+3
Subtract 3 from 2.
d=3
a=-1
4=8a+3
1=27a+3
d=3
a=-1
4=8a+3
1=27a+3
d=3
a=-1
4=8a+3
1=27a+3
Replace all occurrences of a with -1 in each equation.
Replace all occurrences of a in 4=8a+3 with -1.
d=3
a=-1
4=8(-1)+3
1=27a+3
Replace all occurrences of a in 1=27a+3 with -1.
d=3
a=-1
4=8(-1)+3
1=27(-1)+3
d=3
a=-1
4=8(-1)+3
1=27(-1)+3
Simplify.
Simplify 8(-1)+3.
Multiply 8 by -1.
d=3
a=-1
4=-8+3
1=27(-1)+3
d=3
a=-1
4=-5
1=27(-1)+3
d=3
a=-1
4=-5
1=27(-1)+3
Simplify 27(-1)+3.
Multiply 27 by -1.
d=3
a=-1
4=-5
1=-27+3
d=3
a=-1
4=-5
1=-24
d=3
a=-1
4=-5
1=-24
d=3
a=-1
4=-5
1=-24
Since 4≠-5, there are no solutions.
d=3
a=-1
No solution
1=-24
Since 1≠-24, there are no solutions.
d=3
a=-1
No solution
No solution
d=3
a=-1
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=-1, b=0, c=0, d=3, and x=0.
Simplify each term.
Raising 0 to any positive power yields 0.
y=-1⋅0+(0)⋅(02)+(0)⋅(0)+3
Multiply -1 by 0.
y=0+(0)⋅(02)+(0)⋅(0)+3
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)+3
Use the power rule aman=am+n to combine exponents.
y=0+01+2+(0)⋅(0)+3
y=0+01+2+(0)⋅(0)+3
y=0+03+(0)⋅(0)+3
y=0+03+(0)⋅(0)+3
Raising 0 to any positive power yields 0.
y=0+0+(0)⋅(0)+3
Multiply 0 by 0.
y=0+0+0+3
y=0+0+0+3
y=0+0+3
y=0+3
y=3
y=3
y=3
If the table has a cubic function rule, y=y for the corresponding x value, x=0. This check passes since y=3 and y=3.
3=3
Calculate the value of y such that y=ax3+b when a=-1, b=0, c=0, d=3, and x=1.
Simplify each term.
One to any power is one.
y=-1⋅1+(0)⋅(12)+(0)⋅(1)+3
Multiply -1 by 1.
y=-1+(0)⋅(12)+(0)⋅(1)+3
One to any power is one.
y=-1+0⋅1+(0)⋅(1)+3
Multiply 0 by 1.
y=-1+0+(0)⋅(1)+3
Multiply 0 by 1.
y=-1+0+0+3
y=-1+0+0+3
y=-1+0+3
y=-1+3
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=-1, b=0, c=0, d=3, and x=2.
Simplify each term.
Raise 2 to the power of 3.
y=-1⋅8+(0)⋅(22)+(0)⋅(2)+3
Multiply -1 by 8.
y=-8+(0)⋅(22)+(0)⋅(2)+3
Raise 2 to the power of 2.
y=-8+0⋅4+(0)⋅(2)+3
Multiply 0 by 4.
y=-8+0+(0)⋅(2)+3
Multiply 0 by 2.
y=-8+0+0+3
y=-8+0+0+3
y=-8+0+3
y=-8+3
y=-5
y=-5
y=-5
If the table has a cubic function rule, y=y for the corresponding x value, x=2. This check does not pass, since y=-5 and y=4. The function rule can’t be cubic.
-5≠4
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],[0,3],[1,2],[2,4],[3,1]]     