xy03122431
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
Add 0 and 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
Add -2 and 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
Add -3 and 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
Add 0 and 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
Add -1 and 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
Add -2 and 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
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.
Add 0 and 0.
3=0+c
2=a(1)2+b(1)+c
4=a(2)2+b(2)+c
1=a(3)2+b(3)+c
Add 0 and 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
Simplify by adding terms.
Add -4b and 2b.
c=3
a=-b-1
4=-2b-4+3
1=9(-b-1)+3b+3
Add -4 and 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
Simplify by adding terms.
Add -9b and 3b.
c=3
a=-b-1
4=-2b-1
1=-6b-9+3
Add -9 and 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
Add 4 and 1.
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
Simplify by adding zeros.
Add 0 and 0.
y=0+3
Add 0 and 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
Simplify by adding and subtracting.
Subtract 5 from 6.
y=1+3
Add 1 and 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
Add 3 and 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.
The function is not quadratic
The function is not quadratic
The function is not quadratic
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.
Add 0 and 0.
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
Add 0 and 0.
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
Add 0 and 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.
Add a and 0.
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
Add a and 0.
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.
Add 8a and 0.
3=d
2=a+d
4=8a+0+d
1=a(3)3+b(0)2+c(0)+d
Add 8a and 0.
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.
Add 27a and 0.
3=d
2=a+d
4=8a+d
1=27a+0+d
Add 27a and 0.
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
Add -8 and 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
Add -27 and 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
Add 1 and 2.
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
Simplify by adding zeros.
Add 0 and 0.
y=0+0+3
Add 0 and 0.
y=0+3
Add 0 and 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
Simplify by adding numbers.
Add -1 and 0.
y=-1+0+3
Add -1 and 0.
y=-1+3
Add -1 and 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
Simplify by adding numbers.
Add -8 and 0.
y=-8+0+3
Add -8 and 0.
y=-8+3
Add -8 and 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]]
