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Amy Ferguson Moncure

H Algebra 2/Trig · Unit 3

Equation from a Graph

Turn the intercepts into factors, then use the labeled point to find a.

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Key Ideas Easy Medium Hard

End Behavior

Even

x²
−x²

Odd

x³
−x³
What to Remember

Start from the zeros

List each solution. Under it, write the multiplicity.

Then write a factor for each solution. The exponent is the multiplicity.

y = a(x − r)m…

Solve for a

The colored point is on the graph. x is under the first number. y is under the second.

Substitute and solve for a. Then put the entire equation together.

Level 1

Easy

Every factor has multiplicity 1. Use the y-intercept.

Example
Write an equation for the polynomial.
−3−2−1123456−12−10−8−6−4−22468(0, −8)

x = −1

multiplicity 1

x = 2

multiplicity 1

x = 4

multiplicity 1

(x + 1)(x − 2)(x − 4)

y = a(x + 1)(x − 2)(x − 4)

(0x,−8y)

−8 = a(0 + 1)(0 − 2)(0 − 4)

−8 = a(1)(−2)(−4)

−8 = 8a

a = −88

a = −1

Put the entire equation together.

y = −1(x + 1)(x − 2)(x − 4)

Answer: y = −1(x + 1)(x − 2)(x − 4)

1
Write an equation for the polynomial.
−6−5−4−3−2−1123−12−10−8−6−4−22468(0, −8)
Show Answer

x = −4

multiplicity 1

x = −2

multiplicity 1

x = 1

multiplicity 1

(x + 4)(x + 2)(x − 1)

y = a(x + 4)(x + 2)(x − 1)

(0x,−8y)

−8 = a(0 + 4)(0 + 2)(0 − 1)

−8 = a(4)(2)(−1)

−8 = −8a

a = −8−8

a = 1

Put the entire equation together.

y = 1(x + 4)(x + 2)(x − 1)

Answer: y = 1(x + 4)(x + 2)(x − 1)

2
Write an equation for the polynomial.
−5−4−3−2−11234−8−6−4−224681012(0, 6)
Show Answer

x = −3

multiplicity 1

x = −1

multiplicity 1

x = 2

multiplicity 1

(x + 3)(x + 1)(x − 2)

y = a(x + 3)(x + 1)(x − 2)

(0x,6y)

6 = a(0 + 3)(0 + 1)(0 − 2)

6 = a(3)(1)(−2)

6 = −6a

a = 6−6

a = −1

Put the entire equation together.

y = −1(x + 3)(x + 1)(x − 2)

Answer: y = −1(x + 3)(x + 1)(x − 2)

3
Write an equation for the polynomial.
−4−3−2−112345−8−6−4−224681012(0, 6)
Show Answer

x = −2

multiplicity 1

x = 1

multiplicity 1

x = 3

multiplicity 1

(x + 2)(x − 1)(x − 3)

y = a(x + 2)(x − 1)(x − 3)

(0x,6y)

6 = a(0 + 2)(0 − 1)(0 − 3)

6 = a(2)(−1)(−3)

6 = 6a

a = 66

a = 1

Put the entire equation together.

y = 1(x + 2)(x − 1)(x − 3)

Answer: y = 1(x + 2)(x − 1)(x − 3)

4
Write an equation for the polynomial.
−4−3−2−1123456−16−14−12−10−8−6−4−2246810121416(0, −8)
Show Answer

x = −2

multiplicity 1

x = 1

multiplicity 1

x = 4

multiplicity 1

(x + 2)(x − 1)(x − 4)

y = a(x + 2)(x − 1)(x − 4)

(0x,−8y)

−8 = a(0 + 2)(0 − 1)(0 − 4)

−8 = a(2)(−1)(−4)

−8 = 8a

a = −88

a = −1

Put the entire equation together.

y = −1(x + 2)(x − 1)(x − 4)

Answer: y = −1(x + 2)(x − 1)(x − 4)

5
Write an equation for the polynomial.
−3−2−112345−3−2−112345678(0, 6)
Show Answer

x = −1

multiplicity 1

x = 2

multiplicity 1

x = 3

multiplicity 1

(x + 1)(x − 2)(x − 3)

y = a(x + 1)(x − 2)(x − 3)

(0x,6y)

6 = a(0 + 1)(0 − 2)(0 − 3)

6 = a(1)(−2)(−3)

6 = 6a

a = 66

a = 1

Put the entire equation together.

y = 1(x + 1)(x − 2)(x − 3)

Answer: y = 1(x + 1)(x − 2)(x − 3)

Level 2

Medium

Squared factors and cubed factors.

Example
Write an equation for the polynomial.
−3−2−11234−3−2−11234567891011(0, 8)

x = −1

multiplicity 2

x = 2

multiplicity 3

(x + 1)²(x − 2)³

y = a(x + 1)²(x − 2)³

(0x,8y)

8 = a(0 + 1)²(0 − 2)³

8 = a(1)²(−2)³

8 = a(1)(−8)

8 = −8a

a = 8−8

a = −1

Put the entire equation together.

y = −1(x + 1)²(x − 2)³

Answer: y = −1(x + 1)²(x − 2)³

1
Write an equation for the polynomial.
−5−4−3−2−1123−14−12−10−8−6−4−2246(0, −9)
Show Answer

x = −3

multiplicity 2

x = −1

multiplicity 1

x = 1

multiplicity 1

(x + 3)²(x + 1)(x − 1)

y = a(x + 3)²(x + 1)(x − 1)

(0x,−9y)

−9 = a(0 + 3)²(0 + 1)(0 − 1)

−9 = a(3)²(1)(−1)

−9 = a(9)(1)(−1)

−9 = −9a

a = −9−9

a = 1

Put the entire equation together.

y = 1(x + 3)²(x + 1)(x − 1)

Answer: y = 1(x + 3)²(x + 1)(x − 1)

2
Write an equation for the polynomial.
−4−3−2−1123−11−10−9−8−7−6−5−4−3−2−1123(0, −8)
Show Answer

x = −2

multiplicity 3

x = 1

multiplicity 1

(x + 2)³(x − 1)

y = a(x + 2)³(x − 1)

(0x,−8y)

−8 = a(0 + 2)³(0 − 1)

−8 = a(2)³(−1)

−8 = a(8)(−1)

−8 = −8a

a = −8−8

a = 1

Put the entire equation together.

y = 1(x + 2)³(x − 1)

Answer: y = 1(x + 2)³(x − 1)

3
Write an equation for the polynomial.
−4−3−2−11234−12−10−8−6−4−2246(0, −8)
Show Answer

x = −2

multiplicity 2

x = 1

multiplicity 1

x = 2

multiplicity 1

(x + 2)²(x − 1)(x − 2)

y = a(x + 2)²(x − 1)(x − 2)

(0x,−8y)

−8 = a(0 + 2)²(0 − 1)(0 − 2)

−8 = a(2)²(−1)(−2)

−8 = a(4)(−1)(−2)

−8 = 8a

a = −88

a = −1

Put the entire equation together.

y = −1(x + 2)²(x − 1)(x − 2)

Answer: y = −1(x + 2)²(x − 1)(x − 2)

4
Write an equation for the polynomial.
−3−2−11234−3−2−11234567891011(0, 8)
Show Answer

x = −1

multiplicity 1

x = 2

multiplicity 3

(x + 1)(x − 2)³

y = a(x + 1)(x − 2)³

(0x,8y)

8 = a(0 + 1)(0 − 2)³

8 = a(1)(−2)³

8 = a(1)(−8)

8 = −8a

a = 8−8

a = −1

Put the entire equation together.

y = −1(x + 1)(x − 2)³

Answer: y = −1(x + 1)(x − 2)³

5
Write an equation for the polynomial.
−4−3−2−1123−11−10−9−8−7−6−5−4−3−2−1123(0, −4)
Show Answer

x = −2

multiplicity 2

x = 1

multiplicity 3

(x + 2)²(x − 1)³

y = a(x + 2)²(x − 1)³

(0x,−4y)

−4 = a(0 + 2)²(0 − 1)³

−4 = a(2)²(−1)³

−4 = a(4)(−1)

−4 = −4a

a = −4−4

a = 1

Put the entire equation together.

y = 1(x + 2)²(x − 1)³

Answer: y = 1(x + 2)²(x − 1)³

Level 3

Hard

x = 0 is a solution. Use the other labeled point.

Example
Write an equation for the polynomial.
−4−3−2−112345−8−6−4−224681012(1, 6)

x = −2

multiplicity 1

x = 0

multiplicity 1

x = 3

multiplicity 1

(x + 2)(x)(x − 3)

y = a(x + 2)(x)(x − 3)

(1x,6y)

6 = a(1 + 2)(1)(1 − 3)

6 = a(3)(1)(−2)

6 = −6a

a = 6−6

a = −1

Put the entire equation together.

y = −1(x + 2)(x)(x − 3)

Answer: y = −1(x + 2)(x)(x − 3)

1
Write an equation for the polynomial.
−2−1123456−224681012(2, 8)
Show Answer

x = 0

multiplicity 2

x = 4

multiplicity 1

(x)²(x − 4)

y = a(x)²(x − 4)

(2x,8y)

8 = a(2)²(2 − 4)

8 = a(2)²(−2)

8 = a(4)(−2)

8 = −8a

a = 8−8

a = −1

Put the entire equation together.

y = −1(x)²(x − 4)

Answer: y = −1(x)²(x − 4)

2
Write an equation for the polynomial.
−2−11234−6−5−4−3−2−1123(3, −3)
Show Answer

x = 0

multiplicity 1

x = 2

multiplicity 2

(x)(x − 2)²

y = a(x)(x − 2)²

(3x,−3y)

−3 = a(3)(3 − 2)²

−3 = a(3)(1)²

−3 = a(3)(1)

−3 = 3a

a = −33

a = −1

Put the entire equation together.

y = −1(x)(x − 2)²

Answer: y = −1(x)(x − 2)²

3
Write an equation for the polynomial.
−5−4−3−2−112−12−10−8−6−4−22468(1, 4)
Show Answer

x = −3

multiplicity 1

x = 0

multiplicity 3

(x + 3)(x)³

y = a(x + 3)(x)³

(1x,4y)

4 = a(1 + 3)(1)³

4 = a(4)(1)³

4 = a(4)(1)

4 = 4a

a = 44

a = 1

Put the entire equation together.

y = 1(x + 3)(x)³

Answer: y = 1(x + 3)(x)³

4
Write an equation for the polynomial.
−3−2−11234−6−5−4−3−2−1123(1, −4)
Show Answer

x = −1

multiplicity 1

x = 0

multiplicity 1

x = 2

multiplicity 1

(x + 1)(x)(x − 2)

y = a(x + 1)(x)(x − 2)

(1x,−4y)

−4 = a(1 + 1)(1)(1 − 2)

−4 = a(2)(1)(−1)

−4 = −2a

a = −4−2

a = 2

Put the entire equation together.

y = 2(x + 1)(x)(x − 2)

Answer: y = 2(x + 1)(x)(x − 2)

5
Write an equation for the polynomial.
−4−3−2−11234−2−112345678910(1, 6)
Show Answer

x = −2

multiplicity 1

x = 0

multiplicity 2

x = 2

multiplicity 1

(x + 2)(x)²(x − 2)

y = a(x + 2)(x)²(x − 2)

(1x,6y)

6 = a(1 + 2)(1)²(1 − 2)

6 = a(3)(1)²(−1)

6 = a(3)(1)(−1)

6 = −3a

a = 6−3

a = −2

Put the entire equation together.

y = −2(x + 2)(x)²(x − 2)

Answer: y = −2(x + 2)(x)²(x − 2)

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