Creating Polynomial Equations
Alg2 CC
Name _________________________
As we’ve discovered, if we know the roots of a polynomial function and one other point on the function, then
we can write the equation for the function.
{r1 , r2 , r3 , r4 ,...rn } represents the roots (zeros) of a polynomial, then the polynomial can be
written as: y  a( x  r1 )( x  r2 )( x  r3 )...( x  rn ) where a is some constant.
If the set
Note:You need another point on the function to determine the value of a. Why?
What effects could
a have on the graph of the polynomial?
Express your answer in standard form (𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐)
Let’s explore how “a” affects a polynomial function.
2. Consider quadratic polynomials of the form 𝑦 = 𝑎(𝑥 + 2)(𝑥 − 5), where 𝑎 ≠ 0.
a) What are the x-intercepts of this parabola?
b) Sketch on the axes given the following equations:
𝑦 = (𝑥 + 2)(𝑥 − 5)
𝑦 = 2(𝑥 + 2)(𝑥 − 5)
𝑦 = 4(𝑥 + 2)(𝑥 − 5)
Adapted from:
5.
6. Create the equation of a cubic in standard form that has a double zero at −2 and another zero at 4. The cubic
has a y-intercept of 16. Sketch your cubic on the axes below to verify your result.
Practice:
7. Identify the leading coefficient, degree, and end behavior.
A) 𝑃(𝑥) = −4𝑥 4 − 3𝑥 3 + 𝑥 2 + 4
B) 𝑄(𝑥) = −2𝑥 7 + 6𝑥 5 + 2𝑥 3
C) 𝑅(𝑥) = 𝑥 5 − 4𝑥 2 + 3𝑥 − 1
D) 𝑆(𝑥) = 3𝑥 2 + 6𝑥 − 10
8. Identify whether the function graphed has an odd or even degree and a positive or negative leading
coefficient.
A)
B)
C)
D)
9. Sketch the graph of the following without a calculator. Include x-intercepts, y-intercept, and end
behavior.
A) 𝒚 = 𝒙𝟑 − 𝟑𝒙𝟐 + 𝟐𝒙
B) 𝒚 = −𝒙𝟐 (𝒙𝟐 − 𝟒)
C) 𝒚 = −𝟑𝒙𝟑 − 𝟖𝒙𝟐 + 𝟗𝒙 + 𝟓𝒙𝟐