Graphs and Equations of Polynomial Functions
You are getting more comfortable writing the equation for a polynomial in factored form:
𝑓(𝑥) = 𝑎(𝑥 − 𝑧1 )(𝑥 − 𝑧2 )(𝑥 − 𝑧3 ) … when given a graph. You are also pretty confident
sketching the graph of a polynomial using it’s factored form. Today we are going to reinforce
those skills, while integrating the use of technology and our general knowledge of the
behavior of polynomials.
1.
Graph to Equation Given the graph of the
polynomial function, answer the following.
a. Just by looking at the graph, what degree should it
have and why? 3, 3 zeros and 2 turning points
b. Just by looking at the graph, what should the sign
(+ or -) of the leading coefficient, “a”, be? Why?
Negative, positive x values have negative y values.
c. Using the zeros, write an equation for the graph. Be
sure to include an unknown ‘a’ value.
y = a(x + 2)(x – 1)(x – 0)
d. Using the fact that (−1, −4) is on the graph, find the value of the leading coefficient
(the “a” value). a = -2
y = -2(x + 2)(x – 1)(x – 0)
e. Using your TI-84 calculator, find the exact coordinates of the maximum and minimum
values (the turning points).
Max: (0.549,1.262)
Min: (-1.215, -4.225)
2.
Equation to Graph Consider the polynomial function 𝑓(𝑥) = (𝑥 − 1)(𝑥 + 1)(𝑥 − 2)(𝑥 + 2).
a. Find the y-intercept of the function
(0,4)
b. ACCURATELY sketch a graph of the function using
the x and y-intercepts.
Verify using Desmos or calculator
c. Using your calculator, find the exact coordinates of all the turning points. Please specify
whether they are a max or a min.
Max: (0,4)
Min: (-1.581, -2.25) and (1.581, -2.25)
3.
Factoring a polynomial Consider the polynomial 𝑦 = 𝑥 3 − 𝑥 2 − 6𝑥.
a. Find the y-intercept of the function.
(0,0)
b. Before you make any calculations, how many x-intercepts (at most) could this polynomial
have and why? 3, it’s a 3rd degree polynomial.
c. Completely factor the polynomial. Hint: Start by first factoring out a GCF from all three terms.
y = x(x – 3)(x + 2)
d. Find all x-intercepts of the polynomial.
(0,0) , (3,0) , (-2,0)
e. Using the x and y intercepts, sketch a graph of the
polynomial. Verify with Desmos.
The remaining problems are to be completed for homework.
4.
Given the graph of the polynomial function, answer
the following.
a. Just by looking at the graph, what degree should
it have and why? 4, 4 zeros and 3 turns.
b. Just by looking at the graph, what should the sign
(+ or -) of the leading coefficient be? Why?
+, the positive x values have positive y values.
c. Using the zeros, write an equation for the graph.
Be sure to include an unknown ‘a’ value.
y = a(x + 3)(x + 1)(x – 1)(x – 3)
d. Using the fact that (0, 9) is the y-intercept, find the value of the leading coefficient (the “a”
value).
y = (x + 3)(x + 1)(x – 1)(x – 3)
e. Using your TI-84 calculator, find the exact coordinates of the maximum and minimum values
(the turning points).
Max: (0,9)
Min: (-2.236, -16) and (2.236, -16)
5.
Consider the polynomial function 𝑝(𝑥) = 𝑥 3 − 2𝑥 2 − 8𝑥.
a. Find the y-intercept of the function
(0,0)
b. Completely factor the polynomial. Hint: Start by first
factoring out a GCF from all three terms.
p(x) = x(x – 4)(x + 2)
c. ACCURATELY sketch a graph of the function using the x and y-intercepts.
Verify with Desmos
d. Using your calculator, find the exact coordinates of all the turning points. Please
specify whether they are a max or a min.
Max: (-1.097, 5.049)
Min: 2.431, -16.901)
6.
Equation to Graph Consider the polynomial function 𝑔(𝑥) = −𝑥(𝑥 + 2)(𝑥 − 2)(𝑥 − 3).
a.
Is the graph going open up or open down? Why?
Down, there is a negative “a” value.
b.
Find the y-intercept of the function.
(0,0)
c.
ACCURATELY sketch a graph of the function using the x
and y-intercepts. Rescale the axes as necessary.
Verify with Desmos
d.
Using your calculator, find the exact coordinates of all the turning points. Please specify
whether they are a max or a min.
Min: (0.93, -6.035) Max (-1.254, 12.949) and (2.574, 2.879)