Algebra 2/Pre-Calculus
Name__________________
Finding Equations for Graphs (Day 9, Polynomials)
In this handout, we will learn how to write the equation for a polynomial from information that
we are given about that polynomial. We will begin with a few problems to review graphing.
1. Sketch the graph of each of the following polynomials without using your calculator. Your
sketches should include the coordinates of the x-intercepts, the y-intercept, and the correct
general shape of the graph. Check your answers on the calculator.
a. y = 5(x + 4)(x -1)2
c.
y = 2x 3 + 2x 2 - 60x
b. y = -3(x -1)2 (x + 2)2
d. y = -x 4 + 5x 3 + 4x 2 - 20x
2. Here’s the graph of a cubic polynomial. Note: “Cubic” means degree three.
a. The coordinates of the x-intercepts are (2,0) , (1,0) , and ( 4,0) . What are three factors
of this polynomial? What theorem allows us to make this conclusion?
b. The factor theorem tells us that x  2 , x  1 , and x  4 are all factors of this
polynomial. Suppose we are told that the graph of this polynomial goes through the
point (5,21) . Find the equation for the polynomial. Note: If you need a hint, look ahead
to part c, but try it on your own first.
c. The equation of the polynomial can be written as p( x )  a ( x  2)( x  1)( x  4) . (Why
is this?) Can we plug in the point (5,21) and solve for a?
In problem 2, you should have found that p( x)  43 ( x  2)( x  1)( x  4) . Here’s the way to find
that a  43 .
p( x )  a ( x  2)( x  1)( x  4)
p(5)  a (5  2)(5  1)(5  4)
21  (7)( 4)(1)a
21  28a
3
4
a
Hence, p( x)  43 ( x  2)( x  1)( x  4) .
3. Here’s the graph of another polynomial.
Suppose this is a cubic polynomial whose x-intercepts are (5,0) , (3,0) , and (1,0) , and
whose y-intercept is (0,45) . Find the equation for the polynomial. Hint: Use the method
you developed in problem 2.
Answer p( x )  3( x  5)( x  3)( x  1)
4. Here are the graphs of two quartic polynomials. The one on the left is called f ( x ) and the
one on the right is called g ( x ) . Note: “Quartic” means degree four.
a. Suppose f ( 2)  80 . Find a formula for f ( x ) .
b. Suppose g (1)  80 . Find a formula for g ( x ) .
Answers a. f ( x )  2 x( x  3)( x  2)( x  1) b. g ( x )  5( x  2) 2 ( x  3) 2
5. In this problem, you will continue finding equations for polynomials. Note: Answers are
provided at the end of this problem.
a. Suppose p ( x ) is a cubic polynomial such that p( 4)  p(0)  p( 8)  0 and p ( 2)  60 .
Find a formula for p ( x ) .
b. Suppose f ( x ) is a quartic polynomial. It’s only x-intercepts are ( 4,0) and (10,0) .
f ( 2)  64 and f ( x ) is never positive. Find the equation for f ( x ) . Hint: Start by
sketching the graph of f ( x ) .
Answers a. p( x )   23 x( x  4)( x  8) b. f ( x )   14 ( x  4) 2 ( x  10) 2
6. Suppose g ( x ) is a cubic polynomial such that g (1)  g ( 1)  g (6)  0 . Also, when g ( x )
is divided by x  4 the remainder is  75 .
a. Can you find the equation for g ( x ) ? Note: If you need a hint, look at parts b and c. But
try it on your own first.
b. What is the value of g ( 4 ) ? Hint: Remainder theorem!
c. By the remainder theorem, g ( 4)  75 . Now find the equation for g ( x ) .
Answer g ( x )  25 ( x  1)( x  1)( x  6)
7. Suppose f ( x ) is a polynomial such that f (3)  0 , f (5)  0 , and f (7)  0 .
a. What can you say about the degree of this polynomial?
b. Suppose f ( x ) is a cubic polynomial. Write three possible equations for f ( x ) .
c. Suppose f ( x ) is a quartic polynomial. Write three possible equations for f ( x ) .
8. (Optional Challenge) Suppose h(x ) is a cubic polynomial such that h (1)  0 , h(3)  0 ,
h ( 2)  16 , and h (0)  72 . Find a formula for h(x ) . Hint: There are two variables that
you will need to find. To find them, set up and solve a system of equations.
9. (Optional Challenge) Suppose f ( x ) is a polynomial that has the following values:
f (1)  2 , f ( 2)  2 , and f (3)  8 . Our goal is to write the equation for a function that
has these values. Note: In our next lesson, we will learn a method for doing this type of
problem, but for now, this is a pretty hard problem. See what you can do with it. Don’t be
afraid to guess and check!