Polynomial Functions & Their Graphs — Practice
The degree and the leading coefficient decide the shape before you plot a single point.
What these questions cover
These questions use 8 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Exponent — How many times the base is used as a factor. In 53 the exponent is 3.
- Factor — A number that divides evenly into another.
- y-intercept — The output when x = 0 — where the line crosses the vertical axis.
- Polynomial — A sum of terms, each a number times a whole-number power of the variable.
- Roots — The x values where y = 0.
- Leading coefficient — The number in front of the highest power. Its sign decides which way the right end goes.
- Multiplicity — How many times a root repeats. It decides whether the graph crosses or just touches.
- Turning point — Where the graph changes from rising to falling or back. At most n - 1 of them.
Problem 1
What is the degree of 4x3 - 2x5 + 7?
Worked answer
5
The powers are 3, 5 and 0
The highest is 5
Problem 2
Describe the ends of y = x4 - 3x.
- Both up
- Both down
- Down left, up right
- Up left, down right
Worked answer
Both up
Degree 4 is even, so the ends agree
The lead +1 sends them both up
Problem 3
Describe the ends of y = -x3 + 2.
- Up left, down right
- Down left, up right
- Both up
- Both down
Worked answer
Up left, down right
Odd degree means the ends disagree
Negative lead flips the usual pattern
Problem 4
Give the roots of y=(x-2)(x+5), smallest first.
Worked answer
-5,2
x-2=0 gives 2
x+5=0 gives -5
Problem 5
Give the roots of y=x(x-4)(x+1), smallest first.
Worked answer
-1,0,4
x=0, x=4, x=-1
Problem 6
At x=3, what does the graph of y=(x-3)2(x+1) do?
- Touches and turns back
- Crosses straight through
- Has a hole
- Is undefined
Worked answer
Touches and turns back
(x-3) appears twice, so multiplicity 2
Even multiplicity bounces off the axis
Problem 7
At x=-1, what does y=(x-3)2(x+1) do?
- Crosses straight through
- Touches and turns back
- Has a hole
- Flattens out
Worked answer
Crosses straight through
Multiplicity 1 is odd
Odd multiplicity crosses
Problem 8
What is the greatest number of turning points a degree 6 polynomial can have?
Worked answer
5
At most n-1 = 5
Problem 9
A graph has 5 turning points. What is the smallest degree it could be?
Worked answer
6
n-1 ≥ 5
So n ≥ 6
Problem 10
Give the degree of y=(x-1)2(x+3)3.
Worked answer
5
2+3=5
Problem 11
Find the y-intercept of y=(x-2)(x+3)(x-1).
Worked answer
6
(-2)(3)(-1)=6
Problem 12
How many real roots can a degree 4 polynomial have at most?
Worked answer
4
Degree n gives at most n real roots
Problem 13
Write a polynomial of least degree with roots -2, 1 and 4, leading coefficient 1. Give it factored.
Worked answer
(x+2)(x-1)(x-4)
Root -2 gives (x+2)
Root 1 gives (x-1)
Root 4 gives (x-4)
Problem 14
Write the factored polynomial of least degree that touches at x=2 and crosses at x=-3, leading coefficient 1.
Worked answer
(x-2)^2(x+3)
Touch at 2 means (x-2)2
Cross at -3 means (x+3)
Problem 15
A graph falls on the left and falls on the right, and crosses the axis twice. What could its degree be?
- 4
- 3
- 5
- 1
Worked answer
4
Ends agree, so the degree is even
Falling both ways needs a negative lead
Degree 4 fits
Problem 16
For y=2(x-1)(x+2)2, find the y-intercept.
Worked answer
-8
2(-1)(2)2
=2 × (-1) × 4 = -8