Graphs of polynomials

What you should be familiar with before taking this lesson

• As $x\to +\mathrm{\infty }$‍, what does $f(x)$‍ approach?
• As $x\to -\mathrm{\infty }$‍, what does $f(x)$‍ approach?

What you will learn in this lesson

Analyzing polynomial functions

Finding the $y$‍-intercept

Finding the $x$‍-intercepts

Finding the end behavior

Sketching a graph

• As $x\to +\mathrm{\infty }$‍, $f(x)\to +\mathrm{\infty }$‍.
• As $x\to -\mathrm{\infty }$‍, $f(x)\to -\mathrm{\infty }$‍.

• The graph touches the $x$‍-axis at $(-2,0)$‍, since $-2$‍ is a zero of even multiplicity.
• The graph crosses the $x$‍-axis at $({\displaystyle \frac{2}{3}},0)$‍, since ${\displaystyle \frac{2}{3}}$‍ is a zero of odd multiplicity.

Positive and negative intervals

Check your understanding

a) What is the $y$‍-intercept of the graph of $g(x)=(x+1)(x-2)(x+5)$‍?

$(0$‍,

• Your answer should be
• an integer, like $6$‍
• a simplified proper fraction, like $3/5$‍
• a simplified improper fraction, like $7/4$‍
• a mixed number, like $13/4$‍
• an exact decimal, like $0.75$‍
• a multiple of pi, like $12\text{pi}$‍ or $2/3\text{pi}$‍

$)$‍

Check

I need help!

To find the $y$‍-intercept we can find $g(0)$‍.

$\begin{array}{rl}g(x)& =(x+1)(x-2)(x+5)\\ \\ g(0)& =(0+1)(0-2)(0+5)\\ \\ g(x)& =(1)(-2)(5)\\ \\ g(x)& =-10\end{array}$‍

The $y$‍-intercept of the graph of $y=g(x)$‍ is $(0,-10)$‍.

b) What is the end behavior of the graph of $g(x)=(x+1)(x-2)(x+5)$‍?

Choose 1 answer:

Choose 1 answer:

• (Choice A)  

  As $x\to +\mathrm{\infty }$‍, $g(x)\to +\mathrm{\infty }$‍, and as $x\to -\mathrm{\infty }$‍, $g(x)\to +\mathrm{\infty }$‍.

  A

  As $x\to +\mathrm{\infty }$‍, $g(x)\to +\mathrm{\infty }$‍, and as $x\to -\mathrm{\infty }$‍, $g(x)\to +\mathrm{\infty }$‍.
• (Choice B)  

  As $x\to +\mathrm{\infty }$‍, $g(x)\to +\mathrm{\infty }$‍, and as $x\to -\mathrm{\infty }$‍, $g(x)\to -\mathrm{\infty }$‍.

  B

  As $x\to +\mathrm{\infty }$‍, $g(x)\to +\mathrm{\infty }$‍, and as $x\to -\mathrm{\infty }$‍, $g(x)\to -\mathrm{\infty }$‍.
• (Choice C)  

  As $x\to +\mathrm{\infty }$‍, $g(x)\to -\mathrm{\infty }$‍, and as $x\to -\mathrm{\infty }$‍, $g(x)\to +\mathrm{\infty }$‍.

  C

  As $x\to +\mathrm{\infty }$‍, $g(x)\to -\mathrm{\infty }$‍, and as $x\to -\mathrm{\infty }$‍, $g(x)\to +\mathrm{\infty }$‍.
• (Choice D)  

  As $x\to +\mathrm{\infty }$‍, $g(x)\to -\mathrm{\infty }$‍, and as $x\to -\mathrm{\infty }$‍, $g(x)\to -\mathrm{\infty }$‍.

  D

  As $x\to +\mathrm{\infty }$‍, $g(x)\to -\mathrm{\infty }$‍, and as $x\to -\mathrm{\infty }$‍, $g(x)\to -\mathrm{\infty }$‍.

Check

I need help!

To find the end behavior of a function, we can examine the leading term when the function is written in standard form.

Let's write the equation in standard form.

$\begin{array}{rl}g(x)& =(x+1)(x-2)(x+5)\\ \\ g(x)& =(x+1)(x^2+3x-10)\\ \\ g(x)& =x^3+3x^2-10x+x^2+3x-10\\ \\ g(x)& =x^3+4x^2-7x-10\end{array}$‍

The leading term of the polynomial is $x^3$‍, and so the end behavior of function $g$‍ will be the same as the end behavior of $x^3$‍.

Since the degree is odd and the leading coefficient is positive, the end behavior will be: as $x\to +\mathrm{\infty }$‍, $g(x)\to +\mathrm{\infty }$‍ and as $x\to -\mathrm{\infty }$‍, $g(x)\to -\mathrm{\infty }$‍.

c) What are the $x$‍-intercepts of the graph of $g(x)=(x+1)(x-2)(x+5)$‍?

Choose 1 answer:

Choose 1 answer:

• (Choice A)  

  $(1,0)$‍, $(-2,0)$‍, and $(5,0)$‍

  A

  $(1,0)$‍, $(-2,0)$‍, and $(5,0)$‍
• (Choice B)  

  $(-1,0)$‍, $(2,0)$‍, and $(-5,0)$‍

  B

  $(-1,0)$‍, $(2,0)$‍, and $(-5,0)$‍

Check

I need help!

To find the $x$‍-intercepts, we can solve $g(x)=0$‍.

$\begin{array}{rl}g(x)& =(x+1)(x-2)(x+5)\\ \\ 0& =(x+1)(x-2)(x+5)\end{array}$‍
$\begin{array}{rlr}\swarrow & \downarrow & \searrow \\ \\ x+1=0& x-2=0& \text{or}& x+5=0& \text{Zero product property}\\ \\ x=-1& x=2& \text{or}& x=-5\end{array}$‍

The $x$‍-intercepts of the graph of $y=g(x)$‍ are $(-1,0)$‍, $(2,0)$‍, and $(-5,0)$‍.

d) Which of the following graphs could be the graph of $g(x)=(x+1)(x-2)(x+5)$‍?

Choose 1 answer:

Choose 1 answer:

• (Choice A)  

  A
• (Choice B)  

  B
• (Choice C)  

  C
• (Choice D)  

  D

Check

I need help!

We can pull together the above information to sketch a graph of $y=g(x)$‍.

Sketching the $y$‍-intercept, end behavior of the polynomial, and what we know about the zeros would give us:

The parts of a polynomial are graphed on an x y coordinate plane. The first end curves up from left to right from the third quadrant. The other end curves up from left to right from the first quadrant. A point is on the x-axis at (negative five, zero), (negative one, zero) and at (two, zero). A part of the polynomial is graphed curving up crossing the x-axis at (negative five, zero). Another part curves down crossing the point at (negative one, zero). Another part curves up through (two, zero). A point is at (zero, negative ten).

We could then fill in the gaps with a smooth, continuous curve to graph the polynomial.

The parts of a polynomial are graphed on an x y coordinate plane. The first end curves up from left to right from the third quadrant. The other end curves up from left to right from the first quadrant. A point is on the x-axis at (negative five, zero), (negative one, zero) and at (two, zero). A part of the polynomial is graphed curving up crossing the x-axis at (negative five, zero). Another part curves down crossing the point at (negative one, zero). Another part curves up through (two, zero) The parts of the polynomial are connected by dashed portions of the graph, passing through the y-intercept at (zero, negative ten)

This corresponds with graph $A$‍.