If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Algebra II (2018 edition)

### Course: Algebra II (2018 edition)>Unit 4

Lesson 13: Graphs of polynomials

# Graphs of polynomials

Analyze polynomials in order to sketch their graph.

#### What you should be familiar with before taking this lesson

The end behavior of a function f describes the behavior of its graph at the "ends" of the x-axis. Algebraically, end behavior is determined by the following two questions:
• As x, right arrow, plus, infinity, what does f, left parenthesis, x, right parenthesis approach?
• As x, right arrow, minus, infinity, what does f, left parenthesis, x, right parenthesis approach?
If this is new to you, we recommend that you check out our end behavior of polynomials article.
The zeros of a function f correspond to the x-intercepts of its graph. If f has a zero of odd multiplicity, its graph will cross the x-axis at that x value. If f has a zero of even multiplicity, its graph will touch the x-axis at that point.
If this is new to you, we recommend that you check out our zeros of polynomials article.

#### What you will learn in this lesson

In this lesson, we will use the above features in order to analyze and sketch graphs of polynomials. We will then use the sketch to find the polynomial's positive and negative intervals.

## Analyzing polynomial functions

We will now analyze several features of the graph of the polynomial f, left parenthesis, x, right parenthesis, equals, left parenthesis, 3, x, minus, 2, right parenthesis, left parenthesis, x, plus, 2, right parenthesis, squared.

### Finding the $y$y-intercept

To find the y-intercept of the graph of f, we can find f, left parenthesis, 0, right parenthesis.
\begin{aligned} f(x)&=(3x-2)(x+2)^2 \\\\ f(\tealD0)&= (3(\tealD 0)-2)(\tealD0+2)^2\\ \\ f(0)&= (-2)(4)\\\\ f(0)&=-8 \end{aligned}
The y-intercept of the graph of y, equals, f, left parenthesis, x, right parenthesis is left parenthesis, 0, comma, minus, 8, right parenthesis.

### Finding the $x$x-intercepts

To find the x-intercepts, we can solve the equation f, left parenthesis, x, right parenthesis, equals, 0.
\begin{aligned} f(x)&=(3x-2)(x+2)^2 \\\\ \tealD 0&= (3x-2)(x+2)^2\\ \\ \end{aligned}
\begin{aligned}&\swarrow&\searrow\\\\ 3x-2&=0&\text{or}\quad x+2&=0&\small{\gray{\text{Zero product property}}}\\\\ x&=\dfrac{2}{3}&\text{or}\qquad x&=-2\end{aligned}
The x-intercepts of the graph of y, equals, f, left parenthesis, x, right parenthesis are left parenthesis, start fraction, 2, divided by, 3, end fraction, comma, 0, right parenthesis and left parenthesis, minus, 2, comma, 0, right parenthesis.
Our work also shows that start fraction, 2, divided by, 3, end fraction is a zero of multiplicity 1 and minus, 2 is a zero of multiplicity 2. This means that the graph will cross the x-axis at left parenthesis, start fraction, 2, divided by, 3, end fraction, comma, 0, right parenthesis and touch the x-axis at left parenthesis, minus, 2, comma, 0, right parenthesis.

### Finding the end behavior

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{aligned}f(x)&=(3x-2)(x+2)^2\\ \\ f(x)&=(3x-2)(x^2+4x+4)\\ \\ f(x)&=3x^3+12x^2+12x-2x^2-8x-8\\ \\ f(x)&=\goldD{3x^3}+10x^2+4x-8 \end{aligned}
The leading term of the polynomial is start color #e07d10, 3, x, cubed, end color #e07d10, and so the end behavior of function f will be the same as the end behavior of 3, x, cubed.
Since the degree is odd and the leading coefficient is positive, the end behavior will be: as x, right arrow, plus, infinity, f, left parenthesis, x, right parenthesis, right arrow, plus, infinity and as x, right arrow, minus, infinity, f, left parenthesis, x, right parenthesis, right arrow, minus, infinity.

### Sketching a graph

We can use what we've found above to sketch a graph of y, equals, f, left parenthesis, x, right parenthesis.
• As x, right arrow, plus, infinity, f, left parenthesis, x, right parenthesis, right arrow, plus, infinity.
• As x, right arrow, minus, infinity, f, left parenthesis, x, right parenthesis, right arrow, minus, infinity.
This means that in the "ends," the graph will look like the graph of y, equals, x, cubed.
The ends of a polynomial are graphed on an x y coordinate plane. The first end curves up from left to right from the third quadrant. It is labeled As x goes to negative infinity, f of x goes to negative infinity. The other end curves up from left to right from the first quadrant. It is labeled As x goes to positive infinity, f of x goes to positive infinity.
• The graph touches the x-axis at left parenthesis, minus, 2, comma, 0, right parenthesis, since minus, 2 is a zero of even multiplicity.
• The graph crosses the x-axis at left parenthesis, start fraction, 2, divided by, 3, end fraction, comma, 0, right parenthesis, since start fraction, 2, divided by, 3, end fraction is a zero of odd multiplicity.
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 two, zero) and at (two over three, zero). A part of the polynomial is graphed curving up to touch (negative two, zero) before curving back down. Another part of the polynomial is graphed curving up and crossing the x-axis at the point (two over three, zero).
Finally, let's finish this process by plotting the y-intercept left parenthesis, 0, comma, minus, 8, right parenthesis and filling in the gaps with a smooth, continuous curve.
While we don't know exactly where the turning points are, we still have a good idea of the overall shape of the function's graph!
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 two, zero) and at (two over three, zero). A part of the polynomial is graphed curving up to touch (negative two, zero) before curving back down. Another part of the polynomial is graphed curving up and crossing the x-axis at the point (two over three, zero). There is a point at (zero, negative eight) labeled the y-intercept. The parts of the polynomial are connected by dashed portions of the graph, passing through the y-intercept.

### Positive and negative intervals

Now that we have a sketch of f's graph, it is easy to determine the intervals for which f is positive, and those for which it is negative.
A polynomial is graphed on an x y coordinate plane. The graph curves up from left to right touching the x-axis at (negative two, zero) before curving down. It curves back up and passes through the x-axis at (two over three, zero). Where x is less than negative two, the section below the x-axis is shaded and labeled negative. Where x is greater than negative two and less than two over three, the section below the x-axis is shaded and labeled negative. Where x is greater than two over three, the section above the x-axis is shaded and labeled positive.
We see that f is positive when x, is greater than, start fraction, 2, divided by, 3, end fraction and negative when x, is less than, minus, 2 or minus, 2, is less than, x, is less than, start fraction, 2, divided by, 3, end fraction.

1) You will now work towards a sketch of g, left parenthesis, x, right parenthesis, equals, left parenthesis, x, plus, 1, right parenthesis, left parenthesis, x, minus, 2, right parenthesis, left parenthesis, x, plus, 5, right parenthesis on your own.
a) What is the y-intercept of the graph of g, left parenthesis, x, right parenthesis, equals, left parenthesis, x, plus, 1, right parenthesis, left parenthesis, x, minus, 2, right parenthesis, left parenthesis, x, plus, 5, right parenthesis?
left parenthesis, 0,
right parenthesis

b) What is the end behavior of the graph of g, left parenthesis, x, right parenthesis, equals, left parenthesis, x, plus, 1, right parenthesis, left parenthesis, x, minus, 2, right parenthesis, left parenthesis, x, plus, 5, right parenthesis?