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## Course: Algebra 2 > Unit 5

- Intro to end behavior of polynomials

## End behavior of polynomials

What's "end behavior", check your understanding.

- (Choice A) As x → + ∞ , g ( x ) → + ∞ , and as x → − ∞ , g ( x ) → + ∞ . A As x → + ∞ , g ( x ) → + ∞ , and as x → − ∞ , g ( x ) → + ∞ .
- (Choice B) As x → + ∞ , g ( x ) → + ∞ , and as x → − ∞ , g ( x ) → − ∞ . B As x → + ∞ , g ( x ) → + ∞ , and as x → − ∞ , g ( x ) → − ∞ .
- (Choice C) As x → + ∞ , g ( x ) → − ∞ , and as x → − ∞ , g ( x ) → + ∞ . C As x → + ∞ , g ( x ) → − ∞ , and as x → − ∞ , g ( x ) → + ∞ .
- (Choice D) As x → + ∞ , g ( x ) → − ∞ , and as x → − ∞ , g ( x ) → − ∞ . D As x → + ∞ , g ( x ) → − ∞ , and as x → − ∞ , g ( x ) → − ∞ .

## Determining end behavior algebraically

- As x → + ∞ , what does f ( x ) approach?
- As x → − ∞ , what does f ( x ) approach?

## Investigation: End behavior of monomials

- (Choice A) f ( x ) is a large positive number. A f ( x ) is a large positive number.
- (Choice B) f ( x ) is a large negative number. B f ( x ) is a large negative number.
- (Choice A) g ( x ) is a large positive number. A g ( x ) is a large positive number.
- (Choice B) g ( x ) is a large negative number. B g ( x ) is a large negative number.
- (Choice A) h ( x ) is a large positive number. A h ( x ) is a large positive number.
- (Choice B) h ( x ) is a large negative number. B h ( x ) is a large negative number.
- (Choice A) j ( x ) is a large positive number. A j ( x ) is a large positive number.
- (Choice B) j ( x ) is a large negative number. B j ( x ) is a large negative number.

## Concluding the investigation

- (Choice A) As x → + ∞ , f ( x ) → + ∞ , and as x → − ∞ , f ( x ) → + ∞ . A As x → + ∞ , f ( x ) → + ∞ , and as x → − ∞ , f ( x ) → + ∞ .
- (Choice B) As x → + ∞ , f ( x ) → + ∞ , and as x → − ∞ , f ( x ) → − ∞ . B As x → + ∞ , f ( x ) → + ∞ , and as x → − ∞ , f ( x ) → − ∞ .
- (Choice C) As x → + ∞ , f ( x ) → − ∞ , and as x → − ∞ , f ( x ) → + ∞ . C As x → + ∞ , f ( x ) → − ∞ , and as x → − ∞ , f ( x ) → + ∞ .
- (Choice D) As x → + ∞ , f ( x ) → − ∞ , and as x → − ∞ , f ( x ) → − ∞ . D As x → + ∞ , f ( x ) → − ∞ , and as x → − ∞ , f ( x ) → − ∞ .

## Why does the leading term determine the end behavior?

Challenge problems.

- (Choice A) A
- (Choice B) B
- (Choice C) C
- (Choice D) D

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- FlippedMath.com
- List of Lessons
- 1.1 Multiply and Factor Polynomials
- 1.2 Advanced Factoring
- 1.3 Dividing Polynomials
- 1.4 Zeroes of Polynomials
- Unit 1 Review
- 2.1 Remainder Theorem
- 2.2 End Behavior
- 2.3 Sketching Polynomials
- 2.4 Graphing Polynomial Functions (Calc)
- Unit 2 Review
- 3.1 Simplify Rational Expressions
- 3.2 Multiply and Divide Rational Expressions
- 3.3 Add and Subtract Rational Expressions
- 3.4 Solve Rational Equations
- Unit 3 Review
- 4.1 Add/Sub Radicals
- 4.2 Mult/Div Radicals
- 4.3 Radical Equations
- Unit 4 Review
- 5.1 Transformations of Functions, Pt 1
- 5.2 Transformations of Functions, Pt 2
- 5.3 Systems of Equations
- Unit 5 Review
- 6.1 Imaginary Numbers
- 6.2 Complex Numbers
- 6.3 Quadratic Formula
- Unit 6 Review
- 7.1 Rational Exponents
- 7.2 Exponential Growth and Decay
- 7.3 Percent Change
- 7.4 Exponential Modeling
- Unit 7 Review
- 8.1 Intro to Logs
- 8.2 Logarithmic Graphs
- 8.3 Properties of Logarithms
- 8.4 Solving Exponential Equations
- Unit 8 Review
- 9.1 Intro to Trig
- 9.2 Reference Triangles
- 9.3 Unit Circle and Radians
- Unit 9 Review
- 10.1 Amplitude and Midline
- 10.2 Period
- 10.3 Modeling with Sine and Cosine
- Unit 10 Review
- 11.1 Sequences
- 11.2 Series
- Unit 11 Review
- 12.1 Introduction to Probability
- 12.2 Compound Events and Independence
- 12.3 Normal Distributions
- Unit 12 Review
- 13.1 Sampling Methods
- 13.2 Experimental Design
- 13.3 Simulations
- Unit 13 Review
- Teacher Resources

## 2.2 End Behavior

## Practice Solutions

## Corrective Assignment

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## End Behavior of a Function

The end behavior of a polynomial function is the behavior of the graph of f ( x ) as x approaches positive infinity or negative infinity.

The degree and the leading coefficient of a polynomial function determine the end behavior of the graph.

The leading coefficient is significant compared to the other coefficients in the function for the very large or very small numbers. So, the sign of the leading coefficient is sufficient to predict the end behavior of the function.

To predict the end-behavior of a polynomial function, first check whether the function is odd-degree or even-degree function and whether the leading coefficient is positive or negative.

Find the end behavior of the function x 4 − 4 x 3 + 3 x + 25 .

The degree of the function is even and the leading coefficient is positive. So, the end behavior is:

f ( x ) → + ∞ , as x → − ∞ f ( x ) → + ∞ , as x → + ∞

The graph looks as follows:

## Download our free learning tools apps and test prep books

## Module 9: Power and Polynomial Functions

Graphs of polynomial functions, learning outcomes.

- Identify zeros of polynomial functions with even and odd multiplicity.
- Draw the graph of a polynomial function using end behavior, turning points, intercepts, and the Intermediate Value Theorem.
- Write the equation of a polynomial function given its graph.

The revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in the table below .

The revenue can be modeled by the polynomial function

[latex]R\left(t\right)=-0.037{t}^{4}+1.414{t}^{3}-19.777{t}^{2}+118.696t - 205.332[/latex]

where R represents the revenue in millions of dollars and t represents the year, with t = 6 corresponding to 2006. Over which intervals is the revenue for the company increasing? Over which intervals is the revenue for the company decreasing? These questions, along with many others, can be answered by examining the graph of the polynomial function. We have already explored the local behavior of quadratics, a special case of polynomials. In this section we will explore the local behavior of polynomials in general.

## Multiplicity and Turning Points

Graphs behave differently at various x -intercepts. Sometimes the graph will cross over the x-axis at an intercept. Other times the graph will touch the x-axis and bounce off.

Suppose, for example, we graph the function [latex]f\left(x\right)=\left(x+3\right){\left(x - 2\right)}^{2}{\left(x+1\right)}^{3}[/latex].

Notice in the figure below that the behavior of the function at each of the x -intercepts is different.

The behavior of a graph at an x-intercept can be determined by examining the multiplicity of the zero.

The x -intercept [latex]x=-3[/latex] is the solution to the equation [latex]\left(x+3\right)=0[/latex]. The graph passes directly through the x -intercept at [latex]x=-3[/latex]. The factor is linear (has a degree of 1), so the behavior near the intercept is like that of a line; it passes directly through the intercept. We call this a single zero because the zero corresponds to a single factor of the function.

The x -intercept [latex]x=2[/latex] is the repeated solution to the equation [latex]{\left(x - 2\right)}^{2}=0[/latex]. The graph touches the axis at the intercept and changes direction. The factor is quadratic (degree 2), so the behavior near the intercept is like that of a quadratic—it bounces off of the horizontal axis at the intercept.

[latex]{\left(x - 2\right)}^{2}=\left(x - 2\right)\left(x - 2\right)[/latex]

The factor is repeated, that is, the factor [latex]\left(x - 2\right)[/latex] appears twice. The number of times a given factor appears in the factored form of the equation of a polynomial is called the multiplicity . The zero associated with this factor, [latex]x=2[/latex], has multiplicity 2 because the factor [latex]\left(x - 2\right)[/latex] occurs twice.

The x- intercept [latex]x=-1[/latex] is the repeated solution of factor [latex]{\left(x+1\right)}^{3}=0[/latex]. The graph passes through the axis at the intercept but flattens out a bit first. This factor is cubic (degree 3), so the behavior near the intercept is like that of a cubic with the same S-shape near the intercept as the function [latex]f\left(x\right)={x}^{3}[/latex]. We call this a triple zero, or a zero with multiplicity 3.

For zeros with even multiplicities, the graphs touch or are tangent to the x -axis at these x-values. For zeros with odd multiplicities, the graphs cross or intersect the x -axis at these x-values. See the graphs below for examples of graphs of polynomial functions with multiplicity 1, 2, and 3.

For higher even powers, such as 4, 6, and 8, the graph will still touch and bounce off of the x-axis, but for each increasing even power the graph will appear flatter as it approaches and leaves the x -axis.

For higher odd powers, such as 5, 7, and 9, the graph will still cross through the x-axis, but for each increasing odd power, the graph will appear flatter as it approaches and leaves the x -axis.

## A General Note: Graphical Behavior of Polynomials at x -Intercepts

If a polynomial contains a factor of the form [latex]{\left(x-h\right)}^{p}[/latex], the behavior near the x -intercept h is determined by the power p . We say that [latex]x=h[/latex] is a zero of multiplicity p .

The graph of a polynomial function will touch the x -axis at zeros with even multiplicities. The graph will cross the x -axis at zeros with odd multiplicities.

The sum of the multiplicities is the degree of the polynomial function.

## How To: Given a graph of a polynomial function of degree [latex]n[/latex], identify the zeros and their multiplicities.

- If the graph crosses the x -axis and appears almost linear at the intercept, it is a single zero.
- If the graph touches the x -axis and bounces off of the axis, it is a zero with even multiplicity.
- If the graph crosses the x -axis at a zero, it is a zero with odd multiplicity.
- The sum of the multiplicities is the degree n .

## Example: Identifying Zeros and Their Multiplicities

Use the graph of the function of degree 6 to identify the zeros of the function and their possible multiplicities.

The polynomial function is of degree n which is 6. The sum of the multiplicities must be 6.

Starting from the left, the first zero occurs at [latex]x=-3[/latex]. The graph touches the x -axis, so the multiplicity of the zero must be even. The zero of –3 has multiplicity 2.

The next zero occurs at [latex]x=-1[/latex]. The graph looks almost linear at this point. This is a single zero of multiplicity 1.

The last zero occurs at [latex]x=4[/latex]. The graph crosses the x -axis, so the multiplicity of the zero must be odd. We know that the multiplicity is 3 and that the sum of the multiplicities must be 6.

Use the graph of the function of degree 7 to identify the zeros of the function and their multiplicities.

The graph has a zero of –5 with multiplicity 3, a zero of –1 with multiplicity 2, and a zero of 3 with multiplicity 2.

## Determining End Behavior

As we have already learned, the behavior of a graph of a polynomial function of the form

[latex]f\left(x\right)={a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}+…+{a}_{1}x+{a}_{0}[/latex]

will either ultimately rise or fall as x increases without bound and will either rise or fall as x decreases without bound. This is because for very large inputs, say 100 or 1,000, the leading term dominates the size of the output. The same is true for very small inputs, say –100 or –1,000.

Recall that we call this behavior the end behavior of a function. As we pointed out when discussing quadratic equations, when the leading term of a polynomial function, [latex]{a}_{n}{x}^{n}[/latex], is an even power function, as x increases or decreases without bound, [latex]f\left(x\right)[/latex] increases without bound. When the leading term is an odd power function, as x decreases without bound, [latex]f\left(x\right)[/latex] also decreases without bound; as x increases without bound, [latex]f\left(x\right)[/latex] also increases without bound. If the leading term is negative, it will change the direction of the end behavior. The table below summarizes all four cases.

## Graphing Polynomial Functions

We can use what we have learned about multiplicities, end behavior, and turning points to sketch graphs of polynomial functions. Let us put this all together and look at the steps required to graph polynomial functions.

## How To: Given a polynomial function, sketch the graph

- Find the intercepts.
- Check for symmetry. If the function is an even function, its graph is symmetric with respect to the y -axis, that is, f (– x ) = f ( x ). If a function is an odd function, its graph is symmetric with respect to the origin, that is, f (– x ) = – f ( x ).
- Use the multiplicities of the zeros to determine the behavior of the polynomial at the x -intercepts.
- Determine the end behavior by examining the leading term.
- Use the end behavior and the behavior at the intercepts to sketch the graph.
- Ensure that the number of turning points does not exceed one less than the degree of the polynomial.
- Optionally, use technology to check the graph.

## Example: Sketching the Graph of a Polynomial Function

Sketch a possible graph for [latex]f\left(x\right)=-2{\left(x+3\right)}^{2}\left(x - 5\right)[/latex].

This graph has two x- intercepts. At x = –3, the factor is squared, indicating a multiplicity of 2. The graph will bounce off the x -intercept at this value. At x = 5, the function has a multiplicity of one, indicating the graph will cross through the axis at this intercept.

The y -intercept is found by evaluating f (0).

[latex]\begin{array}{l}\hfill \\ f\left(0\right)=-2{\left(0+3\right)}^{2}\left(0 - 5\right)\hfill \\ \text{}f\left(0\right)=-2\cdot 9\cdot \left(-5\right)\hfill \\ \text{}f\left(0\right)=90\hfill \end{array}[/latex]

The y -intercept is (0, 90).

Additionally, we can see the leading term, if this polynomial were multiplied out, would be [latex]-2{x}^{3}[/latex], so the end behavior, as seen in the following graph, is that of a vertically reflected cubic with the outputs decreasing as the inputs approach infinity and the outputs increasing as the inputs approach negative infinity.

To sketch the graph, we consider the following:

- As [latex]x\to -\infty [/latex] the function [latex]f\left(x\right)\to \infty [/latex], so we know the graph starts in the second quadrant and is decreasing toward the x -axis.
- Since [latex]f\left(-x\right)=-2{\left(-x+3\right)}^{2}\left(-x - 5\right)[/latex] is not equal to f ( x ), the graph does not have any symmetry.
- At [latex]\left(-3,0\right)[/latex] the graph bounces off of the x -axis, so the function must start increasing after this point.
- At (0, 90), the graph crosses the y -axis.

Somewhere after this point, the graph must turn back down or start decreasing toward the horizontal axis because the graph passes through the next intercept at (5, 0).

As [latex]x\to \infty [/latex] the function [latex]f\left(x\right)\to \mathrm{-\infty }[/latex], so we know the graph continues to decrease, and we can stop drawing the graph in the fourth quadrant.

The complete graph of the polynomial function [latex]f\left(x\right)=-2{\left(x+3\right)}^{2}\left(x - 5\right)[/latex] is as follows:

Sketch a possible graph for [latex]f\left(x\right)=\frac{1}{4}x{\left(x - 1\right)}^{4}{\left(x+3\right)}^{3}[/latex].

## The Intermediate Value Theorem

In some situations, we may know two points on a graph but not the zeros. If those two points are on opposite sides of the x -axis, we can confirm that there is a zero between them. Consider a polynomial function f whose graph is smooth and continuous. The Intermediate Value Theorem states that for two numbers a and b in the domain of f , if a < b and [latex]f\left(a\right)\ne f\left(b\right)[/latex], then the function f takes on every value between [latex]f\left(a\right)[/latex] and [latex]f\left(b\right)[/latex].

We can apply this theorem to a special case that is useful for graphing polynomial functions. If a point on the graph of a continuous function f at [latex]x=a[/latex] lies above the x -axis and another point at [latex]x=b[/latex] lies below the x -axis, there must exist a third point between [latex]x=a[/latex] and [latex]x=b[/latex] where the graph crosses the x -axis. Call this point [latex]\left(c,\text{ }f\left(c\right)\right)[/latex]. This means that we are assured there is a value c where [latex]f\left(c\right)=0[/latex].

In other words, the Intermediate Value Theorem tells us that when a polynomial function changes from a negative value to a positive value, the function must cross the x -axis. The figure below shows that there is a zero between a and b .

The Intermediate Value Theorem can be used to show there exists a zero.

## A General Note: Intermediate Value Theorem

Let f be a polynomial function. The Intermediate Value Theorem states that if [latex]f\left(a\right)[/latex] and [latex]f\left(b\right)[/latex] have opposite signs, then there exists at least one value c between a and b for which [latex]f\left(c\right)=0[/latex].

## Example: Using the Intermediate Value Theorem

Show that the function [latex]f\left(x\right)={x}^{3}-5{x}^{2}+3x+6[/latex] has at least two real zeros between [latex]x=1[/latex] and [latex]x=4[/latex].

To start, evaluate [latex]f\left(x\right)[/latex] at the integer values [latex]x=1,2,3,\text{ and }4[/latex].

We see that one zero occurs at [latex]x=2[/latex]. Also, since [latex]f\left(3\right)[/latex] is negative and [latex]f\left(4\right)[/latex] is positive, by the Intermediate Value Theorem, there must be at least one real zero between 3 and 4.

We have shown that there are at least two real zeros between [latex]x=1[/latex] and [latex]x=4[/latex].

## Analysis of the Solution

We can also graphically see that there are two real zeros between [latex]x=1[/latex] and [latex]x=4[/latex].

Show that the function [latex]f\left(x\right)=7{x}^{5}-9{x}^{4}-{x}^{2}[/latex] has at least one real zero between [latex]x=1[/latex] and [latex]x=2[/latex].

## Writing Formulas for Polynomial Functions

Now that we know how to find zeros of polynomial functions, we can use them to write formulas based on graphs. Because a polynomial function written in factored form will have an x -intercept where each factor is equal to zero, we can form a function that will pass through a set of x -intercepts by introducing a corresponding set of factors.

## A General Note: Factored Form of Polynomials

If a polynomial of lowest degree p has zeros at [latex]x={x}_{1},{x}_{2},\dots ,{x}_{n}[/latex], then the polynomial can be written in the factored form: [latex]f\left(x\right)=a{\left(x-{x}_{1}\right)}^{{p}_{1}}{\left(x-{x}_{2}\right)}^{{p}_{2}}\cdots {\left(x-{x}_{n}\right)}^{{p}_{n}}[/latex] where the powers [latex]{p}_{i}[/latex] on each factor can be determined by the behavior of the graph at the corresponding intercept, and the stretch factor a can be determined given a value of the function other than the x -intercept.

## How To: Given a graph of a polynomial function, write a formula for the function

- Identify the x -intercepts of the graph to find the factors of the polynomial.
- Examine the behavior of the graph at the x -intercepts to determine the multiplicity of each factor.
- Find the polynomial of least degree containing all of the factors found in the previous step.
- Use any other point on the graph (the y -intercept may be easiest) to determine the stretch factor.

## Example: Writing a Formula for a Polynomial Function from Its Graph

Write a formula for the polynomial function.

This graph has three x -intercepts: x = –3, 2, and 5. The y -intercept is located at (0, 2). At x = –3 and x = 5, the graph passes through the axis linearly, suggesting the corresponding factors of the polynomial will be linear. At x = 2, the graph bounces off the x-axis at the intercept suggesting the corresponding factor of the polynomial will be second degree (quadratic). Together, this gives us

[latex]f\left(x\right)=a\left(x+3\right){\left(x - 2\right)}^{2}\left(x - 5\right)[/latex]

To determine the stretch factor, we utilize another point on the graph. We will use the y -intercept (0, –2), to solve for a .

[latex]\begin{array}{l}f\left(0\right)=a\left(0+3\right){\left(0 - 2\right)}^{2}\left(0 - 5\right)\hfill \\ \text{ }-2=a\left(0+3\right){\left(0 - 2\right)}^{2}\left(0 - 5\right)\hfill \\ \text{ }-2=-60a\hfill \\ \text{ }a=\frac{1}{30}\hfill \end{array}[/latex]

The graphed polynomial appears to represent the function [latex]f\left(x\right)=\frac{1}{30}\left(x+3\right){\left(x - 2\right)}^{2}\left(x - 5\right)[/latex].

Given the graph below, write a formula for the function shown.

[latex]f\left(x\right)=-\frac{1}{8}{\left(x - 2\right)}^{3}{\left(x+1\right)}^{2}\left(x - 4\right)[/latex]

## Local and Global Extrema

With quadratics, we were able to algebraically find the maximum or minimum value of the function by finding the vertex. For general polynomials, finding these turning points is not possible without more advanced techniques from calculus. Even then, finding where extrema occur can still be algebraically challenging. For now, we will estimate the locations of turning points using technology to generate a graph.

Each turning point represents a local minimum or maximum. Sometimes, a turning point is the highest or lowest point on the entire graph. In these cases, we say that the turning point is a global maximum or a global minimum . These are also referred to as the absolute maximum and absolute minimum values of the function.

A local maximum or local minimum at x = a (sometimes called the relative maximum or minimum, respectively) is the output at the highest or lowest point on the graph in an open interval around x = a . If a function has a local maximum at a , then [latex]f\left(a\right)\ge f\left(x\right)[/latex] for all x in an open interval around x = a . If a function has a local minimum at a , then [latex]f\left(a\right)\le f\left(x\right)[/latex] for all x in an open interval around x = a .

A global maximum or global minimum is the output at the highest or lowest point of the function. If a function has a global maximum at a , then [latex]f\left(a\right)\ge f\left(x\right)[/latex] for all x . If a function has a global minimum at a , then [latex]f\left(a\right)\le f\left(x\right)[/latex] for all x .

We can see the difference between local and global extrema below.

Do all polynomial functions have a global minimum or maximum?

No. Only polynomial functions of even degree have a global minimum or maximum. For example, [latex]f\left(x\right)=x[/latex] has neither a global maximum nor a global minimum.

## Example: Using Local Extrema to Solve Applications

An open-top box is to be constructed by cutting out squares from each corner of a 14 cm by 20 cm sheet of plastic then folding up the sides. Find the size of squares that should be cut out to maximize the volume enclosed by the box.

We will start this problem by drawing a picture like the one below, labeling the width of the cut-out squares with a variable, w .

Notice that after a square is cut out from each end, it leaves a [latex]\left(14 - 2w\right)[/latex] cm by [latex]\left(20 - 2w\right)[/latex] cm rectangle for the base of the box, and the box will be w cm tall. This gives the volume

[latex]\begin{array}{l}V\left(w\right)=\left(20 - 2w\right)\left(14 - 2w\right)w\hfill \\ \text{}V\left(w\right)=280w - 68{w}^{2}+4{w}^{3}\hfill \end{array}[/latex]

Notice, since the factors are w , [latex]20 - 2w[/latex] and [latex]14 - 2w[/latex], the three zeros are 10, 7, and 0, respectively. Because a height of 0 cm is not reasonable, we consider only the zeros 10 and 7. The shortest side is 14 and we are cutting off two squares, so values w may take on are greater than zero or less than 7. This means we will restrict the domain of this function to [latex]0<w<7[/latex]. Using technology to sketch the graph of [latex]V\left(w\right)[/latex] on this reasonable domain, we get a graph like the one above. We can use this graph to estimate the maximum value for the volume, restricted to values for w that are reasonable for this problem, values from 0 to 7.

On this graph, we turn our focus to only the portion on the reasonable domain, [latex]\left[0,\text{ }7\right][/latex]. We can estimate the maximum value to be around 340 cubic cm, which occurs when the squares are about 2.75 cm on each side. To improve this estimate, we could use advanced features of our technology, if available, or simply change our window to zoom in on our graph to produce the graph below.

From this zoomed-in view, we can refine our estimate for the maximum volume to about 339 cubic cm which occurs when the squares measure approximately 2.7 cm on each side.

## Key Concepts

- Polynomial functions of degree 2 or more are smooth, continuous functions.
- To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero.
- Another way to find the x- intercepts of a polynomial function is to graph the function and identify the points where the graph crosses the x -axis.
- The multiplicity of a zero determines how the graph behaves at the x -intercept.
- The graph of a polynomial will cross the x-axis at a zero with odd multiplicity.
- The graph of a polynomial will touch and bounce off the x-axis at a zero with even multiplicity.
- The end behavior of a polynomial function depends on the leading term.
- The graph of a polynomial function changes direction at its turning points.
- A polynomial function of degree n has at most n – 1 turning points.
- To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most n – 1 turning points.
- Graphing a polynomial function helps to estimate local and global extremas.
- The Intermediate Value Theorem tells us that if [latex]f\left(a\right) \text{and} f\left(b\right)[/latex] have opposite signs, then there exists at least one value c between a and b for which [latex]f\left(c\right)=0[/latex].
- Revision and Adaptation. Provided by : Lumen Learning. License : CC BY: Attribution
- Question ID 121830. Authored by : Lumen Learning. License : CC BY: Attribution . License Terms : IMathAS Community License CC-BY + GPL
- College Algebra. Authored by : Abramson, Jay et al.. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected] . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]
- Question ID 29470, 29478. Authored by : Caren McClure. License : CC BY: Attribution . License Terms : IMathAS Community License CC-BY + GPL

End Behavior (Part 1)

## 8.1: Notice and Wonder: A Different View (5 minutes)

CCSS Standards

Building Towards

- HSF-IF.C.7.c

Routines and Materials

Instructional Routines

- Notice and Wonder

The purpose of this warm-up is to elicit student language about the general shape of graphs of polynomials of even and odd degree with a focus on the end behavior of the graphs. When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). This activity is meant only to introduce the idea of end behavior. Throughout this lesson and the next, students will continue to refine their language use.

Display the graphs for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

## Student Facing

What do you notice? What do you wonder?

\(y=x^3+4x^2-x-4\)

Description: <p>Graph of polynomial function, xy-plane. Horizontal axis, scale -20 to 20, by 10’s. Vertical axis, scale -8,000 to 8,000, by 2,000’s. Polynomial graph comes from Quadrant 3, passes through about (-20 comma -8,000), ascending in a smooth curve flattened near and through the origin, flattened up into Quadrant 1 and ascends in a smooth curve to the upper right through about (20 comma 8,000).</p>

\(y=x^4-10x^2+9\)

Description: <p>Graph of polynomial function, xy-plane. Horizontal axis, scale -20 to 20, by 10’s. Vertical axis, scale -8,000 to 8,000, by 2,000’s. Polynomial graph comes from Quadrant 2, passes through about (-10 comma 8,000), descending in a smooth curve flattened near and through the origin, flattened up into Quadrant 1 and ascends in a smooth curve to the upper right through about (10 comma 8,000).</p>

## Student Response

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## Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, point out contradicting information, etc.

At the end of the discussion, tell students that today, they are going to investigate the end behavior of polynomial functions. The end behavior of a polynomial is what happens to the output as the input gets larger and larger in either the positive or negative directions. For end behavior, we are using “larger” to refer to magnitude, that is, distance from 0. End behavior does not describe the zeros of a function or the intercepts of the graph, but rather, the trend of the input-output pairs of the function as we move farther from \(x=0\) , pairs that are often outside of the graphing windows we use.

## 8.2: Polynomial End Behavior (20 minutes)

- HSA-APR.B.3
- MLR2: Collect and Display

Required Materials

- Graphing technology

The goal of this activity is for students to understand why the leading term of a polynomial determines the end behavior.

Arrange students in groups of 2–3 and assign each group one of the polynomials:

\(y=x^2+1\)

\(y=x^3+1\)

\(y=x^4+1\)

\(y=x^5+1\)

Display a blank version of the table for all to see. As groups finish completing their assigned column of the table, select 1 group for each polynomial to fill in their values on the displayed table before the discussion. Provide access to devices that can run Desmos or other graphing technology.

For your assigned polynomial, complete the column for the different values of \(x\) . Discuss with your group what you notice.

Sketch what you think the end behavior of your polynomial looks like, then check your work using graphing technology.

## Are you ready for more?

Mai is studying the function \(p(x) = \text-\frac{1}{100}x^3 + 25,\!422x^2 + 8x + 26\) . She makes a table of values for \(p\) with \(x = \pm 1, \pm 5, \pm 10, \pm 20\) and thinks that this function has large positive output values in both directions on the \(x\) -axis. Do you agree with Mai? Explain your reasoning.

Teachers with a valid work email address can click here to register or sign in for free access to Extension Student Response.

## Anticipated Misconceptions

Students may use parentheses incorrectly for the exponents when calculating negative values in the table. When this happens, encourage students to consider what happens when a negative number is raised to an even power.

The purpose of this discussion is for students to understand why the end behavior of polynomials with a leading term of odd degree differs from polynomials with a leading term of even degree. Discuss:

- “Describe the similarities and differences between the polynomials listed in the table.” (The equations with an even exponent have very large positive values of \(y\) for large negative values of \(x\) . The equations with an odd exponent have very large negative values of \(y\) for large negative values of \(x\) .)
- “How would the end behavior of the polynomials change if there was another term in the equations, like \(100x\) ?” (For values of \(x\) near 0, there would be a change, but the end behavior would not change, since 100 is a small number compared to 1 million or -999,999,999.)

Tell students that one way to describe the end behavior is by stating what happens to the output values as the input values move away from 0. For example, for \(y= x^3 + 1\) , we can say that as \(x\) gets larger and larger in the negative direction, \(y\) gets larger and larger in the negative direction. As \(x\) gets larger and larger in the positive direction, \(y\) gets larger and larger in the positive direction. For \(y=x^4+1\) , as \(x\) gets larger and larger in the negative direction, \(y\) gets larger and larger in the positive direction. As \(x\) gets larger and larger in the positive direction, \(y\) gets larger and larger in the positive direction. The “larger and larger” language is used to be clear that no matter how large you go (since you can never zoom out far enough to see the entire polynomial), you know what the output values are doing. Students will have more practice using this language in the future, and do not need to be fluent with the exact words at this time.

If time allows, ask students to consider the end behavior of linear functions, that is, polynomials with degree 1.

## 8.3: Two Polynomial Equations (10 minutes)

- MLR1: Stronger and Clearer Each Time

This activity is meant to continue the work begun in the previous activity, investigating the end behavior of polynomials and how it can be determined from the structure of an expression. Starting with a 5th-degree polynomial that has 6 terms, students consider the equation for the polynomial in both factored and standard forms before considering the value of the individual terms at specific values of \(x\) . At each \(x\) -value, a different term has the greatest magnitude. This is meant to reinforce that, as the name implies, end behavior is about the input-output pairs of the function at the ends of the graph away from any intercepts or other unique features.

Tell students to close their books or devices and then display the the equation \(y=(x+1)(x^2-4)(x-5)(2x+3)\) for all to see. Ask students to consider what features of the polynomial they can identify from the equation. After 1–2 minutes, select students to share their observations. (Sample responses: the polynomial has 5 zeros at -1, -2, 2, 5, and -1.5. The graph intercepts the vertical axis at 60. The degree of the polynomial is 5.)

If time allows, ask students to rewrite the equation in standard form before starting the rest of the activity. While students work on the question about which term is greatest, if they wonder where the negative signs went, let them know that the purpose is to consider the magnitude of the term at different inputs, so they were all written as positive terms.

Consider the polynomial \(y=2x^5 - 5x^4 - 30x^3 + 5x^2 + 88x + 60\) .

- Identify the degree of the polynomial.
- Describe the end behavior of the polynomial.

If students have trouble deciding what the end behavior will be, remind them of the previous activity, in which the leading term determined the end behavior. Encourage students to connect this to the question in this activity about which term is greatest, in which the leading term has greater magnitude than all the other terms when \(x\) is not near 0.

The purpose of this discussion is for students to use mathematically correct language about the end behavior of a polynomial and to consider what changes to an equation will, and will not, affect the end behavior of the polynomial. Select 2–3 students to share their descriptions about the end behavior of the polynomial. Discuss:

- “How would the end behavior change if the leading term was \(20x^5\) instead of \(2x^5\) ?” (Since the exponent on the \(x\) did not change, the end behavior wouldn’t change.)
- “How can we change the term \(5x^4\) in order to change the end behavior of the polynomial?” (Changing the exponent to 6 would change the end behavior.)

Consider displaying a graph of the equation for all to see and using graphing technology to change the equation in ways students suggest and verifying the end behavior.

## Lesson Synthesis

Ask students to stand up to play a game. Suggest that students wiggle their arms and do some stretches while explaining the rules of the game.

- A series of polynomial equations will be displayed one at a time.
- After an equation is displayed, there will be a brief quiet think time to identify the end behavior. Give a hand signal when you are ready.
- When you hear “Pose!” (or a different word chosen by the class), use your arms to show the end behavior of the function. For example, for \(y=x^2\) , you put both hands up in the air. For something like \(y=x^3\) , you have your left arm down and your right arm up.

Here are some polynomial equations to use for the game. Have a graph of each ready to show for students to check their pose against.

- \(y=3x^2+2x-2000000\)
- \(y=3x-7+x^2\)
- \(y=x^5+6x^4-x^5-2x+8\)
- \(y=(x-3)(x+4)(2x+1)\)
- \(y=x^{10}\)
- \(y=x^{25}\)
- \(y=x^{50}\)
- \(y=x^{101}\)

## 8.4: Cool-down - Identifying End Behavior (5 minutes)

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## Student Lesson Summary

We know that if the expression for a polynomial function \(f\) written in factored form has the factor \((x-a)\) , then \(a\) is a zero of \(f\) (that is, \(f(a)=0\) ) and the point \((a, 0)\) is on the graph of the function. But what about other values of \(x\) ? In particular, as we consider values of \(x\) that get larger and larger in either the negative or positive direction, what happens to the values of \(f(x)\) ?

The answer to this question depends on the degree of the polynomial, because any negative real number raised to an even power results in a positive number. For example, if we graph \(y=x^2\) , \(y=x^3\) and \(y=x^4\) and zoom out, we see the following:

Description: <p>Graph of polynomial function y = x squared, xy-plane, origin O. Horizontal axis, scale -75 to 75, by 25’s. Vertical axis, scale -2,000 to 4,000, by 1,000’s. Polynomial graph comes from Quadrant 2, passes down through (-50 comma 2,500), descending in a smooth curve through the origin, up into Quadrant 1 ascending in a smooth curve to the upper right through (50 comma 2,500).</p>

Description: <p>Graph of polynomial function y = x cubed, xy-plane, origin O. Horizontal axis, scale -75 to 75, by 25’s. Vertical axis, scale -8,000 to 8,000, by 2,000’s. Polynomial graph comes from Quadrant 3, passes up through (-20 comma 8,000), ascending in a smooth curve through the origin, up into Quadrant 1 ascending in a smooth curve to the upper right through (20 comma 8,000).</p>

Description: <p>Graph of polynomial function, xy-plane, origin O. Horizontal axis, scale -75 to 75, by 25’s. Vertical axis, scale -2,000 to 8,000, by 2,000’s. Polynomial graph comes from Quadrant 3, passes through (-10 comma 10,000), descending in a smooth curve through the origin, up into Quadrant 1 and ascends in a smooth curve to the upper right through (10 comma 10,000).</p>

For both \(y=x^2\) and \(y=x^4\) , large positive values of \(x\) or large negative values of \(x\) each result in large positive values of \(y\) . But for \(y=x^3\) , large positive values of \(x\) result in large positive values of \(y\) , while large negative values of \(x\) result in large negative values of \(y\) .

Consider the polynomial \(P(x)=x^4-30x^3-20x^2+1000\) . The leading term, \(x^4\) , almost seems smaller than the other 3 terms. For certain values of \(x\) , this is even true. But, for values of \(x\) far away from zero, the leading term will always have the greatest value. Can you see why?

The value of the leading term \(x^4\) determines the end behavior of the function, that is, how the outputs of the function change as we look at input values farther and farther from 0. In the case of \(P(x)\) , as \(x\) gets larger and larger in the positive and negative directions, the output of the function gets larger and larger in the positive direction.

## IMAGES

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