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Precalculus for Team-Based Inquiry Learning 2025 Edition — Instructor Version

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Section 4.3 Graphs of Polynomial Functions (PR3)

Subsection 4.3.1 Activities

Remark 4.3.1.

Just like with quadratic functions, we should be able to determine key characteristics that will help guide us in creating a sketch of any polynomial function. We can start by finding both \(x \) and \(y \)-intercepts and then explore other characteristics polynomial functions can have. Recall that the zeros of a function are the \(x\)-intercepts - i.e., the values of \(x\) that cross or touch the \(x\)-axis. Just like with quadratic functions, we can find the zeros of a function by setting the function equal to \(0\) and solving for \(x\text{.}\)
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Activity 4.3.2.

Given the function, \(f(x)=(x-2)(x+1)(x-3)^2 \text{,}\) determine the following characteristics.
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(a)
How many zeros does \(f(x) \) have?
  1. \(\displaystyle 1 \)
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  2. \(\displaystyle 2 \)
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  3. \(\displaystyle 3 \)
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  4. \(\displaystyle 4 \)
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Answer.
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(b)
What are the zeros of \(f(x) \text{?}\)
  1. \(\displaystyle 1, 2, 3 \)
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  2. \(\displaystyle -1, 2, 3 \)
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  3. \(\displaystyle 1, -2, -3 \)
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  4. \(\displaystyle -1, 2, -3 \)
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Answer.
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(c)
What is the \(y\)-intercept of \(f(x) \text{?}\)
  1. \(\displaystyle -1 \)
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  2. \(\displaystyle -6 \)
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  3. \(\displaystyle 6 \)
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  4. \(\displaystyle -18 \)
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Answer.
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(d)
Now that we have found both the \(x\) and \(y\)-intercepts of \(f(x)\text{,}\) do we have enough information to draw a possible sketch of \(f(x)\text{?}\) What other characteristics would be useful to help us draw an accurate sketch of \(f(x) \text{?}\)
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Answer.
Answers may include, \(x\)-intercepts, end behavior, max and min
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Definition 4.3.3.

The end behavior of a polynomial function describes the behavior of the graph at the "ends" of the function. In other words, as we move to the right of the graph (as the \(x\) values increase), what happens to the \(y \) values? Similarly, as we move to the left of the graph (as the \(x\) values decrease), what happens to the \(y \) values?
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Activity 4.3.4.

(a)
How would you describe the behavior of Graph A as you approach the ends?
  1. Graph A rises on the left and on the right.
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  2. Graph A rises on the left, but falls on the right.
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  3. Graph A rises on the right, but falls on the left.
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  4. Graph A falls on the left and on the right.
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Answer.
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(b)
How would you describe the behavior of Graph B as you approach the ends?
  1. Graph B rises on the left and on the right.
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  2. Graph B rises on the left, but falls on the right.
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  3. Graph B rises on the right, but falls on the left.
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  4. Graph B falls on the left and on the right.
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Answer.
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(c)
How would you describe the behavior of Graph C as you approach the ends?
  1. Graph C rises on the left and on the right.
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  2. Graph C rises on the left, but falls on the right.
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  3. Graph C rises on the right, but falls on the left.
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  4. Graph C falls on the left and on the right.
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Answer.
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(d)
How would you describe the behavior of Graph D as you approach the ends?
  1. Graph D rises on the left and on the right.
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  2. Graph D rises on the left, but falls on the right.
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  3. Graph D rises on the right, but falls on the left.
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  4. Graph D falls on the left and on the right.
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Answer.
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Definition 4.3.5.

Typically, when given an equation of a polynomial function, we look at the degree and leading coefficient to help us determine the behavior of the ends. The degree is the highest exponential power in the polynomial. The leading coefficient is the number written in front of the variable with the highest exponential power.
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Activity 4.3.6.

Let’s refer back to the graphs in Activity 4.3.4 and look at the equations of those polynomial functions. Let’s apply Definition 4.3.5 to see if we can determine how the degree and leading coefficients of those graphs affect their end behavior.
  • Graph A: \(f(x)=-11x^3+32+42x^2+x^4-64x\)
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  • Graph B: \(g(x)=2x^2+3-4x^3\)
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  • Graph C: \(h(x)=x^7+2x^3-5x^2+2\)
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  • Graph D: \(j(x)=3x^2-2x^4-5\)
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(a)
What is the degree and leading coefficient of Graph A?
  1. Degree: \(-64\text{;}\) Leading Coefficient: \(4\)
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  2. Degree: \(4\text{;}\) Leading Coefficient: \(0\)
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  3. Degree: \(1\text{;}\) Leading Coefficient: \(-64\)
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  4. Degree: \(4\text{;}\) Leading Coefficient: \(1\)
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Answer.
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(b)
What is the degree and leading coefficient of Graph B?
  1. Degree: \(3\text{;}\) Leading Coefficient: \(-4\)
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  2. Degree: \(-4\text{;}\) Leading Coefficient: \(3\)
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  3. Degree: \(2\text{;}\) Leading Coefficient: \(3\)
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  4. Degree: \(3\text{;}\) Leading Coefficient: \(4\)
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Answer.
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(c)
What is the degree and leading coefficient of Graph C?
  1. Degree: \(-5\text{;}\) Leading Coefficient: \(2\)
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  2. Degree: \(0\text{;}\) Leading Coefficient: \(7\)
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  3. Degree: \(-5\text{;}\) Leading Coefficient: \(3\)
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  4. Degree: \(7\text{;}\) Leading Coefficient: \(1\)
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Answer.
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(d)
What is the degree and leading coefficient of Graph D?
  1. Degree: \(-2\text{;}\) Leading Coefficient: \(4\)
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  2. Degree: \(3\text{;}\) Leading Coefficient: \(2\)
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  3. Degree: \(4\text{;}\) Leading Coefficient: \(-2\)
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  4. Degree: \(-5\text{;}\) Leading Coefficient: \(4\)
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Answer.
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(e)
Notice that Graph A and Graph D have their ends going in the same direction. What conjectures can you make about the relationship between their degrees and leading coefficients with the behavior of their graphs?
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Answer.
Even degree go the same way
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(f)
Notice that Graph B and Graph C have their ends going in opposite directions. What conjectures can you make about the relationship between their degrees and leading coefficients with the behavior of their graphs?
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Answer.
Odd degree go the opposite direction
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Remark 4.3.7.

From Activity 4.3.6, we saw that the degree and leading coefficient of a polynomial function can give us more clues about the behavior of the function. In summary, we know:
  • If the degree is even, the ends of the polynomial function will be going in the same direction. If the leading coefficient is positive, both ends will be pointing up. If the leading coefficient is negative, both ends will be pointing down.
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  • If the degree is odd, the ends of the polynomial function will be going in opposite directions. If the leading coefficient is positive, the left end will fall and the right end will rise. If the leading coefficient is negative, the left end will rise and the right end will fall.
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Definition 4.3.8.

When describing end behavior, mathematicians typically use arrow notation. Just as the name suggests, arrows are used to indicate the behavior of certain values on a graph.
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For end behavior, students are often asked to determine the behavior of \(y\)-values as \(x\)-values either increase or decrease. The statement "As \(x\to \infty\text{,}\) \(f(x)\to -\infty\)" can be translated to "As \(x\) approaches infinity (or as \(x\) increases), \(f(x)\) (or the \(y\)-values) go to negative infinity (i.e., it decreases)."
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Activity 4.3.9.

Use the graph of \(f(x)\) to answer the questions below.
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A polynomial function with one even and one odd multiplicity
Figure 4.3.10.
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(a)
How would you describe the end behavior of \(f(x) \text{?}\)
  1. \(f(x) \) rises on the left and on the right.
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  2. \(f(x) \) rises on the left, but falls on the right.
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  3. \(f(x) \) rises on the right, but falls on the left.
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  4. \(f(x) \) falls on the left and on the right.
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Answer.
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(b)
How would you describe the end behavior of \(f(x) \) using arrow notation?
  1. As \(x\to -\infty\text{,}\) \(f(x)\to -\infty\)
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    As \(x\to \infty\text{,}\) \(f(x)\to -\infty\)
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  2. As \(x\to -\infty\text{,}\) \(f(x)\to -\infty\)
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    As \(x\to \infty\text{,}\) \(f(x)\to \infty\)
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  3. As \(x\to -\infty\text{,}\) \(f(x)\to \infty\)
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    As \(x\to \infty\text{,}\) \(f(x)\to -\infty\)
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  4. As \(x\to -\infty\text{,}\) \(f(x)\to \infty\)
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    As \(x\to \infty\text{,}\) \(f(x)\to \infty\)
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Answer.
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Activity 4.3.11.

Let’s now look at the graph of \(f(x)=(x-2)(x+1)(x-3)^2\) to answer the questions below.
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A polynomial function with different multiplicities
Figure 4.3.12.
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(a)
What are the zeros of \(f(x) \text{?}\)
  1. \(\displaystyle 1, 2, 3 \)
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  2. \(\displaystyle -1, 2, 3 \)
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  3. \(\displaystyle 1, -2, -3 \)
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  4. \(\displaystyle -1, 2, -3 \)
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Answer.
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(b)
Describe the behavior at each zero. What do you notice?
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Answer.
At \(-1\) and \(2\) the graph crosses the \(x\)-axis. At \(3\) the graph touches the axis and turns around.
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Definition 4.3.13.

The multiplicity of a polynomial function is the number of times a given factor appears in the factored form of the equation of a polynomial.
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The zero, \(x=3\) in Activity 4.3.11 has multiplicity \(2\) because the factor \((x−3)\) occurs twice (the exponent is a \(2\)).
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Activity 4.3.14.

Use the function, \(f(x)=(x-3)^2(x-1)\) to answer the following questions:
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(a)
What are the zeros of \(f(x) \text{?}\)
  1. \(\displaystyle -1, -3 \)
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  2. \(\displaystyle -1, 3 \)
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  3. \(\displaystyle 1, -3 \)
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  4. \(\displaystyle 1, 3 \)
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Answer.
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(b)
Using Definition 4.3.13, determine the multiplicities of the zeros you found in part (a).
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Answer.
\(3\) has multiplicity 2. \(1\) has multiplicity 1.
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(c)
Now graph \(f(x)\) and look at the zeros on the graph. What do you notice about the behavior of the graph at each zero?
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Answer.
\(3\) touches the axis and turns around. \(1\) crosses the axis.
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Remark 4.3.15.

Refer back to Activity 4.3.11 and Activity 4.3.14. Notice that when the graph crosses the \(x \)-axis at the zero, the mulitplicity of that zero is odd. When the graph touches the \(x \)-axis at the zero, the multiplicity is even. In other words, factors of \(f(x)\) with odd exponents will cross the \(x \)-axis and factors of \(f(x)\) with even exponents will touch the \(x\)-axis.
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Definition 4.3.16.

When graphing polynomial functions, you may notice that these functions have some "hills" and "valleys." These characteristics of the graph are known as the local maxima and local minima of the graph - similar to what we’ve already seen with quadratic functions. Unlike quadratic functions, however, a polynomial graph can have many local maxima/minima (quadratic functions only have one).
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Activity 4.3.17.

Now that we know all the different characteristics of polynomials, we should also be able to identify them from a graph. Use the graph below to find the given characteristics.
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Polynomial Function
Figure 4.3.18.
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(a)
What are the \(x\)-intercept(s) of the polynomial function? Select all that apply.
  1. \(\displaystyle (1, 0) \)
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  2. \(\displaystyle (-1, 0) \)
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  3. \(\displaystyle (2, 0)\)
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  4. \(\displaystyle (0, -2) \)
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Answer.
A and C
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(b)
What are the \(y\)-intercept(s) of the polynomial function?
  1. \(\displaystyle (1, 0) \)
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  2. \(\displaystyle (-1, 0) \)
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  3. \(\displaystyle (2, 0)\)
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  4. \(\displaystyle (0, -2) \)
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Answer.
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(c)
How many zeros does this polynomial function have?
  1. \(\displaystyle 0 \)
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  2. \(\displaystyle 1 \)
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  3. \(\displaystyle 2 \)
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  4. \(\displaystyle 3 \)
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Answer.
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(d)
At what point is the local minimum located?
  1. \(\displaystyle (2, -4) \)
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  2. \(\displaystyle (-1, 0) \)
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  3. \(\displaystyle (-2, 0)\)
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  4. \(\displaystyle (1, -4) \)
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  5. \(\displaystyle (2, 0) \)
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Answer.
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(e)
At what point is the local maximum located?
  1. \(\displaystyle (2, -4) \)
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  2. \(\displaystyle (-1, 0) \)
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  3. \(\displaystyle (-2, 0)\)
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  4. \(\displaystyle (1, -4) \)
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  5. \(\displaystyle (2, 0) \)
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Answer.
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(f)
How do you describe the behavior of the polynomial function as \(x\to \infty\text{?}\)
  1. the \(y\)-values go to negative infinity
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  2. \(\displaystyle f(x) \to \infty\)
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  3. the \(y\)-values go to positive infinity
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  4. \(\displaystyle f(x) \to -\infty\)
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Answer.
B and C
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(g)
How do you describe the behavior of the polynomial function as \(x\to -\infty\text{?}\)
  1. the \(y\)-values go to negative infinity
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  2. \(\displaystyle f(x) \to \infty\)
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  3. the \(y\)-values go to positive infinity
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  4. \(\displaystyle f(x) \to -\infty\)
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Answer.
A and D
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Activity 4.3.19.

Sketch the function, \(f(x)=(x-2)(x+1)(x-3)^2 \text{,}\) by first finding the given characteristics.
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(a)
Find the zeros of \(f(x)\text{.}\)
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Answer.
\(2\text{,}\) \(1\text{,}\) and \(3\)
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(b)
Find the multiplicities and describe the behavior at each zero.
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Answer.
\(2\) and \(-1\) multiplicity 1 and crosses axis. \(3\) multiplicity 2 and touches the axis and turns around
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(c)
Find the \(y\)-intercept of \(f(x)\text{.}\)
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Answer.
\((0,-18)\)
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(d)
Describe the end behavior of \(f(x)\text{.}\)
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Answer.
As \(x \to \infty\text{,}\) \(y \to \infty\text{.}\) As \(x \to -\infty\text{,}\) \(y \to -\infty\text{.}\)
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(e)
Estimate where any local maximums and minimums may occur.
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Answer.
Minimum occurs between \(x=-1\) and \(x=2\text{.}\) Then a maximum at \(x=3\)
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Activity 4.3.20.

Sketch the graph of a function \(f(x) \) that meets all of the following criteria. Be sure to scale your axes and label any important features of your graph.
  • The \(x\)-intercepts of \(f(x)\) are \(0, 2,\) and \(5 \text{.}\)
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  • \(0\) and \(2\) have even multiplicity and \(5\) has odd multiplicity.
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  • The end behavior of \(f(x)\) is given as: As \(x\to \infty\text{,}\) \(f(x)\to\infty\) and As \(x\to -\infty\text{,}\) \(f(x)\to-\infty\)
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Answer.
One such graph is
Polynomial Function
Figure 4.3.21.
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Activity 4.3.22.

Use the given function, \(f(x)=(x+1)^2(x-5) \text{,}\) to answer the following questions.
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(a)
What are the zeros of \(f(x) \text{?}\)
  1. \(\displaystyle -1, -5 \)
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  2. \(\displaystyle -1, 5 \)
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  3. \(\displaystyle 1, -5 \)
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  4. \(\displaystyle 1, 5 \)
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Answer.
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(b)
What are the multiplicities at each zero?
  1. At \(x=-1\text{,}\) the multiplicity is even.
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    At \(x=5\text{,}\) the multiplicity is even.
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  2. At \(x=-1\text{,}\) the multiplicity is even.
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    At \(x=5\text{,}\) the multiplicity is odd.
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  3. At \(x=-1\text{,}\) the multiplicity is odd.
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    At \(x=5\text{,}\) the multiplicity is even.
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  4. At \(x=-1\text{,}\) the multiplicity is odd.
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    At \(x=5\text{,}\) the multiplicity is odd.
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Answer.
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(c)
What is the end behavior of \(f(x) \text{?}\)
  1. \(f(x) \) rises on the left and on the right.
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  2. \(f(x) \) rises on the left, but falls on the right.
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  3. \(f(x) \) rises on the right, but falls on the left.
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  4. \(f(x) \) falls on the left and on the right.
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Answer.
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(d)
Using what you know about the zeros, multiplicities, and end behavior, where on the graph can we estimate the local maxima and minima to be?
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Answer.
Between \(x=-1\) and \(x=5\) since the graph must touch the \(x\)-axis at both places.
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(e)
Now look at the graph of \(f(x)\text{.}\) At which zero does a local maximum or local minimum occur? Explain how you know.
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Answer.
At \(x=-1\) since the graph touches the axis and turns around.
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Remark 4.3.23.

We can estimate where these local maxima and minima occur by looking at other characteristics, such as multiplicities and end behavior.
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From Activity 4.3.22, we saw that when the function touches the \(x\)-axis at a zero, then that zero could be either a local maximum or a local minimum of the graph. When the function crosses the \(x\)-axis, however, the local maximum or local minimum occurs between the zeros.
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Subsection 4.3.2 Exercises

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