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Lesson Menu Five-Minute Check (over Lesson 5–3) CCSS Then/Now New Vocabulary Example 1:Graph of a Polynomial Function Key Concept: Location Principle Example 2:Locate Zeros of a Function Example 3:Maximum and Minimum Points Example 4:Real-World Example: Graph a Polynomial Model

Over Lesson 5–3 5-Minute Check 1 A.5; –4 B.5; 4 C.4; 5 D.8; 6 State the degree and leading coefficient of –4x 5 + 2x

Over Lesson 5–3 5-Minute Check 2 A.1; –10 B.2; –12 C.3; –13 D.4; –14 Find p(3) and p(–5) for p(x) = 12 – x 2.

Over Lesson 5–3 5-Minute Check 3 A.37; 35 B.23; –20 C.–15; 20 D.37; –35 Find p(3) and p(–5) for p(x) = x 3 – 10x + 40.

Over Lesson 5–3 5-Minute Check 4 A.x 2 + 3x + 6 B.x 2 + x + 2 C.x – 6 D.x + 4 If p(x) = x 2 – 3x + 4, find p(x + 2).

Over Lesson 5–3 5-Minute Check 5 A.sometimes B.always C.never Determine whether the statement is sometimes, always, or never true. The graph of a polynomial of degree three will intersect the x-axis three times.

Over Lesson 5–3 5-Minute Check 6 A.as x → –∞, f(x) → –∞ and as x → +∞, f(x) → –∞ B.as x → –∞, f(x) → –∞ and as x → +∞, f(x) → +∞ C.as x → –∞, f(x) → +∞ and as x → +∞, f(x) → +∞ D.as x → –∞, f(x) → +∞ and as x → +∞, f(x) → –∞ Describe the end behavior of the graph of function f(x) = –x

CCSS Content Standards F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. F.IF.7.c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Mathematical Practices 3 Construct viable arguments and critique the reasoning of others.

Then/Now You used maxima and minima and graphs of polynomials. Graph polynomial functions and locate their zeros. Find the relative maxima and minima of polynomial functions.

Vocabulary Location Principle relative maximum relative minimum extrema turning points

Example 1 Graph of a Polynomial Function Graph f(x) = –x 3 – 4x by making a table of values. Answer:

Example 1 Graph of a Polynomial Function This is an odd degree polynomial with a negative leading coefficient, so f(x)  +  as x  –  and f(x)  –  as x  + . Notice that the graph intersects the x-axis at 3 points indicating that there are 3 real zeros. Graph f(x) = –x 3 – 4x by making a table of values. Answer:

A.B. C.D. Example 1 Which graph is the graph of f(x) = x 3 + 2x 2 + 1?

Concept

Example 2 Locate Zeros of a Function Determine consecutive values of x between which each real zero of the function f(x) = x 4 – x 3 – 4x is located. Then draw the graph. Make a table of values. Since f(x) is a 4th degree polynomial function, it will have 0, 2, or 4 real zeros.

Example 2 Locate Zeros of a Function Look at the values of f(x) to locate the zeros. Then use the points to sketch the graph of the function. There are zeros between x = –2 and –1, x = –1 and 0, x = 0 and 1, and x = 2 and 3. Answer:

Example 2 A.x = –1 and x = 0 B.x = –1 and x = 0, x = 0 and x = 1, x = 3 and x = 4 C.x = –1 and x = 0, x = 3 and x = 4 D.x = –1 and x = 0, x = 0 and x = 1, x = 2 and x = 3 Determine consecutive values of x between which each real zero of the function f(x) = x 3 – 4x is located.

Example 3 Maximum and Minimum Points Graph f(x) = x 3 – 3x Estimate the x-coordinates at which the relative maxima and relative minima occur. Make a table of values and graph the function.

Example 3 Maximum and Minimum Points Answer: The value of f(x) at x = 0 is greater than the surrounding points, so there must be a relative maximum near x = 0. The value of f(x) at x = 2 is less than the surrounding points, so there must be a relative minimum near x = 2.

Example 3 Maximum and Minimum Points Check:You can use a graphing calculator to find the relative maximum and relative minimum of a function and confirm your estimate. Enter y = x 3 – 3x in the Y= list and graph the function. Use the CALC menu to find each maximum and minimum. When selecting the left bound, move the cursor to the left of the maximum or minimum. When selecting the right bound, move the cursor to the right of the maximum or minimum.

Example 3 Maximum and Minimum Points The estimates for a relative maximum near x = 0 and a relative minimum near x = 2 are accurate. Press twice. ENTER

Example 3 A.relative minimum: x = 0 relative maximum: x = –2 B.relative minimum: x = –2 relative maximum: x = 0 C.relative minimum: x = –3 relative maximum: x = 1 D.relative minimum: x = 0 relative maximum: x = 2 Consider the graph of f(x) = x 3 + 3x Estimate the x-coordinates at which the relative maximum and relative minimum occur.

Example 4 Graph a Polynomial Model A. HEALTH The weight w, in pounds, of a patient during a 7-week illness is modeled by the function w(n) = 0.1n 3 – 0.6n , where n is the number of weeks since the patient became ill. Graph the equation. Make a table of values for weeks 0 through 7. Plot the points and connect with a smooth curve.

Example 4 Graph a Polynomial Model Answer:

Example 4 Graph a Polynomial Model B. Describe the turning points of the graph and its end behavior. Answer: There is a relative minimum at week 4. w(n) → ∞ as n → ∞.

Example 4 Graph a Polynomial Model C. What trends in the patient’s weight does the graph suggest? Answer: The patient lost weight for each of 4 weeks after becoming ill. After 4 weeks, the patient gained weight and continues to gain weight.

Example 4 Graph a Polynomial Model D. Is it reasonable to assume the trend will continue indefinitely? Answer: The trend may continue for a few weeks, but it is unlikely that the patient’s weight will rise indefinitely.

Example 4 A. WEATHER The rainfall r, in inches per month, in a Midwestern town during a 7-month period is modeled by the function r(m) = 0.01m 3 – 0.18m m , where m is the number of months after March 1. Graph the equation. A.B. C.D.

Example 4 A.There is a relative minimum at Month 2. r(m) decreases as m increases. B.There is a relative maximum at Month 2. r(m) decreases as m increases. C.There is a relative maximum at Month 2. r(m) increases as m increases. D.There is a relative minimum at Month 2. r(m) decreases as m decreases. B. WEATHER Describe the turning points of the graph and its end behavior.

Example 4 A.The rainfall decreased the first two months, then increased. B.The rainfall increased the first two months, then decreased. C.The rainfall continued to increase throughout the entire 8 months. D.The rainfall continued to decrease throughout the entire 8 months. C. WEATHER What trends in the amount of rainfall received by the town does the graph suggest?

Example 4 A.yes B.no D. Is it reasonable to assume the trend will continue indefinitely?

End of the Lesson