Five-Minute Check (over Lesson 2–3) Mathematical Practices Then/Now Key Concept: Key Features for Graphing a Linear Function Example 1: Sketch a Linear Graph Key Concept: Key Features for Graphing a Nonlinear Function Example 2: Sketch a Nonlinear Graph Example 3: Real-World Example: Sketch a Real-World Function Lesson Menu

Describe the end behavior of the function shown in the graph. 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) → –∞ 5-Minute Check 1

Which is not a zero of the function shown in the graph? B. –2 C. 2 D. 4 5-Minute Check 2

Use the table of values for f(x) = x4 – 12x2 + 5 Use the table of values for f(x) = x4 – 12x2 + 5. Estimate the x-coordinates at which any relative maxima and relative minima occur. Which is not a possible relative maximum or relative minimum? A. x = –2.5 B. x = 0 C. x = 1.5 D. x = 2.5 5-Minute Check 3

Estimate the x-value at which the relative minimum of the function occurs. B. 0 C. –0.5 D. –1.5 5-Minute Check 4

For which part(s) of its domain does this function have negative f(x) values? B. (–4, –3), (1, 3) C. (–∞, –3), (1, ∞) D. (–∞, –2), (1, 2) 5-Minute Check 5

Mathematical Practices 2 Reason abstractly and quantitatively. 7 Look for and make use of structure. Content Standards A.CED.2 Create equations in two or more variable to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 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. MP

You analyzed characteristics of functions. Use the key features of functions to sketch graphs of linear functions. Use the key features of functions to sketch graphs of nonlinear functions. Then/Now

Key Concept

Use the given key features to sketch a linear graph. The y-intercept is 2. The function is positive for x > –1. The function is increasing for all values of x. As x → ∞, f (x) → ∞ and as x → –∞, f (x) → –∞. The y-intercept is 2 Plot a point on the y-axis at 2. The function is positive for x > –1 Plot a point on the x- axis at –1. The graph will be above the x-axis for all values greater than –1. The function is increasing for all values of x. The slope of the line is positive. Example 1

Sketch a Linear Graph As x → +∞, f(x) → +∞ As the value of x approaches positive infinity the value of y will continue to increase and the graph will continue to increase and approach positive infinity. As x → –∞, f(x) → –∞ As the value of x approaches negative infinity the value of y will continue to decrease and the graph will continue to decrease and approach negative infinity. Determine the shape of the function. Use the information to graph the function. Example 1

Sketch a Linear Graph Answer: Example 1

Key Concept

Sketch a Nonlinear Graph Use the given key features to sketch a nonlinear graph. Sample answers shown. A. The y-intercept is 1. The function is continuous. The function is positive for –1 < x < 3. The function has a maximum at (1, 2). The function is increasing for x < 1. As x → ∞, f (x) → –∞ and as x → –∞, f (x) → –∞. Example 2A

Sketch a Nonlinear Graph The y-intercept is 1 Plot a point on the y-axis at 1. The function is positive for –1 < x < 3 Plot the points at 3 and –1 on the x- axis. The graph will be above the x-axis for all values of x between –1 and 3. The function has a maximum at (1, 2) Plot a point at (1, 2). Example 2A

Sketch a Nonlinear Graph The function is increasing The slope is for x < 1. positive until x = 1. As x → +∞, f(x) → –∞ As the value of x approaches positive infinity the value of y will continue to decrease and the graph will continue to decrease and approach negative infinity. Example 2A

Sketch a Nonlinear Graph As x → –∞, f(x) → –∞ As the value of x approaches negative infinity the value of y will continue to decrease and the graph will continue to decrease and approach negative infinity. Determine the shape of Use the information the function to graph the function. Example 2A

Sketch a Nonlinear Graph Answer: Example 2A

Sketch a Nonlinear Graph Use the given key features to sketch a nonlinear graph. Sample answers shown. B. The function is continuous and symmetric about the line x = 1. The function has a minimum at (1, 0). As x →∞, f (x) → ∞ and as x → –∞, f (x) → ∞. Example 2B

Sketch a Nonlinear Graph The function is symmetric Sketch a vertical line at x = 1 about the line x = 1 The graph will be the same on both sides of the line. The function has a Plot a point at (1, 0). minimum at (1, 0) Example 2B

Sketch a Nonlinear Graph As x → +∞, f(x) → +∞ As the value of x approaches positive infinity the value of y will continue to increase and the graph will continue to increase and approach positive infinity. As x → –∞, f(x) → +∞ As the value of x approaches negative infinity the value of y will continue to increase and the graph will continue to increase and approach positive infinity. Example 2B

Answer: Sketch a Nonlinear Graph Determine the shape of the function Use the information to graph the function. Answer: Example 2B

Increasing: Leah’s speed increases steadily for the first minute. Sketch a Real-World Function BIKING Use the given key features to sketch a graph. Leah goes for a bike ride on a bike path near her house. y-intercept: Leah starts at 0 mi/h. Linear or nonlinear: The function that models the situation is nonlinear. Extrema: Leah’s maximum speed is 15 mi/h, which she reaches 1 minute after she starts riding her bike. Increasing: Leah’s speed increases steadily for the first minute. Real-World Example 3

Sketch a Real-World Function Decreasing: At the 10-minute mark, Leah decreases her speed for 1 minute, then she stays at 10 mi/h for 5 minutes. At the 16-minute mark, she again decreases her speed for 1 minute until she reaches a stop. Real-World Example 3

Sketch a Real-World Function The y-intercept is 0 Plot a point on the y-axis at 0. The function is nonlinear The graph will not be a straight line. The function is positive for –1 < x < 3 Plot the points at 3 and –1 on the x-axis. The graph will be above the x-axis for all values of x between –1 and 3. The function has a maximum at 15 mph Plot a point at (1, 15). Real-World Example 3

Sketch a Real-World Function The function is increasing for one minute The slope is positive until x = 1. The function is constant for 10 minutes The slope is 0 until x = 10. The function is decreasing for one minute The slope is negative until x =11. The function is constant for 5 minutes The slope is 0 until x = 16. The function is decreasing for one minute The slope is negative until x = 17. Determine the shape of function Use the information to the graph the function. Real-World Example 3

Sketch a Real-World Function Answer: Real-World Example 3