Graph Square Root Functions

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Presentation transcript:

Graph Square Root Functions Warm Up Lesson Presentation Lesson Quiz

Warm-Up 1. Graph the function y = 2x. ANSWER 2. Evaluate 3 x when x = 4. √ ANSWER 6 3. Evaluate x + 5 when x = 11. √ ANSWER 4

Example 1 Graph the function y = 3 x and identify its domain and range. Compare the graph with the graph of y = x . SOLUTION STEP 1 Make a table. Because the square root of a negative number is undefined, x must be non-negative. So, the domain is x ≥ 0. STEP 2 Plot the points. STEP 3 Draw a smooth curve through the points. From either the table or the graph, you can see the range of the function is y ≥ 0.

Example 1 STEP 4 Compare the graph with the graph of y = x . The graph of y = 3 x is a vertical stretch (by a factor of 3) of the graph of y = x .

Example 2 Graph the function y = –0.5 x and identify its domain and range. Compare the graph with the graph of y = x . SOLUTION To graph the function, make a table, plot the points, and draw a smooth curve through the points. The domain is x ≥ 0. The range is y ≤ 0.The graph of y = –0.5 x is a vertical shrink (by a factor of 0.5) with a reflection in the x-axis of the graph of y = x .

Example 3 Graph the function y = x + 2 and identify its domain and range. Compare the graph with the graph of y = x . SOLUTION To graph the function, make a table, then plot and connect the points. The domain is x ≥ 0. The range is y ≥ 2.The graph of y = x + 2 is a vertical translation (of 2 units up) of the graph of y = x .

Guided Practice Graph the function and identify its domain and range. Compare the graph with the graph of y = x . 1. y = 2 x ANSWER Domain: x ≥ 0, Range: y ≥ 0 Vertical stretch by a factor of 2

Guided Practice Graph the function and identify its domain and range. Compare the graph with the graph of y = x . 2. y = –2 x ANSWER Domain: x ≥ 0, Range: y ≤ 0 Vertical stretch by a factor of 2 and a reflection in the x-axis

Guided Practice Graph the function and identify its domain and range. Compare the graph with the graph of y = x . 3. y = x – 1 ANSWER Domain: x ≥ 0, Range: y ≥ 0 – 1 Vertical translation of 1 unit down

Guided Practice Graph the function and identify its domain and range. Compare the graph with the graph of y = x . 4. y = x + 3 ANSWER Domain: x ≥ 0, Range: y ≤ 0 Vertical translation of 3 units up

Example 4 Graph the function y = and identify its domain and range. Compare the graph with the graph of y = x . x – 4 SOLUTION To graph the function, make a table, then plot and connect the points. To find the domain, find the values of x for which the radicand, x – 4 , is nonnegative. The domain is x ≥ 4.

Example 4 The range is y ≥ 0.The graph of y = x – 4 is a horizontal translation (of 4 units to the right) of the graph of y = . x

Example 5 Graph the function y = 2 – 1 . x + 4 SOLUTION STEP 1 Sketch the graph of y = 2 . x STEP 2 Shift the graph h units horizontally and k units vertically. Notice that y = 2 – 1 = 2 x – (–4) + (–1). x + 4 So, h = –4 and k = –1. Shift the graph left 4 units and down 1 unit.

Guided Practice 5. Graph the function y = x + 3 and identify its domain and range. Compare the graph with the graph of y = x . ANSWER Domain: x ≥ – 3; Range: y ≥ 0; Horizontal translation 3 units to the left 6. Identify the domain and range of the function in Example 5. The domain is x ≥ – 4. The range is y ≥ –1. ANSWER

Example 6 MICROPHONE SALES For the period 1988–2002, the amount of sales y (in millions of dollars) of microphones in the United States can be modeled by the function y = 93 where x is the number of years since 1988. Graph the function on a graphing calculator. In what year were microphone sales about $325 million? x + 2.2 The graph of the function is shown. Using the trace feature, you can see that y 325 when x = 10. So, microphone sales were about $325 million 10 years after 1988, or in 1998. SOLUTION

Guided Practice 7. MICROPHONE SALES: Use the function in Example 6 to find the year in which microphone sales were about $250 million. ANSWER 1993

Lesson Quiz 1. Graph the function y = 2 x + 1. ANSWER

Lesson Quiz 1 2 2. Graph the function y = x – 2 + 1. ANSWER

Lesson Quiz 1 4 3. Graph the function t = h, where t is time (in seconds) and h is distance (in feet) to model a sky diver’s freefall. How many feet did a sky diver freefall if the freefall lasted 9 seconds? 1296 ft ANSWER