Lesson 6.5, For use with pages 446-451 State the domain and range of the function. 1. f(x) = 2x – 5. ANSWER domain and range: all real numbers 2. g(x) = – 𝑥 2 + 6 ANSWER domain: all real numbers: range: y ≤ 6

3. y = 𝑥+4 2 − 8. Lesson 6.5, For use with pages 446-451 State the domain and range of the function. 2 3. y = 𝑥+4 2 − 8. 1 – domain: all real numbers: range: y ≤ –8 ANSWER

Graphing Square Roots (Radicals) Alg II 6.5 PP Graphing Square Roots (Radicals)

EXAMPLE 1 Graph a square root function Graph y = 𝒙 and state the domain and range. Compare the graph with the graph of y = 𝑥 1 2 SOLUTION Make a table of values and sketch the graph.

EXAMPLE 1 Graph a square root function The radicand of a square root must be nonnegative. So, the domain is x ≥ 0. The range is y ≥ 0. The graph of y = 𝑿 is a vertical shrink of the graph of y = by a factor of . 1 2  x

EXAMPLE 2 Graph a cube root function Graph y = –3 3 𝑥 , and state the domain and range. Compare the graph with the graph of y = 3 𝑥 . SOLUTION Make a table of values and sketch the graph.

EXAMPLE 2 Graph a cube root function The domain and range are all real numbers. The graph of y = – 3 is a vertical stretch of the graph of y = by a factor of 3 followed by a reflection in the x-axis. x  3

EXAMPLE 3 Solve a multi-step problem Pendulums The period of a pendulum is the time the pendulum takes to complete one back-and-forth swing. The period T (in seconds) can be modeled by T = 1.11 𝑙 where l is the pendulum’s length (in feet). • Use a graphing calculator to graph the model. • How long is a pendulum with a period of 3 seconds?

EXAMPLE 3 Solve a multi-step problem SOLUTION STEP 1 Graph the model. Enter the equation y = 1.11 𝑥 . The graph is shown below. Use the trace feature to find the value of x when y = 3.The graph shows x 7.3. STEP 2

EXAMPLE 3 Solve a multi-step problem ANSWER A pendulum with a period of 3 seconds is about 7.3 feet long.

GUIDED PRACTICE for Examples 1, 2 and 3 1. y = – 3 𝑥 SOLUTION Make a table of values and sketch the graph. ANSWER Domain: Range: x > 0 y < 0

Make a table of values and sketch the graph. GUIDED PRACTICE for Examples 1, 2 and 3 2. y = 1 4  x SOLUTION Make a table of values and sketch the graph. x 1 2 3 4 y 0.25 0.35 0.43 0.5 ANSWER Domain: Range: x > 0 y < 0

GUIDED PRACTICE for Examples 1, 2 and 3 3. y =  x 3 1 2 – SOLUTION Make a table of values and sketch the graph. x -3 -2 -1 1 2 3 y 0.72 0.63 0.5 -0.5 -0.6 -0.7 ANSWER Domain: Range: All real numbers

GUIDED PRACTICE for Examples 1, 2 and 3 4. g(x) = 4 3 𝑥 SOLUTION Make a table of values and sketch the graph. x -3 -2 -1 1 2 3 y -5.77 -5.04 -4 4 5.04 5.77 ANSWER Domain: Range: All real numbers

GUIDED PRACTICE for Examples 1, 2 and 3 5. What if ? Use the model in example 3 to find the length of a pendulum with a period of 1 second. ANSWER About 0.8 ft

EXAMPLE 4 Graph a translated square root function Graph y = -2 (𝑥 −3) +2. Then state the domain and range. SOLUTION STEP 1 Sketch the graph of y = –2 𝑥 (shown in blue). Notice that it begins at the origin and passes through the point (1, –2).

EXAMPLE 4 Graph a translated square root function STEP 2 Translate the graph. For y = - 2 (𝑥 −3) +2 , h = 3 and k = 2. So, shift the graph of y = –2 𝑥 right 3 units and up 2 units. The resulting graph starts at (3, 2) and passes through (4, 0). From the graph, you can see that the domain of the function is x ≥ 3 and the range of the function is y ≤ 2.

EXAMPLE 5 Graph a translated cube root function Graph y = 3 3 x + 4 – 1. Then state the domain and range. SOLUTION STEP 1 Sketch the graph of y = 3 3 x (shown in blue). Notice that it begins at the origin and passes through the point (–1, –3) and (1, 3).

EXAMPLE 5 Graph a translated cube root function STEP 2 Translate the graph. Note the for y = –3 3 x + 4 – 1, h = –4 and k = –1. So, shift the graph of y = 3 3 x left 4 units and up 1 unit. The resulting graph starts at (–5, –4),(–4, –1) and passes through (–3, 2). From the graph, you can see that the domain and range of the function are both all real numbers.

GUIDED PRACTICE for Examples 4 and 5 Graph the function. Then state the domain and range. 6. y = –4 x + 2 ANSWER Domain : x > 0 ,range : y < 0.

GUIDED PRACTICE for Examples 4 and 5 Graph the function. Then state the domain and range. 7. y = x + 1 2 ANSWER Domain : x > –1 ,range : y > 0.

GUIDED PRACTICE for Examples 4 and 5 Graph the function. Then state the domain and range. 8. f(x) = x – 3 – 1 2 1 ANSWER Domain : x > 3 ,range : y > –1.

GUIDED PRACTICE for Examples 4 and 5 Graph the function. Then state the domain and range. 9. x – 4 2 3 y = ANSWER Domain :all real numbers, range: all real numbers.

GUIDED PRACTICE for Examples 4 and 5 Graph the function. Then state the domain and range. 10. x – 5 3 y = ANSWER Domain :all real numbers, range: all real numbers.

GUIDED PRACTICE for Examples 4 and 5 Graph the function. Then state the domain and range. 11. g(x) = x + 2 – 3 3 ANSWER Domain :all real numbers, range: all real numbers.