1. Is the graph an increasing or decreasing function? Explain your - PowerPoint PPT Presentation

D AY 97 – E XPONENTIAL F UNCTIONS : D OMAIN & R ANGE

E XAMPLE Part I Using a graphing calculator, graph the function and sketch the graph on the grid provided below.

E XAMPLE Part I Using a graphing calculator, graph the function and sketch the graph on the grid provided below.

1. Is the graph an increasing or decreasing function? Explain your answer. 2. Trace or use the table feature on your calculator to fill out the tables below. As the value of x gets very large, what happens to the value of 2 x ? x y 0 1 5 10 20

1. Is the graph an increasing or decreasing function? Explain your answer. Increasing function as x increases, y increases 2. Trace or use the table feature on your calculator to fill out the tables below. As the value of x gets very large, what happens to the value of 2 x ? x 2 x 0 1 1 2 5 32 10 1024 20 1.048576x10 6

As the value of x gets very small, what happens to the value of 2 x ? x y -1 -3 -5 -10 -20

As the value of x gets very small, what happens to the value of 2 x ? x 2 x -1 0.5 -3 0.125 -5 0.03125 -10 9.765625x10 -4 -20 9.536743x10 -7

3. Will the value of 2 x ever equal 0? Explain your answer. 4. Are there any values of x that would make 2 x undefined? Explain your answer  5. State the domain and range for f ( x ) 2 x Domain: Range:

3. Will the value of 2 x ever equal 0? Explain your answer. approaches 0 4. Are there any values of x that would make 2 x undefined? Explain your answer no.  5. State the domain and range for f ( x ) 2 x Domain: x R Range: y > 0

Part II Using a graphing calculator, graph the function   along with the graph of from f ( x ) 3 x f ( x ) 2 x Part I, and sketch the graph of on the grid provided below

Part II Using a graphing calculator, graph the function   along with the graph of from f ( x ) 3 x f ( x ) 2 x Part I, and sketch the graph of on the grid provided below

1.Is the graph an increasing or decreasing function? Explain your answer increasing function as x increases y increases 2. As the value of x gets very large, what happens to the value of 3x ? also gets large 3. As the value of x gets very small, what happens to the value of 3x ? 3 x gets very small 4. How does the graph of compare to the graph of ? 3 x increases faster that 2 x

 x 5. a. Given the general form (where a > 1), ( ) f x a what effect does increasing the value of "a" have upon the graph? b. What effect does decreasing the value of "a" have upon the graph?

 x 5. a. Given the general form (where a > 1), ( ) f x a what effect does increasing the value of "a" have upon the graph? The larger the a, the faster the graph will rise b. What effect does decreasing the value of "a" have upon the graph?

Part III Use a graphing calculator to graph the function   x along with the graph of from x ( ) 2 f x ( ) ( 0 . 5 ) f x  Part I, and sketch the graph of on the x f ( x ) ( 0 . 5 ) grid provided below.

Part III Use a graphing calculator to graph the function   x along with the graph of from x ( ) 2 f x ( ) ( 0 . 5 ) f x  Part I, and sketch the graph of on the x f ( x ) ( 0 . 5 ) grid provided below.

1. Is the graph an increasing or decreasing function? Explain your answer. 2. Trace or use the table feature on your calculator to fill out the tables below. As the value of x gets very large, what happens to the value of (0.5) x ? x (0.5) x 0 1 5 10 20

1. Is the graph an increasing or decreasing function? Explain your answer. Decreasing, as x increases y decreases 2. Trace or use the table feature on your calculator to fill out the tables below. As the value of x gets very large, what happens to the value of (0.5) x ? x (0.5) x 0 1 1 0.5 5 0.03125 10 9.765625x10 -4 20 9.536743x10 -7

As the value of x gets very small, what happens to the value of (0.5) x ? x (0.5) x -1 -3 -5 -10 -20 3. Will the value of (0.5) x ever equal 0? Explain your answer.

As the value of x gets very small, what happens to the value of (0.5) x ? x (0.5) x -1 1 -3 2 -5 8 -10 32 -20 1024 3. Will the value of (0.5) x ever equal 0? Explain your answer. no, it will get very close, but never reach 0.

4. Are there any values of x that would make (0.5) x undefined? Explain your answer. 5. State the domain and range for Domain: Range:

4. Are there any values of x that would make (0.5) x undefined? Explain your answer. no, we are never dividing by 0. 5. State the domain and range for Domain: x R Range: y > 0

6. How does the graph of compare to the graph  x ( ) ( 0 . 5 ) f x of ?  x ( ) 2 f x

6. How does the graph of compare to the graph  x ( ) ( 0 . 5 ) f x of ?  x ( ) 2 f x The graphs are reflected over the y-axis

Part IV Use the graphing calculator to graph the function   x x along with the graph of from ( ) ( 0 . 5 ) f x f ( x ) ( 0 . 8 )  Part III, and sketch the graph of on x ( ) ( 0 . 8 ) f x the grid provided below.

Part IV Use the graphing calculator to graph the function   x x along with the graph of from ( ) ( 0 . 5 ) f x f ( x ) ( 0 . 8 )  Part III, and sketch the graph of on x ( ) ( 0 . 8 ) f x the grid provided below.

1. Is the graph an increasing or decreasing function? Explain your answer. 2. As the value of x gets very large, what happens to the value of (0.8) x ? 3. As the value of x gets very small, what happens to the value of (0.8) x ?

1. Is the graph an increasing or decreasing function? Explain your answer. Decreasing, as x increases y decreases 2. As the value of x gets very large, what happens to the value of (0.8) x ? (0.5) x gets smaller 3. As the value of x gets very small, what happens to the value of (0.8) x ? (0.5) x gets larger

 4. How does the graph of compare to the x ( ) ( 0 . 8 ) f x  graph of ? x ( ) ( 0 . 5 ) f x  5. a. Given the general form (where 0 < a < 1), x f ( x ) a what effect does increasing the value of "a" have upon the graph? b. What effect does decreasing the value of "a" have upon the graph?

 4. How does the graph of compare to the x ( ) ( 0 . 8 ) f x  graph of ? x ( ) ( 0 . 5 ) f x The graph (0.8) decreases slower that (0.5) x  5. a. Given the general form (where 0 < a < 1), x f ( x ) a what effect does increasing the value of "a" have upon the graph? The higher the value or a closer a gets to 1, the more the graph will look like horizontal line where y=1 b. What effect does decreasing the value of "a" have upon the graph?

E XAMPLE   Sketch the graph of State the domain x ( ) 2 1 . g x and range. The graph of f(x) = 2 x this function is a vertical translation of the graph f(x)=2 x f(x) = 2 x -1 down one unit. Domain: ( – ∞, ∞) Range: ( –1, ∞) y = – 1

E XAMPLE   Sketch the graph of State the domain x ( ) 2 . g x and range. The graph of f(x) = 2 x this function is a reflection the graph f(x)=2 x in the y-axis. f(x) = 2 - x Domain: ( – ∞, ∞) Range: (0, ∞)