1. Properties of Exponential Functions Lesson 7-2 Part 1
Algebra 2
Properties of Exponential Functions
Lesson 7-2 Part 1

2. Goals
Goal
Rubric
To explore the properties of functions of the form y = abx.
Level 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary
None

4. Essential Question Big Idea: Modeling
What are the transformations on exponential functions?

5. Properties of Exponential Functions
The domain of f (x) = bx consists of all real numbers. The range of f (x) = bx consists of all positive real numbers.
The graphs of all exponential functions pass through the point (0, 1) because f (0) = b0 = 1.
If b > 1, f (x) = bx has a graph that goes up to the right and is an increasing function.
If 0 < b < 1, f (x) = bx has a graph that goes down to the right and is a decreasing function.
f (x) = bx is a one-to-one function and has an inverse that is a function.
The graph of f (x) = bx approaches but does not cross the x-axis. The x-axis is a horizontal asymptote.
f (x) = bx
b > 1
f (x) = bx
0 < b < 1

6. Changing the base of y = bx, when b > 1

7. Changing the base of y = bx, when 0 < b < 1

8. Exponential Function: Change of Base Summary
For b > 1, as b increases the curve moves closer to the y-axis. The y-intercept (0,1) does not change.
Y=2x
Y=3x

9. Exponential Function: Change of Base Summary
For 0 < b < 1, as b decreases the curve moves closer to the y-axis. The y-intercept (0,1) does not change.
Y=(1/3)x
Y=(1/2)x

10. Exponential Function Transformations
You can perform the same transformations on exponential functions that you performed on linear, quadratic, and absolute value functions.

11. Transformations Involving Exponential Functions
Shifts the graph of f (x) = bx upward k units if k > 0.
Shifts the graph of f (x) = bx downward k units if k < 0.
g(x) = bx + k
Vertical translation
Reflects the graph of f (x) = bx about the x-axis.
Reflects the graph of f (x) = bx about the y-axis.
g(x) = -bx
g(x) = b-x
Reflecting
Multiplying y-coordintates of f (x) = bx by a,
Stretches the graph of f (x) = bx if a > 1.
Shrinks the graph of f (x) = bx if 0 < a < 1.
g(x) = a bx
Vertical stretching or shrinking
Shifts the graph of f (x) = bx to the left h units if h < 0.
Shifts the graph of f (x) = bx to the right h units if h > 0.
g(x) = bx-h
Horizontal translation
Description
Equation
Transformation

12. It may help you remember the direction of the shift if you think of “h is for horizontal.”
Helpful Hint

13. Example:
Make a table of values, and graph g(x) = 2–x + 1. Describe the asymptote. Tell how the graph is transformed from the graph of the function f(x) = 2x. x –3 –2 –1 1 2
g(x) 9 5 3
1.5
1.25
The asymptote is y = 1, and the graph approaches this line as the value of x increases. The transformation reflects the graph across the y-axis and moves the graph 1 unit up.

14. Your Turn:
Make a table of values, and graph f(x) = 2x – 2. Describe the asymptote. Tell how the graph is transformed from the graph of the function f(x) = 2x. x –2 –1 1 2
f(x) 1 16 8 4 2
The asymptote is y = 0, and the graph approaches this line as the value of x decreases. The transformation moves the graph 2 units right.

15. Example:
Graph the function. Find y-intercept and the asymptote. Describe how the graph is transformed from the graph of its parent function.
g(x) = (1.5x) 2 3
The graph of g(x) is a vertical compression of the parent function f(x) 1.5x by a factor of
parent function: f(x) = 1.5x
y-intercept: 2 3
asymptote: y = 0 2 3

16. Your Turn:
Graph the function. Find y-intercept and the asymptote. Describe how the graph is transformed from the graph of its parent function.
h(x) = 2.7–x + 1
parent function: f(x) = 2.7x
y-intercept: 2.7
asymptote: y = 0
The graph of h(x) is a reflection of the parent function f(x) = 2.7x across the y-axis and a shift of 1 unit to the right. The range is {y|y > 0}.

17. Your Turn:
Graph the exponential function. Find y-intercept and the asymptote. Describe how the graph is transformed from the graph of its parent function.
h(x) = (5x) 1 3
parent function: f(x) = 5x
The graph of h(x) is a vertical compression of the parent function f(x) = 5x by a factor of
y-intercept 1 3
asymptote: 0 1 3

18. Your Turn:
Graph the exponential function. Find y-intercept and the asymptote. Describe how the graph is transformed from the graph of its parent function.
g(x) = 2(2–x)
parent function: f(x) = 2x
y-intercept: 2
asymptote: y = 0
The graph of g(x) is a reflection of the parent function f(x) = 2x across the y-axis and vertical stretch by a factor of 2.

19. Summary of Exponential Function Transformations
f(x) = ±a·b±x-h+k
Reflection over
the y-axis
(neg. x makes
decay curve out
of growth curve)
Horizontal translation
by of h units(opposite
direction of sign)
Vertical translation
by of k units(same
direction of sign)
y = k equation of
horizontal asymptote
Reflection over
the x-axis
Vertical Stretch
by factor of a
Base
The order of transformations: horizontal translation, reflection (horz., vert.), vertical stretch/compression, vertical translation.

20. Your Turn: Horizontal shift right 1. Horizontal shift left 2.
Reflect about the x-axis.
Vertical shrink ½ .
Vertical shift up 1.
Vertical shift down 3.

21. Example:
The parent function for the graph shown is of the form y = abx. Write the parent function. Then write a function for the translation indicated.
When x = 0, y = -1.
-1 = a(b0) and a = -1
When x = 1, y = -3.
-3 = (-1)(b1) and b = 3
Parent Function: y = -3x
Transformed Function: y = -3(x – 8) + 2

22. Your Turn:
The parent function for the graph shown is of the form y = abx. Write the parent function. Then write a function for the translation indicated.
When x = 0, y = -3.
-3 = a(b0) and a = -3
When x = 1, y = -1.
-1 = (-3)(b1) and b = 1/3

23. Essential Question Big Idea: Modeling
What are the transformations on exponential functions?
Every exponential function can be described as a transformation of the parent exponential function y = bx. The general form is given by y = ab(x – h) + k. The value of h and k represent translations. The factor a performs stretches, compressions, and reflections.

24. Assignment
Section 7-2 Part 1, Pg 474 – 475; #1 – 5 all, 6 – 18 even, 22, 24.