Transcript Slide 1

Algebra 2
Properties of Exponential Functions
Lesson 7-2 Part 1
Goals
Goal
• To explore the properties of
functions of the form
y = abx.
Rubric
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.
Vocabulary
• None
Essential Question
Big Idea: Modeling
•
What are the transformations on exponential
functions?
Properties of Exponential Functions
1.
2.
3.
4.
5.
6.
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.
bx
f (x) =
0<b<1
f (x) = bx
b>1
Changing the base of y = bx, when b > 1
Changing the base of y = bx, when 0 < b < 1
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=3x
Y=2x
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
Exponential Function
Transformations
You can perform the same transformations on
exponential functions that you performed on
linear, quadratic, and absolute value functions.
Transformations Involving
Exponential Functions
Transformation
Equation
Description
Horizontal translation
g(x) = bx-h
• 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.
Vertical stretching or
shrinking
g(x) = a bx
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.
Reflecting
g(x) = -bx
g(x) = b-x
• Reflects the graph of f (x) = bx about the x-axis.
• Reflects the graph of f (x) = bx about the y-axis.
Vertical translation
g(x) = bx + k
• 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.
Helpful Hint
It may help you remember the direction of the shift if you
think of “h is for horizontal.”
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
0
1
2
g(x)
9
5
3
2
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.
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
f(x)
–2
–1
0
1
16
1
8
1
4
1
1
2
2
1
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.
Example:
Graph the function. Find y-intercept
and the asymptote. Describe how the
graph is transformed from the graph
of its parent function.
2
g(x) = 3 (1.5x)
parent function: f(x) = 1.5x
y-intercept:
asymptote: y = 0
2
3
The graph of g(x) is a vertical
compression of the parent
function f(x) 1.5x by a factor of
2
.
3
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}.
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) =
1
3
(5x)
parent function: f(x) = 5x
y-intercept
1
3
asymptote: 0
The graph of h(x) is a vertical
compression of the parent
function f(x) = 5x by a factor of
1
.
3
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.
Summary of Exponential
Function Transformations
f(x) =
Reflection over
the x-axis
Vertical Stretch
by factor of a
±x-h
±a·b +k
Base
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
The order of transformations: horizontal translation, reflection (horz., vert.), vertical
stretch/compression, vertical translation.
Your Turn:
y  (2)
x 1
1
Horizontal shift right 1.
Reflect about the x-axis.
y  (3)
1
2
x2
3
Horizontal shift left 2.
Vertical shrink ½ .
Vertical shift up 1.
Vertical shift down 3.
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
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
1
Parent Function: y  3 
 3
x
1
Transformed Function: y  3  
 3
x 15
1
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.
Assignment
• Section 7-2 Part 1, Pg 474 – 475; #1 – 5 all,
6 – 18 even, 22, 24.