Exponential Functions Topic 1: Graphs and Equations of
Exponential Functions
Slide 2
I can describe the characteristics of an exponential function
by analyzing its graph. I can describe the characteristics of an
exponential function by analyzing its equation. I can match
equations in a given set to their corresponding graphs. Interpret
the graph of an exponential function that models a situation, and
explain the reasoning.
Slide 3
Use technology to draw the graph of the polynomial functions
below. Set your windows to X: [10, 10, 1] and Y: [0, 10, 1].
Complete the table in each section. Work through all of the Explore
activity before proceeding to the next slides, which will outline
what you should have noticed Explore
Slide 4
You Should Notice
Slide 5
Slide 6
Slide 7
Information An exponential function has the form, where x is
the exponent b > 0 b 1 In Math 30-2, the a-value will always be
positive The ratio that exists between y-values that correspond to
consecutive x-values is known as the common ratio.
Slide 8
Information The graphs of exponential functions have many
characteristics. The characteristics that will be explored in this
topic are the number of x-intercepts, the value of the y-intercept,
domain, range, and end behaviour. The end behaviour of a function
is the description of the graphs behaviour at the far left and the
far right.
Slide 9
Example 1 Identify which of the following functions is an
exponential function. Explain. a) b) c) d) e) f) Identifying an
exponential function This one is not an exponential function since
the base (b-value)is negative. This one is not an exponential
function since the base (b-value)contains a variable (x). This one
is not an exponential function since there is no variable. It is a
constant function. a, b, and c are exponential functions. They all
have x as the exponent, and a b value greater than 0 (but not
1).
Slide 10
Example 2 Identify which of the following tables of values
represent an exponential function. Explain. a) b) Determine if a
table of values represents an exponential function Note: To
identify whether or not a set of data values represents an
exponential function, you must see a common ratio. 1.Ensure that
the x-values are increasing by a constant value. 2.Check if there
is a common ratio (multiplier) between y- values. If so, the data
represents an exponential function.
Slide 11
Example 2 Identify which of the following tables of values
represent an exponential function. For those that do, create an
equation that represents the function. a) b) Determine if a table
of values represents an exponential function x-values are
increasing by 1. Each value is obtained by adding 2 to the previous
y-value. Since we are adding instead of multiplying, this does not
show an exponential function. Each value is obtained by multiplying
the previous y-value by 2. Since we are multiplying by a common
ratio (2), the data represents an exponential function.
Slide 12
Example 2 Identify which of the following tables of values
represent an exponential function. For those that do, create an
equation that represents the function. c) d) Determine if a table
of values represents an exponential function x-values are
increasing by 1. There is no common ratio, so this data does not
represent an exponential function. Each value is obtained by
multiplying the previous y-value by 1/2. Since we are multiplying
by a common ratio, the data represents an exponential
function.
Slide 13
Example 3 Using an equation to determine characteristics of a
graph On the next slide, determine the characteristics of the given
exponential functions, when a > 0.
Slide 14
Example 3 a = 1 b = e none 1 Q2 Q1 increasing a = 9 none 9 Q2
Q1 decreasing
Slide 15
Example 3 c) Which characteristics are common to all
exponential functions with positive a-values.
Slide 16
Example 4 For each set of characteristics below, write an
equation of a possible exponential function. a) y-intercept of 3
decreasing exponential function b)y-intercept of 1 increasing
exponential function c) y-intercept of 7 end behaviour extends from
quadrant II to quadrant I Reasoning about the characteristics of
the graphs of exponential functions Answers will vary! Note: The y-
intercept is given by the a-value. Note: The b-value tells us that
the function is increasing (b >1) or decreasing (0 < b <
1).
Slide 17
Example 5 Match each graph with the correct exponential
function. Justify your reasoning. Matching exponential functions to
their graphs This graph has a y-intercept (a-value) or 2 and is
increasing (b-value is greater than 1). This graph has y-intercept
(a-value) of 3, and is decreasing (b-value is between 0 and
1).
Slide 18
Example 5 Match each graph with the correct exponential
function. Justify your reasoning. Matching exponential functions to
their graphs This graph has a y-intercept (a-value) of 4, and is
increasing (b-value is greater than 1). This graph has a
y-intercept (a-value) of 4, and is decreasing (b-value is between 0
and 1).
Slide 19
Example 6 How can you recognize when a graph, a table of
values, or an equation represents an exponential function?
Recognizing an exponential function From a Table of Values: There
must be a common ratio. 1.Ensure that the x-values are increasing
by 1. 2.Check if there is a common ratio (multiplier) between
y-values. From a Equation: Equation characteristics The equation
must be of the form y = ab x, where b is positive (but not 1).
Slide 20
Need to Know An exponential function has the form f(x) = a(b)
x, where x is the exponent, b > 0, and b 1. In Math 30-2, all
exponential functions have a > 0. The parameter a is the
y-intercept, where a > 0. The parameter b is the base, where b
> 0 and b 1. It is the constant ratio, in a table of values,
between consecutive y-values when the x-values increase by the same
amount. If b > 1, then the exponential function is increasing.
If 0 < b < 1, then the exponential function is
decreasing.
Slide 21
Need to Know All exponential functions of the form f(x) = a(b)
x, where a > 0, b > 0, and b 1 have the following
characteristics:
Slide 22
Need to Know There are two different shapes of the graphs of an
exponential function of the form f(x) = a(b) x, where a > 0, b
> 0, and b 1.
Slide 23
Need to Know To determine if a table of values is an
exponential function, look for a pattern in the ordered pairs. As
the value of x increases or decreases by a constant amount, the
value of y changes by a constant ratio (common factor). Youre
ready! Try the homework from this section.