7 5 Practice Exponential Functions Glencoe Algebra 1

A Table Of Values Gives Us

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Example: Graphing The Stretch Of An Exponential Function

Sketch a graph of f\left=4\right)}^. State the domain, range, and asymptote.

Before graphing, identify the behavior and key points on the graph.

• Since b=\frac is between zero and one, the left tail of the graph will increase without bound as x decreases, and the right tail will approach the x-axis as x increases.
• Since a = 4, the graph of f\left=\right)}^ will be stretched vertically by a factor of 4.
• Create a table of points:
  

  x
  f\left=4\left^	32	1	0.5

• Plot the y-intercept, \left, along with two other points. We can use \left and \left.
• Draw a smooth curve connecting the points.

The domain is \left, the range is \left, the horizontal asymptote is y = 0.

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Graphing A Horizontal Shift

The next transformation occurs when we add a constant c to the input of the parent function f\left=^ giving us a horizontal shiftc units in the opposite direction of the sign. For example, if we begin by graphing the parent function f\left=^, we can then graph two horizontal shifts alongside it using c=3: the shift left, g\left=^, and the shift right, h\left=^. Both horizontal shifts are shown in the graph below.

Observe the results of shifting f\left=^ horizontally:

• The domain, \left, remains unchanged.
• The asymptote, y=0, remains unchanged.
• The y-intercept shifts such that:
• When the function is shifted left 3 units to g\left=^, the y-intercept becomes \left. This is because ^=\left^=\left^, so the initial value of the function is 8.
• When the function is shifted right 3 units to h\left=^, the y-intercept becomes \left. Again, see that ^=\left^=\left^, so the initial value of the function is \frac.

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Graphs Of Exponential Functions

• Determine whether an exponential function and its associated graph represents growth or decay.
• Sketch a graph of an exponential function.
• Graph exponential functions shifted horizontally or vertically and write the associated equation.
• Graph a stretched or compressed exponential function.
• Graph a reflected exponential function.
• Write the equation of an exponential function that has been transformed.

As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things by seeing their visual representations, and that is exactly why graphing exponential equations is a powerful tool. It gives us another layer of insight for predicting future events.

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Graphing A Vertical Shift

The first transformation occurs when we add a constant d to the parent function f\left=^ giving us a vertical shiftd units in the same direction as the sign. For example, if we begin by graphing a parent function, f\left=^, we can then graph two vertical shifts alongside it using d=3: the upward shift, g\left=^+3 and the downward shift, h\left=^-3. Both vertical shifts are shown in the figure below.

Observe the results of shifting f\left=^ vertically:

• The domain \left remains unchanged.
• When the function is shifted up 3 units giving g\left=^+3:
• The y-intercept shifts up 3 units to \left.
• The asymptote shifts up 3 units to y=3.
• The range becomes \left.

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Example: Graphing A Shift Of An Exponential Function

Graph f\left=^-3. State the domain, range, and asymptote.

We have an exponential equation of the form f\left=^+d, with b=2, c=1, and d=-3.

Draw the horizontal asymptote y=d, so draw y=-3.

Identify the shift it is \left.

The domain is \left, the range is \left, and the horizontal asymptote is y=-3.

Shift the graph of f\left=^ left 1 unit and down 3 units.The domain is \left the range is \left the horizontal asymptote is y=-3.

In the following video, we show more examples of the difference between horizontal and vertical shifts of exponential functions and the resulting graphs and equations.

A General Note: Characteristics Of The Graph Of The Parent Function F\left=^

An exponential function with the form f\left=^, b> 0, b\ne 1, has these characteristics:

• one-to-one function

• 0.0625

  0.015625

• Plot the y-intercept, \left, along with two other points. We can use \left and \left.

Draw a smooth curve connecting the points.

The domain is \left, the range is \left, and the horizontal asymptote is y=0.

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After Graphing The First Two Examples We Will Take A Look At The Similarities And Differences Between The Two Graphs

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A General Note: Reflecting The Parent Function F\left=^

The function f\left=-^

• reflects the parent function f\left=^ about the x-axis.
• has a y-intercept of \left.
• has a range of \left.
• has a horizontal asymptote of y=0 and domain of \left which are unchanged from the parent function.

The function f\left=^

• reflects the parent function f\left=^ about the y-axis.
• has a y-intercept of \left, a horizontal asymptote at y=0, a range of \left, and a domain of \left which are unchanged from the parent function.

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Example: Writing And Graphing The Reflection Of An Exponential Function

Find and graph the equation for a function, g\left, that reflects f\left=\right)}^ about the x-axis. State its domain, range, and asymptote.

Since we want to reflect the parent function f\left=\right)}^ about the x-axis, we multiply f\left by 1 to get g\left=-\right)}^. Next we create a table of points.

Plot the y-intercept, \left, along with two other points. We can use \left and \left.

Draw a smooth curve connecting the points:

The domain is \left, the range is \left, and the horizontal asymptote is y=0.

Text Of : 1: Cc: A: Rspc: C1: Web Viewglencoe Algebra 2 Chapter 7 32 Glencoe Algebra 1 7

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Graph each function. Find the y-intercept and state the domain and range.

1. y = 2. y = 3. y =

4. y = 4 + 1 5. y = 2 6. y = 0.5

Determine whether the set of data shown below displays exponential behavior. Write yes or no. Explain why or why not.

9. LEARNING Ms. Klemperer told her English class that each week students tend to forget one sixth of the vocabulary words they learned the previous week. Suppose a student learns 60 words. The number of words remembered can be described by the function W = 60 , where x is the number of weeks that pass. How many words will the

student remember after 3 weeks?

10. BIOLOGY Suppose a certain cell reproduces itself in four hours. If a lab researcher begins with 50 cells, how many cells will there be after one day, two days, and three days? .)

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Characteristics Of Graphs Of Exponential Functions

Before we begin graphing, it is helpful to review the behavior of exponential growth. Recall the table of values for a function of the form f\left=^ whose base is greater than one. Well use the function f\left=^. Observe how the output values in the table below change as the input increases by 1.

Each output value is the product of the previous output and the base, 2. We call the base 2 the constant ratio. In fact, for any exponential function with the form f\left=a^, b is the constant ratio of the function. This means that as the input increases by 1, the output value will be the product of the base and the previous output, regardless of the value of a.

Notice from the table that:

• the output values are positive for all values of x
• as x increases, the output values increase without bound
• as x decreases, the output values grow smaller, approaching zero

The graph below shows the exponential growth function f\left=^.

Notice that the graph gets close to the x-axis but never touches it.

The domain of f\left=^ is all real numbers, the range is \left, and the horizontal asymptote is y=0.

To get a sense of the behavior of exponential decay, we can create a table of values for a function of the form f\left=^ whose base is between zero and one. Well use the function g\left=\right)}^. Observe how the output values in the table below change as the input increases by 1.

Notice from the table that:

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Domain And Range Of Exponential Function

We know that the domain of a function y = f is the set of all x-values where it can be computed and the range is the set of all y-values of the function. From the graphs of f = 2x and g = x in the previous section, we can see that an exponential function can be computed at all values of x. Thus, the domain of an exponential function is the set of all real numbers . The range of an exponential function can be determined by the horizontal asymptote of the graph, say, y = d, and by seeing whether the graph is above y = d or below y = d. Thus, for an exponential function f = abx,

• Domain is the set of all real numbers .
• Range is f > d if a > 0 and f < d if a < 0.

To understand this, you can see the example below.

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Stretching Compressing Or Reflecting An Exponential Function

While horizontal and vertical shifts involve adding constants to the input or to the function itself, a stretch or compression occurs when we multiply the parent function f\left=^ by a constant |a|> 0. For example, if we begin by graphing the parent function f\left=^, we can then graph the stretch, using a=3, to get g\left=3^ and the compression, using a=\frac, to get h\left=\frac^.

g\left=3^ stretches the graph of f\left=^ vertically by a factor of 3. h\left=\frac^ compresses the graph of f\left=^ vertically by a factor of \frac.

What Is The Range Of Exponential Function

The range of an exponential function depends upon its horizontal asymptote and also whether the curve lies above or below the horizontal asymptote. i.e., for an exponential function f = abx, the range is

• f > d if a > 0 and
• f < d if a < 0,

where y = d is the horizontal asymptote of the graph of the function.

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A General Note: Shifts Of The Parent Function F\left=^

For any constants c and d, the function f\left=^+d shifts the parent function f\left=^

• shifts the parent function f\left=^ vertically d units, in the same direction as the sign of d.
• shifts the parent function f\left=^ horizontally c units, in the opposite direction as the sign of c.
• has a y-intercept of \left.
• has a horizontal asymptote of y = d.
• has a range of \left.
• has a domain of \left which remains unchanged.

Summarizing Transformations Of The Exponential Function

Now that we have worked with each type of translation for the exponential function, we can summarize them to arrive at the general equation for transforming exponential functions.

• reflected about the x-axis when a < 0.

Note the order of the shifts, transformations, and reflections follow the order of operations.

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At This Point In Our Study Of Algebra We Begin To Look At Transcendental Functions Or Functions That Seem To Transcend Algebra

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