Topics
5. Exponential & Piecewise Functions
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United States of AmericaTennessee
Integrated Math 2

5.01 Characteristics of exponential functions

Lesson
Practice
Lesson

Concept summary

To draw the graph of an exponential function we can fill out a table of values for the function and draw the curve through the points found. We can also identify key features from the equation:

\displaystyle f\left(x\right)=ab^x
\bm{a}
The initial value gives us the the value of the y-intercept
\bm{b}
We can use the constant factor to identify other points on the curve

The constant factor, b, can be found by finding the common ratio.

We can determine the key features of an exponential function from its graph:

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  • The graph is increasing
  • y approaches a minimum value of 0
  • The domain is \left(-\infty, \infty\right)
  • The range is \left(0, \infty\right)
  • The y-intercept is at \left(0,\, 3\right)
  • The common ratio is 4
  • The horizontal asymptote is y=0
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  • The graph is decreasing
  • y approaches a minimum value of 0
  • The domain is \left(-\infty, \infty\right)
  • The range is \left( -\infty, 0\right)
  • The y-intercept is at \left(0,\, 10\right)
  • The common ratio is \dfrac{1}{2}
  • The horizontal asymptote is y=0
Asymptote

A line that a curve or graph approaches as it heads toward positive or negative infinity

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Exponential functions can be dilated, reflected, and translated in a similar way to other functions.

The exponential parent function y=b^{x} can be transformed to y=ab^{c\left(x-h\right)}+k

  • If a is negative, the basic curve is reflected across the x-axis
  • The graph is dilated (stretched/compressed) vertically by a factor of a
  • If c is negative, the basic curve is reflected across the y-axis
  • The graph is dilated (stretched/compressed) horizontally by a factor of c
  • The graph is translated to the right by h units
  • The graph is translated upwards by k units
  • The asymptote, originally y=0, is translated up to y=k
  • The range of the graph becomes \left[k, \infty\right), for a>0, or \left(-\infty, k\right], for a<0, due to the vertical shift of k units

Worked examples

Example 1

For each scenario, find the new equation.

a

The graph of y=3^x is translated up by 3 units and then 2 units to the left.

Approach

We can represent this change graphically:

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  • A translation of 2 units to the left is equivalent to adding 2 to x in the exponent.
  • A translation of 3 units up is equivalent to adding 3 to the equation.

Solution

The new equation is y=3^{x+2}+3

b

The graph of y = \left(\dfrac{1}{2}\right)^{x} is reflected across the x-axis and stretched vertically by a factor of 2.

Approach

We can represent this change graphically:

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  • A reflection across the x-axis is equivalent to multiplying the function by -1.
  • A vertical stretch by a factor of 2 is equivalent to multiplying the function by 2.

Solution

The new equation is y=-2\left(\dfrac{1}{2}\right)^{x}

Example 2

Consider the exponential function f\left(x\right)=5^{-x}-2

a

Find the coordinates of the y-intercept.

Approach

We can see that the parent function y=5^x has been reflected across the y-axis, and then translated down 2 units.

The y-intercept of y=5^{-x}-2 occurs when x=0, so we can substitute x=0, or we can consider what would happen when we perform the transformations on the y-intercept of y=5^x, which is at \left(0,1\right).

Solution

Substituting x=0 into f\left(x\right)=5^{-x}-2 gives f\left(0\right)=5^{-0}-2=-1

The y-intercept of f\left(x\right)=5^{-x}-2 is at \left(0,-1\right).

Reflection

Reflecting across the y-axis does not change the y-intercept.

b

State the domain and range

Approach

The domain of the parent function is \left(-\infty, \infty\right) and the range is \left(0, \infty\right). Neither the reflection across the y-axis, or the vertical translation will alter the domain. The range will be shifted down 2 units.

Solution

The domain of f\left(x\right)=5^{-x}-2 is \left(-\infty, \infty\right) and the range is \left(-2, \infty\right).

c

Sketch a graph of the function.

Approach

The graph of the function will be the graph of the parent function, reflected across the y-axis, and translated 2 units downwards. We know it has a y-intercept at \left(0,-1\right), domain of \left(-\infty, \infty\right) and range of \left(-2, \infty\right). The asymptote will be shifted down 2 units also.

Solution

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Example 3

For the given table, describe the transformations needed to get from f\left(x\right) to g\left(x\right)

xf(x)g(x)
-1\dfrac{1}{3}\dfrac{1}{9}
01\dfrac{1}{3}
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Approach

We can see that g\left(x\right)=f\left(x-1\right).

Solution

The transformation is a horizontal translation of 1 unit to the right

Reflection

We can see this transformation clearly if we draw the graphs of both functions:

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Example 4

The graph of the function f\left(x\right) = 5\cdot 2^{x} is translated 10 units down to give a new function g\left(x\right) :

Complete the table of values for f\left(x\right) and the transformed function g\left(x\right), and then sketch the graphs of both functions.

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f(x)
g(x)

Approach

To find the values for f\left(x\right) we can substitute in the values from the table and evaluate.

As the graph of f\left(x\right) = 5 \cdot 2^{x} is translated 3 down, g\left(x\right)=f\left(x\right)-10, we can subtract 10 from each value in the table for f\left(x\right) to find the corresponding value for g\left(x\right)

Similarly, the graph of g\left(x\right) will be the graph of f\left(x\right) translated down 10 units.

Solution

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f(x)\dfrac{5}{4}\dfrac{5}{2}510204080
g(x)-\dfrac{35}{4}-\dfrac{15}{2}-50103070
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Outcomes

M2.N.Q.A.1

Use units as a way to understand real-world problems.*

M2.F.IF.B.3

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.*

M2.F.IF.B.5

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate and interpret the rate of change from a graph.*

M2.MP1

Make sense of problems and persevere in solving them.

M2.MP2

Reason abstractly and quantitatively.

M2.MP3

Construct viable arguments and critique the reasoning of others.

M2.MP4

Model with mathematics.

M2.MP6

Attend to precision.

M2.MP7

Look for and make use of structure.

M2.MP8

Look for and express regularity in repeated reasoning.

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