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1.01 Exponential functions

Lesson

Exponential functions and their graphs

A base form of an exponential function is $y=a^x$y=ax for $a>0$a>0 and $a\ne1$a1 and the variable $x$x is in the exponent. These graphs take the following form:

$a>1$a>1 $00<a<1

 

Key features:

  • Exponential growth: As the $x$x-values increase, the $y$y-values increase at an increasing rate.

  • The $y$y-intercept is $\left(0,1\right)$(0,1), since when $x=0$x=0$y=a^0=1$y=a0=1, for any positive value $a$a.

  • $y=0$y=0 is a horizontal asymptote for each graph. As $x\ \rightarrow-\infty$x , $y\ \rightarrow0^+$y 0+

  • Domain: $x$x is any real number

  • Range: $y>0$y>0

Key features:

  • Exponential decay: As the $x$x-values increase, the $y$y-values decrease at a decreasing rate.

  • The $y$y-intercept is $\left(0,1\right)$(0,1), since when $x=0$x=0$y=a^0=1$y=a0=1, for any positive value $a$a.

  • $y=0$y=0 is a horizontal asymptote for each graph. As $x\ \rightarrow\infty$x , $y\ \rightarrow0^+$y 0+

  • Domain: $x$x is any real number

  • Range: $y>0$y>0

How did the graph of $y=\left(\frac{1}{2}\right)^x$y=(12)x compare to that of $y=2^x$y=2x? Can you see they are a reflection of each other across the $y$y-axis? In general, for $a>0$a>0 the graph of $g\left(x\right)=\left(\frac{1}{a}\right)^x$g(x)=(1a)x is equivalent to $g\left(x\right)=a^{-x}$g(x)=ax, which is a decreasing exponential function and a reflection of the graph $f\left(x\right)=a^x$f(x)=ax across the $y$y-axis. 

 

Transformations of exponential functions

Let's look at the graphs of $f\left(x\right)=a^x$f(x)=ax and $y=A\times a^{b\left(x-h\right)}+k$y=A×ab(xh)+k and the impact the parameters have on the key features. Use the applet below to observe the impact of $A$A$b$b$h$h and $k$k for a particular $a$a value:

Summary

To obtain the graph of $y=A\times a^{b\left(x-h\right)}+k$y=A×ab(xh)+k from the graph of $y=a^x$y=ax:

  • $A$A dilates (stretches) the graph by a factor of $A$A from the $x$x-axis, parallel to the $y$y-axis
  • When $A<0$A<0 the graph is reflected across the $x$x-axis
  • $b$b dilates (stretches) the graph by a factor of $\frac{1}{b}$1b from the $y$y-axis, parallel to the $x$x-axis
  • When $b<0$b<0 the graph is reflected across the $y$y-axis
  • $h$h translates the graph $h$h units horizontally, the graph shifts $h$h units to the right when $h>0$h>0 and $|h|$|h| units to the left when $h<0$h<0
  • $k$k translates the graph $k$k units vertically, the graph shifts $k$k units upwards when $k>0$k>0 and $|k|$|k| units downwards when $k<0$k<0

We can see in particular, the vertical translation by $k$k units causes the horizontal asymptote to become $y=k$y=k.

 

Practice questions

Question 1

Consider the function $y=\left(\frac{1}{2}\right)^x$y=(12)x

  1. Which two functions are equivalent to $y=\left(\frac{1}{2}\right)^x$y=(12)x ?

    $y=\frac{1}{2^x}$y=12x

    A

    $y=2^{-x}$y=2x

    B

    $y=-2^x$y=2x

    C

    $y=-2^{-x}$y=2x

    D
  2. Sketch a graph of $y=2^x$y=2x on the coordinate plane.

    Loading Graph...

  3. Using the result of the first part, sketch $y=\left(\frac{1}{2}\right)^x$y=(12)x on the same coordinate plane.

    Loading Graph...

Question 2

Consider the function $y=8^{-x}+6$y=8x+6.

  1. What value is $8^{-x}$8x always greater than?

    $0$0

    A

    $1$1

    B

    $8$8

    C
  2. Hence what value is $8^{-x}+6$8x+6 always greater than?

    $6$6

    A

    $14$14

    B

    $8$8

    C
  3. Hence how many $x$x-intercepts does $y=8^{-x}+6$y=8x+6 have?

  4. State the equation of the asymptote of the curve $y=8^{-x}+6$y=8x+6.

  5. What is the domain of the function?

    $x<0$x<0

    A

    $x>0$x>0

    B

    $x>6$x>6

    C

    all real $x$x

    D
  6. What is the range of the function?

Question 3

Beginning with the equation $y=6^x$y=6x, fill in the gaps to find the equation of the new function that results from the given transformations.

  1. The function is dilated by a factor of $5$5 vertically. We get the equation:

    $\editable{}$

    The new function is then translated $3$3 unit up. The resulting equation is: $\editable{}$

  2. What is the horizontal asymptote of the new function?

  3. What is the $y$y-intercept of the new function?

  4. Using the previous parts, pick the correct graph for $y=5\times6^x+3$y=5×6x+3.

    Loading Graph...

    A

    Loading Graph...

    B

    Loading Graph...

    C

    Loading Graph...

    D

Question 4

The graph represents an exponential function that has the form $y=a^x+k$y=ax+k.

Loading Graph...

  1. State the equation for the horizontal asymptote.

  2. State the equation of the function.

Question 5

The graph represents an exponential function that has the form $y=a^{x-h}+k$y=axh+k.

Loading Graph...

  1. State the value of $k$k for this function.

  2. Determine the value of $h$h for this function.

  3. State the equation for the exponential function.

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