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Standard level

4.01 Exponential functions

Lesson

A base form of an exponential function is $f\left(x\right)=a^x$f(x)=ax, where $a$a is a positive number and the variable is in the exponent. Unlike linear functions which increase or decrease by a constant, exponential functions increase or decrease by a constant multiplier. Let's first look at cases for $a>1$a>1, where we have exponential growth and identify key characteristics of such functions.

 

Graphs of $y=a^x$y=ax for $a>1$a>1

Let's create a table for the function $y=2^x$y=2x:

$x$x $-4$4 $-3$3 $-2$2 $-1$1 $0$0 $1$1 $2$2 $3$3 $4$4
$y$y $\frac{1}{16}$116 $\frac{1}{8}$18 $\frac{1}{4}$14 $\frac{1}{2}$12 $1$1 $2$2 $4$4 $8$8 $16$16

We can see our familiar powers of two and as $x$x increases by one, the $y$y values are increasing by a constant multiplier - here they are doubling. This causes the differences between successive $y$y values to grow and hence, $y$y is increasing at an increasing rate. Let's look at what this function looks like when we graph it.

Key features:

  • 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, $y=2^0=1$y=20=1.
  • As $x$x becomes a larger and larger negative number, $y$y becomes a smaller and smaller fraction. In the graph we see as $x$x becomes a larger negative number the graph approaches but does not reach the line $y=0$y=0. Hence, $y=0$y=0 is a horizontal asymptote. We can write this asymptotic behaviour mathematically as follows: As $x\ \rightarrow-\infty,y\ \rightarrow0^+$x ,y 0+ (As $x$x approaches negative infinity, $y$y approaches zero from above).
  • Domain: $x$x is any real number.
  • Range: $y>0$y>0

How does this compare to other values of $a$a? Let's graph $y=2^x$y=2x, $y=3^x$y=3x and $y=5^x$y=5x on the same graph. You can create a table for each to confirm the values sketched in the graph below:

We can see all of the key features mentioned above were not unique to the graph of $y=2^x$y=2x.

Key features:

  • 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 $x\ \rightarrow-\infty,y\ \rightarrow0^+$x ,y 0+ 
  • Domain: $x$x is any real number.
  • Range: $y>0$y>0

The difference is that for $x>0$x>0 the higher the $a$a value the faster the graph increases. Each graph goes through the point $\left(1,a\right)$(1,a) and we can see the larger the $a$a value the higher this point will be.

For $x<0$x<0 the higher the $a$a value the quicker the graph approaches the horizontal asymptote. 

 

Practice questions

question 1

Consider the function $y=3^x$y=3x.

  1. Complete the table of values.

    $x$x $-5$5 $-4$4 $-3$3 $-2$2 $-1$1 $0$0 $1$1 $2$2 $3$3 $5$5 $10$10
    $y$y $\frac{1}{243}$1243 $\frac{1}{81}$181 $\frac{1}{27}$127 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
  2. Is $y=3^x$y=3x an increasing function or a decreasing function?

    Increasing

    A

    Decreasing

    B
  3. How would you describe the rate of increase of the function?

    As $x$x increases, the function increases at a constant rate.

    A

    As $x$x increases, the function increases at a faster and faster rate.

    B

    As $x$x increases, the function increases at a slower and slower rate.

    C
  4. What is the domain of the function?

    all real $x$x

    A

    $x\ge0$x0

    B

    $x<0$x<0

    C

    $x>0$x>0

    D
  5. What is the range of the function?

question 2

Consider the graph of the equation $y=4^x$y=4x.

Loading Graph...
A plot of $y=4^x$y=4x on a Cartesian plane is an upward-sloping curve that represents exponential growth. As x increases, the y values rise rapidly. The graph passes through the point (0, 1), since $4^0=1$40=1, and approaches the x-axis asymptotically from above as x decreases, but never touches the x-axis. The curve is smooth and continuous.
  1. What can we say about the $y$y-value of every point on the graph?

    The $y$y-value of most points of the graph is greater than $1$1.

    A

    The $y$y-value of every point on the graph is positive.

    B

    The $y$y-value of every point on the graph is an integer.

    C

    The $y$y-value of most points on the graph is positive, and the $y$y-value at one point is $0$0.

    D
  2. As the value of $x$x gets large in the negative direction, what do the values of $y$y approach but never quite reach?

    $4$4

    A

    $-4$4

    B

    $0$0

    C
  3. What do we call the horizontal line $y=0$y=0, which $y=4^x$y=4x gets closer and closer to but never intersects?

    A horizontal asymptote of the curve.

    A

    An $x$x-intercept of the curve.

    B

    A $y$y-intercept of the curve.

    C

 

Graphs of $y=a^x$y=ax for $00<a<1

Let's create a table for the function $y=\left(\frac{1}{2}\right)^x$y=(12)x:

$x$x $-4$4 $-3$3 $-2$2 $-1$1 $0$0 $1$1 $2$2 $3$3 $4$4
$y$y $16$16 $8$8 $4$4 $2$2 $1$1 $\frac{1}{2}$12 $\frac{1}{4}$14 $\frac{1}{8}$18 $\frac{1}{16}$116

Again we can see our familiar powers of two but this time as $x$x increases by one the $y$y values are decreasing by a constant multiplier - here they are halving. The differences between successive $y$y values is shrinking and hence, $y$y is decreasing at a decreasing rate. Let's look at what this this function looks like when we graph it.

Key features:

  • As the $x$x-values increase, the $y$y-values decrease at decreasing rate.
  • The $y$y-intercept is still $\left(0,1\right)$(0,1), since when $x=0$x=0, $y=\left(\frac{1}{2}\right)^0=1$y=(12)0=1.
  • As $x$x becomes a larger and larger positive number, $y$y becomes a smaller and smaller fraction. So again we have $y=0$y=0 as a horizontal asymptote, however this time the graph approaches this line as $x$x gets larger. We can write this asymptotic behaviour mathematically as follows: As $x\ \rightarrow\infty,y\ \rightarrow0^+$x ,y 0+ (As $x$x approaches infinity, $y$y approaches zero from above).

And as before:

  • Domain: $x$x is any real number.
  • Range: $y>0$y>0

All graphs of the form $y=a^x$y=ax where $00<a<1 will have these similar key features. They will all be exponential decreasing (decaying) functions, since our multiplier is a fraction.

How did the graph and table of $y=\left(\frac{1}{2}\right)^x$y=(12)x compare that of $y=2^x$y=2x? Can you see they are a reflection of each other in the $y$y-axis? The values in the tables were reversed and the $y$y-value for $y=\left(\frac{1}{2}\right)^x$y=(12)x at $x=k$x=k was the same as $y=2^x$y=2x at $x=-k$x=k. We can see why this is the case by using our index laws to rewrite $y=\left(\frac{1}{2}\right)^x$y=(12)x as follows:

Let $g(x)=\left(\frac{1}{2}\right)^x$g(x)=(12)x and $f(x)=2^x$f(x)=2x

$g(x)$g(x) $=$= $\left(\frac{1}{2}\right)^x$(12)x
  $=$= $\left(2^{-1}\right)^x$(21)x
  $=$= $\left(2\right)^{-x}$(2)x
  $=$= $f\left(-x\right)$f(x)

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 $f(x)=a^x$f(x)=ax in the $y$y-axis. 

 

Practice questions

Question 3

Consider the graphs of the functions $y=4^x$y=4x and $y=\left(\frac{1}{4}\right)^x$y=(14)x.

Loading Graph...

Loading Graph...
  1. Which function is an increasing function?

    $y=\left(\frac{1}{4}\right)^x$y=(14)x

    A

    $y=4^x$y=4x

    B
  2. How would you describe the rate of increase of $y=4^x$y=4x?

    $y$y is increasing at a constant rate

    A

    $y$y is increasing at a decreasing rate

    B

    $y$y is increasing at an increasing rate

    C

Question 4

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 5

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

  1. Can the value of $y$y ever be negative?

    Yes

    A

    No

    B
  2. As the value of $x$x increases towards $\infty$ what value does $y$y approach?

    $8$8

    A

    $-\infty$

    B

    $\infty$

    C

    $0$0

    D
  3. As the value of $x$x decreases towards $-\infty$, what value does $y$y approach?

    $0$0

    A

    $\infty$

    B

    $8$8

    C

    $-\infty$

    D
  4. Can the value of $y$y ever be equal to $0$0?

    Yes

    A

    No

    B
  5. Determine the $y$y-value of the $y$y-intercept of the curve.

  6. How many $x$x-intercepts does the curve have?

  7. Which of the following could be the graph of $y=8^{-x}$y=8x?

    Loading Graph...

    A

    Loading Graph...

    B

    Loading Graph...

    C

    Loading Graph...

    D

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