UK Secondary (7-11)
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Transformations of Exponential Graphs (1 transform only)
Lesson

Any function can be transformed by adding something to it and any function can be transformed by multiplying it by a number. We might write

$f(x)\rightarrow g(x)=f(x)+c$f(x)g(x)=f(x)+c
$f(x)\rightarrow h(x)=af(x)$f(x)h(x)=af(x)

We apply this general principle to exponential functions. Thus, if $f(x)=2^x$f(x)=2x then $g(x)$g(x) might be $2^x+5$2x+5 and $h(x)$h(x) might be $-\frac{1}{2}\cdot2^x$12·2x.

 

Example 1

adding a number

The function $g(x)=2^x+5$g(x)=2x+5 is just the function $f(x)$f(x) with $5$5 added to every function value. The graph of $g(x)$g(x) must look the same as the graph of $f(x)$f(x) but shifted $5$5 units up the vertical axis. The following diagram shows these two functions.

Observe that $f(0)=1$f(0)=1 and $g(0)=1+5=6$g(0)=1+5=6, as expected.

The function $f(x)$f(x) is asymptotic to the horizontal axis and $g(x)$g(x) is asymptotic to the line $y(x)=5$y(x)=5.

At every point $x$x, the distance between $f(x)$f(x) and $g(x)$g(x) is $5$5.

 

Example 2

multiplying by a number

The function  $h(x)=-\frac{1}{2}\cdot2^x$h(x)=12·2x, is the function $f(x)$f(x) with every function value multiplied by $-\frac{1}{2}$12.

Multiplication by $\frac{1}{2}$12 brings all values of $f(x)$f(x) closer to zero by that factor. The graph of $h(x)$h(x) will appear compressed in the vertical direction compared with the graph of $f(x)$f(x).

Since all the values of $f(x)$f(x) are positive, all the values of $h(x)$h(x) must be negative. That is, the graph of $h(x)$h(x) is not only compressed in the vertical direction but is also reflected across the horizontal axis.

The graphs are represented in the following diagram.

Observe that $f(0)=1$f(0)=1 but $h(0)=-\frac{1}{2}$h(0)=12; $f(1)=2$f(1)=2 but $h(1)=-1$h(1)=1; $f(2)=4$f(2)=4 but $h(2)=-2$h(2)=2; and so on, as expected.

Worked Examples

Question 1

Answer the following.

  1. Determine the $y$y-intercept of $y=2^x$y=2x.

  2. Hence or otherwise determine the $y$y-intercept of $y=2^x-2$y=2x2.

  3. Determine the horizontal asymptote of $y=2^x$y=2x.

  4. Hence or otherwise determine the horizontal asymptote of $y=2^x-2$y=2x2.

Question 2

Consider a graph of $y=5^x$y=5x.

  1. How could the graph of $y=-5^x$y=5x be obtained from the graph of $y=5^x$y=5x?

    through a vertical translation

    A

    through a reflection about the $y$y-axis

    B

    through a reflection about the $x$x-axis

    C

    by making it steeper

    D

    through a vertical translation

    A

    through a reflection about the $y$y-axis

    B

    through a reflection about the $x$x-axis

    C

    by making it steeper

    D
  2. Given the graph of $y=5^x$y=5x, sketch $y=-5^x$y=5x on the same coordinate plane.

    Loading Graph...

Question 3

This is a graph of $y=3^x$y=3x.

Loading Graph...

  1. How do we shift the graph of $y=3^x$y=3x to get the graph of $y=3^x-4$y=3x4?

    Move the graph $4$4 units to the right.

    A

    Move the graph downwards by $4$4 units.

    B

    Move the graph $4$4 units to the left.

    C

    Move the graph upwards by $4$4 units.

    D

    Move the graph $4$4 units to the right.

    A

    Move the graph downwards by $4$4 units.

    B

    Move the graph $4$4 units to the left.

    C

    Move the graph upwards by $4$4 units.

    D
  2. Hence plot $y=3^x-4$y=3x4 on the same graph as $y=3^x$y=3x.

    Loading Graph...

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