Lesson 6.2 : Graphs of Exponential Functions

New Concept

Graphing exponential functions like $f(x) = b^x$ gives us a powerful visual tool for making predictions. We'll explore the parent graph's shape and then see how transformations alter its position and key characteristics.

What’s next

Next up are interactive examples and practice cards where you'll graph parent functions and apply various transformations to master this skill.

Property

An exponential function with the form $f(x) = b^x$, $b > 0$, $b \neq 1$, has these characteristics:

• one-to-one function
• horizontal asymptote: $y = 0$
• domain: $(-\infty, \infty)$
• range: $(0, \infty)$
• x-intercept: none
• y-intercept: $(0, 1)$
• increasing if $b > 1$
• decreasing if $b < 1$

To graph the function, create a table of points, plot at least 3 points including the y-intercept $(0, 1)$, and draw a smooth curve.

Examples

• Sketch a graph of $f(x) = 3^x$. This is an exponential growth function. Key points include the y-intercept $(0, 1)$ and $(1, 3)$. The domain is $(-\infty, \infty)$, the range is $(0, \infty)$, and the horizontal asymptote is $y = 0$.

Property

For any constants $c$ and $d$, the function $f(x) = b^{x+c} + d$ shifts the parent function $f(x) = b^x$

• vertically $d$ units, in the same direction of the sign of $d$.
• horizontally $c$ units, in the opposite direction of the sign of $c$.
• The $y$-intercept becomes $(0, b^c + d)$.
• The horizontal asymptote becomes $y = d$.
• The range becomes $(d, \infty)$.
• The domain, $(-\infty, \infty)$, remains unchanged.

Examples

• Graph $f(x) = 2^{x+3}$. This shifts the parent function $y=2^x$ to the left by 3 units. The horizontal asymptote remains $y=0$, but the y-intercept becomes $(0, 8)$.

• Graph $g(x) = 3^x - 2$. This shifts the parent function $y=3^x$ down by 2 units. The horizontal asymptote becomes $y=-2$, and the y-intercept is $(0, -1)$.

Property

For any factor $a > 0$, the function $f(x) = a(b^x)$

• is stretched vertically by a factor of $a$ if $|a| > 1$.
• is compressed vertically by a factor of $a$ if $|a| < 1$.
• has a $y$-intercept of $(0, a)$.
• has a horizontal asymptote at $y = 0$, a range of $(0, \infty)$, and a domain of $(-\infty, \infty)$, which are unchanged from the parent function.

Examples

• Sketch a graph of $f(x) = 5(2^x)$. This stretches the parent function $y=2^x$ vertically by a factor of 5. The y-intercept becomes $(0, 5)$, while the horizontal asymptote remains $y=0$.

• Sketch a graph of $g(x) = \frac{1}{3}(4^x)$. This compresses the parent function $y=4^x$ vertically by a factor of $\frac{1}{3}$. The y-intercept becomes $(0, \frac{1}{3})$.

Property

The function $f(x) = -b^x$

• reflects the parent function $f(x) = b^x$ about the $x$-axis.
• has a $y$-intercept of $(0, -1)$.
• has a range of $(-\infty, 0)$.
• has a horizontal asymptote at $y = 0$ and domain of $(-\infty, \infty)$.

The function $f(x) = b^{-x}$

• reflects the parent function $f(x) = b^x$ about the $y$-axis.
• has a $y$-intercept of $(0, 1)$ and a range of $(0, \infty)$.

Examples

• Graph the function $g(x) = -(4^x)$. This reflects the parent function $f(x)=4^x$ about the x-axis. The range becomes $(-\infty, 0)$, and the y-intercept is $(0, -1)$.

Property

A translation of an exponential function has the form

Where the parent function, $y = b^x$, $b > 1$, is

• shifted horizontally $c$ units to the left.
• stretched vertically by a factor of $|a|$ if $|a| > 0$.
• compressed vertically by a factor of $|a|$ if $0 < |a| < 1$.
• shifted vertically $d$ units.
• reflected about the $x$-axis when $a < 0$.

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

Examples

• Write the equation for $f(x) = e^x$ that is vertically stretched by a factor of 3, reflected across the y-axis, and shifted up 2 units. The equation is $g(x) = 3e^{-x} + 2$.

• Write the equation for $f(x) = 10^x$ that is reflected across the x-axis, shifted left 2 units, and shifted down 5 units. The equation is $h(x) = -10^{x+2} - 5$.