Equations and Their Graphs

5.12.2 Equations and Their Graphs

5.12.2 • Equations and Their Graphs

Activity

Access the Desmos guide PDF for tips on solving problems with the Desmos graphing calculator.

1. Each of the following functions, $f$ , $g$ , $h$ , and $j$ , represents the amount of money in a bank account, in dollars, as a function of time $x$ , in years. They are each written in the form $m (x) = a \cdot b^{x}$ .

$f (x) = 50 \cdot 2^{x}$
$g (x) = 50 \cdot 3^{x}$
$h (x) = 50 \cdot (\frac{3}{2})^{x}$
$j (x) = 50 \cdot (0.5)^{x}$

a. Use graphing technology to graph each function on the same coordinate plane. It may be helpful to use a different color for each function.

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Compare your answers:

A graph with four labeled curves: f (orange), g (blue), h (green), and j (brown). All intersect near (0, 50); f, g, and h curve upward, while j curves downward as x increases from 0 to 4.

b. Explain how changing the value of $b$ changes the graph.

Compare your answer:

When $b$ is greater than 1, larger values of $b$ mean that the function grows more quickly as $x$ increases. A positive value of $b$ that is less than 1 means the function is decreasing as $x$ increases.

2. Here are equations defining functions $p$ , $q$ , and $r$ . They are also written in the form $m (x) = a \cdot b^{x}$ .

$p (x) = 10 \cdot 4^{x}$
$q (x) = 40 \cdot 4^{x}$
$r (x) = 100 \cdot 4^{x}$

a. Use graphing technology to graph each function on the same coordinate plane. It may be helpful to use a different color for each function.

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Compare your answers:

A graph with three exponential curves labeled p, q, and r, each in a different color, increasing steeply to the right on a grid with x- and y-axes marked, and y-values ranging from 0 to over 200.

b. Explain how changing the value of $a$ changes the graph.

Compare your answer:

It changes the $y$ -intercept to $(0, a)$ .

3. Here are equations defining functions $f$ , $g$ , and $h$ written in the form $m (x) = a \cdot b^{x} + d$ .

$f (x) = 4^{x} + 2$
$g (x) = 4^{x}$
$h (x) = 4^{x} - 2$

a. Use graphing technology to graph each function on the same coordinate plane. It may be helpful to use a different color for each function.

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Compare your answers:

A graph showing three curves: f (red), g (black), and h (teal). All pass through (0,1); f increases fastest, h slowest as x increases. The x- and y-axes are labeled.

b. Discuss with a partner how changing the value of $d$ changes the asymptote of the graph.

Compare your answer:

Changing the value of $d$ changes the horizontal asymptote from $y = 0$ to $y = d$ . So function $f (x)$ has a horizontal asymptote of $y = 2$ , and $h (x)$ has a horizontal asymptote of $y = - 2$ .

Are you ready for more?

Extending Your Thinking

As before, consider bank accounts whose balances are given by the following functions:

$f (x) = 10 \cdot 3^{x}$

$g (x) = 3^{x + 2}$

$h (x) = \frac{1}{2} \cdot 3^{x + 3}$

Which function would you choose? Does your choice depend on $x$ ?

Compare your answer:

We can rewrite each of the functions as multiples of $3^{x}$ to more easily compare them:

$f (x) = 10 \cdot 3^{x}$

$g (x) = 3^{x + 2} = 3^{x} \cdot 3^{2} = 9 \cdot 3^{x}$

$h (x) = \frac{1}{2} \cdot 3^{x + 3} = \frac{1}{2} \cdot 3 x \cdot 3^{3} = \frac{27}{2} \cdot 3^{x} = (13.5) \cdot 3^{x}$

Since $9 < 10 < 13.5$ , we see that $h$ has the most money for any value of $x$ .

Video: Exponential Equations and How They Affect the Graphs

Watch the following video to learn more about exponential functions and how they affect the graphs:

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Self Check

What is the horizontal asymptote for the function

f(x)=2 \cdot 3^x-5

$y=3$
$y=-5$
$y=2$
$y=5$

Additional Resources

Vertical Shifts with Exponential Functions

The first transformation occurs when we add a constant $d$ to the parent function $f (x) = b^{x}$ , giving us a vertical shift $d$ units in the same direction as the sign. For example, if we begin by graphing a parent function, $f (x) = 2^{x}$ , we can then graph two vertical shifts alongside it, using $d = 3$ : the upward shift, $g (x) = 2^{x} + 3$ , and the downward shift, $h (x) = 2^{x} - 3$ . Both vertical shifts are shown in the graph below.

Graph of three exponential growth functions. f of x has a y-intercept of 1 and a horizontal asymptote of y equals 0. g of x has a y-intercept of 4 and a horizontal asymptote of y equals 3. h of x has a y-intercept of negative 2 and a horizontal asymptote of y equals negative 3.

Observe the results of shifting $f (x) = 2^{x}$ vertically:

The domain, $(- \infty, \infty)$ , remains unchanged.
When the function is shifted up 3 units to g ( x ) = 2 x + 3 g ( x ) = 2 x + 3 :
- The $y$ -intercept shifts up 3 units to $(0, 4)$ .
- The asymptote shifts up 3 units to $y = 3$ .
- The range becomes $(3, \infty)$ .
When the function is shifted down 3 units to h ( x ) = 2 x − 3 h ( x ) = 2 x − 3 :
- The $y$ -intercept shifts down 3 units to $(0, - 2)$ .
- The asymptote also shifts down 3 units to $y = - 3$ .
- The range becomes $(- 3, \infty)$ .

Try it

Vertical Shifts with Exponential Functions

Where would the horizontal asymptote be located for the graph of $f (x) = 4 \cdot 2^{x} - 6$ ?

Compare your answer:

Here is how to find the equation of the horizontal asymptote:

In the function $f (x) = 4 \cdot 2^{x} - 6$ , the vertical shift is represented by –6.

The parent function, $f (x) = 2^{x}$ , has a horizontal asymptote at $y = 0$ . For the function $f (x) = 4 \cdot 2^{x} - 6$ , the horizontal asymptote will be shifted down 6 to $y = - 6$ .