Analyze Underlying Relationship in a Graph of a Function

5.9.2 Analyze Underlying Relationship in a Graph of a Function

5.9.2 • Analyze Underlying Relationship in a Graph of a Function

Activity

The cost, in dollars, to produce 1 watt of solar energy is a function of the number of years since 1977, $t$ .

From 1977 to 1987, the cost could be modeled by an exponential function $f$ . Here is the graph of the function.

Line graph showing a sharp decline in dollars per watt over 12 years since 1977, dropping from about $80 at the y-intercept to below $5 per watt. The curve smooths as it approaches the lower values.

1. What is the statement $f (9) \approx 6$ saying about this situation?

Compare your answer:

9 years after 1977 (or in 1986), the cost in dollars for one watt of solar energy was about $6.

2. What is $f (4)$ ? What about $f (3.5)$ ? What do these values represent in this context?

Compare your answer:

$f (4) \approx 25$ , and $f (3.5) \approx 30$ . These values represent the cost in dollars of 1 watt of solar energy in 1981 (4 years after 1977) and halfway through 1980 (3.5 years after 1977).

3. When $f (t) = 45$ , what is $t$ ? What does that value of $t$ represent in this context?

Compare your answer:

2 years after 1977 (in 1979), the cost in dollars of 1 watt of solar energy was $45.

4. By what factor did the cost of 1 watt of solar energy change each year? (If you get stuck, consider creating a table.)

Compare your answer:

$\frac{3}{4}$ or 0.75

Video: Understanding Characteristics of Graphs

Watch the following video to learn more about the characteristics of graphs:

Access multimedia content

An imaginary line that a function value will not touch or cross is called an asymptote.

The value of the range that the function will never touch is called the horizontal asymptote.

In the function graphed, the horizontal asymptote is $y = 0$ .

Self Check

The graph below shows the total smartphones, $s$ , in millions that were sold in the years, $t$ , since 2008. What does the point $(4, 125)$ represent in this context?

A graph. Total smartphone sales in millions. Years since 2008.

In 2004, there were 125 million more smartphones than in 2000.
In 2012, there were 125 million smartphones sold.
In 2012, there were 125 million more smartphones than in 2008.
In 2004, there were 125 million smartphones sold.

Additional Resources

Reading Exponential Graphs

The graph below describes the amount of caffeine, $c$ , in a person’s body $t$ hours after an initial measurement of 100 mg.

A graph showing caffeine (mg) decreasing over time (hours); caffeine starts at 100 mg and gradually declines as time increases from 0 to 14 hours. A point P is marked on the curve.

What does the point $c (10) = 35$ mean?

After 10 hours, there is 35 mg of caffeine in a person’s body.

What is the value of $t$ when $c (t) = 80$ ?

When the caffeine concentration is 80 mg, time is 2 hours, so $t = 2$ .

Try it

Reading Exponential Graphs

Using the graph above, what is the value of $c (5)$ , and what does it mean in the context of the situation?

Compare your answer:

Here is how to interpret this exponential graph:

When $t = 5$ , the curve crosses at 60, so $c (5) = 60$ .

This means that after 5 hours, the person has 60 mg of caffeine in their body.