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# Function Rules based on Graphs

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What if you were given the coordinate graph of a function and you wanted to find the function rule shown by the graph? How could you write that rule? After completing this Concept, you'll be able to create a table of values from a graph to help you write function rules like this one.

### Guidance

In the last two Concepts, you learned how to graph a function from a table and from a function rule. Now, you will learn how to find coordinate points on a graph and to interpret the meaning. Recall that each point on the graph has an x-value and y-value. When given an x-value, you will be asked to find its y-value.

#### Example A

The students at a local high school took The Youth Risk Behavior Survey. The graph below shows the percentage of high school students who reported that they were current smokers. (A current smoker is anyone who has smoked one or more cigarettes in the past 30 days.) What percentage of high-school students were current smokers in the following years?

1. 1991
2. 2004

Solution:

1. First, find the value 1991 on the x-axis. This appears to be the first red point on the left side of the graph. By looking at the y-axis, it looks like this point has a y-value of approximately 27. This means that in 1991, approximately 27% of high school students reported that they were current smokers.

2. Find the value 2004 on the x-axis, which appears to be between the two red points on the right. This has an approximate y-value of 22.5. This means that in 2004, approximately 22.5% of high school students reported that they were current smokers.

Write a Function Rule from a Graph

Sometimes you’ll need to find the equation or rule of a function by looking at the graph of the function. Points that are on the graph can give you values of dependent and independent variables that are related to each other by the function rule. However, you must make sure that the rule works for all the points on the curve. In this course you will learn to recognize different kinds of functions and discover the rules for all of them. For now we’ll look at some simple examples and find patterns that will help us figure out how the dependent and independent variables are related.

#### Example B

The graph to the right shows the distance that an ant covers over time. Find the function rule that shows how distance and time are related to each other.

Solution

Let’s make a table of values of several coordinate points to see if we can spot how they are related to each other.

$& \text{Time} \qquad \ 0 \quad \ 1 \quad \ \ 2 \quad \ 3 \quad \ 4 \quad \ 5 \quad \ 6\\& \text{Distance} \quad 0 \quad 1.5 \quad 3 \quad 4.5 \quad 6 \quad 7.5 \quad 9$

We can see that for every second the distance increases by 1.5 feet. We can write the function rule as

$\text{Distance} = 1.5 \times \text{time}$

The equation of the function is $f(x) = 1.5x$ .

#### Example C

Find the function rule that describes the function shown in the graph.

Solution

Again, we can make a table of values of several coordinate points to identify how they are related to each other.

$& x \quad -4 \quad -3 \quad -2 \quad -1 \quad 0 \quad 1 \quad \ 2 \quad \ 3 \quad \ 4\\& y \qquad \ 8 \quad \ \ 4.5 \quad \ \ 2 \quad \ \ .5 \quad 0 \quad .5 \quad 2 \quad 4.5 \quad 8$

Notice that the values of $y$ are half of perfect squares: 8 is half of 16 (which is 4 squared), 4.5 is half of 9 (which is 3 squared), and so on. So the equation of the function is $f(x) = \frac{1}{2} x^2$ .

Watch this video for help with the Examples above.

### Vocabulary

Most often, we want to find the function rule or formula that fits the set of values in the table, so we can use the rule to predict what could happen for values that are not in the graph.

### Guided Practice

Find the function rule that shows the volume of a balloon at different times, based on the following graph:

(Notice that the graph shows negative time. The negative time can represent what happened on days before you started measuring the volume.)

Solution

Once again, we make a table to spot the pattern:

$& \text{Time} \quad \ -1 \quad 0 \quad \ 1 \quad \ \ 2 \quad \ \ 3 \qquad 4 \qquad 5\\& \text{Volume} \quad 10 \quad 5 \quad 2.5 \quad 1.2 \quad 0.6 \quad 0.3 \quad 0.15$

We can see that every day, the volume of the balloon is half what it was the previous day. On day 0, the volume is 5; on day 1, the volume is $5 \times \frac{1}{2}$ ; on day 2, it is $5 \times \frac{1}{2} \times \frac{1}{2}$ , and in general, on day $x$ it is $5 \times \left ( \frac{1}{2} \right )^x$ . The equation of the function is $f(x) = 5 \times \left ( \frac{1}{2} \right )^x$ .

### Explore More

For 1-4, the graph below shows the average life-span of people based on the year in which they were born. This information comes from the National Vital Statistics Report from the Center for Disease Control.

1. What is the average life-span of a person born in 1940?
2. What is the average life-span of a person born in 1955?
3. What is the average life-span of a person born in 1980?
4. What is the average life-span of a person born in 1995?

For 5-8, the graph below shows the median income of an individual based on his/her number of years of education. The top curve shows the median income for males and the bottom curve shows the median income for females. (Source: US Census, 2003.)

1. What is the median income of a male that has 10 years of education?
2. What is the median income of a male that has 17 years of education
3. What is the median income of a female that has 10 years of education?
4. What is the median income of a female that has 17 years of education?

For 9-11, write the function rule for each graph.