Linear Function Constant Rate of Change

5.15.2 Linear Function Constant Rate of Change

5.15.2 • Linear Function Constant Rate of Change

Activity

Here is a graph of $y = f (x)$ where $f (x) = 2 x + 5$ .

Line graph showing a straight line with positive slope passing through points (1,5), (2,7), (3,9), (4,11), and (5,13) on an x-y grid, with labeled axes x (horizontal) for 5 and 10 with grid lines at the unit, and y (vertical) at 5, 10 and 15 with grid lines for each unit.

1. How do the values of $f$ change whenever $x$ increases by 1, for instance, when it increases from 1 to 2, or from 19 to 20? Be prepared to explain or show how you know.

Compare your answer:

The value of $f$ increases by 2 whenever the input increases by 1. For example:

a. From the graph, we can see the $y$ -value increasing by 2 when the $x$ -value increases by 1.

b. $f (1) = 7$ and $f (2) = 9$ . The difference between $f (1)$ and $f (2)$ is 2. Likewise, $f (20) = 45$ and $f (19) = 43$ , and $f (20) - f (19) = 2$ .

2. Here is an expression we can use to find the difference in the values of $f$ when the input changes from $x$ to $x + 1$ .

$[2 (x + 1) + 5] - [2 x + 5]$

Does this expression have the same value as what you found in the previous questions? Show your reasoning.

Compare your answer:

Yes. $2 (x + 1) = 2 x + 2$ . Replacing $2 (x + 1)$ with $2 x + 2$ and performing arithmetic gives $f (x + 1) - f (x) = 2$ .

3. How do the values of $f$ change whenever $x$ increases by 4? Explain or show how you know.

Compare your answer:

The values of $f$ change by 8 if the input is increased by 4. Increasing the input by 4 is the same as increasing the input by 1 four successive times, and each time the value of $f$ increased by 2.

4. Write an expression that shows the change in the values of $f$ when the input value changes from $x$ to $x + 4$ .

Compare your answer:

$f (x + 4) - f (x)$ can be expressed as $[2 (x + 4) + 5] - [2 x + 5]$ .

5. Show or explain how that expression has a value of 8.

Compare your answer:

Applying the distributive property to $[2 (x + 4) + 5] - [2 x + 5]$ gives us: $2 x + 8 + 5 - 2 x - 5$ or $2 x - 2 x + 8 + 5 - 5$ , which equals 8.

Self Check

Here is a graph of $y= f(x)$ where $f(x) = 3x+2$ .

$GRAPH OF AN EXPONENTIAL GROWTH FUNCTION WITH A $y$-intercepts OF 1 AND PASSING THROUGH THE POINTS 1 COMMA 2, 2 COMMA 4, AND 3 COMMA 8.$

How do the values of $f$ change whenever $x$ increases by 2?

The values of $f$ increase by 1 whenever $x$ increases by 2.
The values of $f$ increase by 6 whenever $x$ increases by 2.
The values of $f$ increase by 3 whenever $x$ increases by 2.
The values of $f$ increase by 2 whenever $x$ increases by 2.

Additional Resources

Determining Rate of Change of Linear Functions

For the table below, assume the function $f$ is defined for all real numbers. Calculate $△ f = f (x + 1) - f (x)$ in the last column. (The symbol $△$ in this context means “change in.”) What do you notice about $△ f$ ? Could the function be linear or exponential? Write a linear or an exponential function formula that generates the same input–output pairs as given in the table.

$x$	$f (x)$	$△ f = f (x + 1) - f (x)$
1	-3
2	1
3	5
4	9
5	13

For each row, you subtract $f (x)$ values. The first row would be $1 - (- 3) = 4$ , then $5 - 1 = 4$ , then $9 - 5 = 4$ , and then $13 - 9 = 4$ . Think back to the lessons on the slope or rate of change of a linear function. The numerator was always the change in $y$ or output, which is the same as the change in $f (x)$ . Notice that all the changes in consecutive terms are the same, 4. This means that there is a constant rate of change and this particular function is linear.

Since this function is linear, an equation will have the form $f (x) = r a t e o f c h a n g e \cdot x + i n i t i a l / v a l u e$ . The rate of change is 4. The initial value is the same as the $y$ -intercept, or where $x = 0$ . $f (0)$ would be 4 less than the first term listed in this case, which means that $f (0) = - 7$ . The linear equation for this table would be $f (x) = 4 x - 7$ .

Try it

Determining Rate of Change of Linear Functions

For the graph provided, assume that the function is defined for all real numbers. What is the rate of change? Write an equation that would define this function.

Graph of a linear function with a y-intercepts of 2 and passing through the points 2 comma 3, 4 comma 4, 6 comma 5, and 8 comma 6.

Compare your answer:

Here is how to write the equation of the graph:

The graph seems to be linear. If you find the rate of change between consecutive points, you find the graph is linear and the rate of change is $\frac{1}{2}$ since the $y$ -values increase by 1 every time the $x$ -values increase by 2. Since the graph crosses the $y$ -axis at $(0, 2)$ , that is the initial value. The equation for this graph is $f (x) = \frac{1}{2} x + 2$ .