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Algebra 1

5.15.2 Linear Function Constant Rate of Change

Algebra 15.15.2 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 ) y = f ( x ) where f ( x ) = 2 x + 5 f ( x ) = 2 x + 5 .

1. How do the values of f f change whenever x 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.

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

[ 2 ( x + 1 ) + 5 ] − [ 2 x + 5 ] [ 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.

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

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

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

Self Check

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

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

  1. The values of f increase by 1 whenever x increases by 2.
  2. The values of f increase by 6 whenever x increases by 2.
  3. The values of f increase by 3 whenever x increases by 2.
  4. 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 f is defined for all real numbers. Calculate △ f = f ( x + 1 ) − f ( x ) △ f = f ( x + 1 ) − f ( x ) in the last column. (The symbol △ △ in this context means “change in.”) What do you notice about △ f △ 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 x f ( x ) f ( x ) △ f = f ( x + 1 ) − 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 ) f ( x ) values. The first row would be 1 − ( − 3 ) = 4 1 − ( − 3 ) = 4 , then 5 − 1 = 4 5 − 1 = 4 , then 9 − 5 = 4 9 − 5 = 4 , and then 13 − 9 = 4 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 y or output, which is the same as the change in f ( x ) 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 · x + i n i t i a l / v a l u e f ( x ) = r a t e   o f   c h a n g e · 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 y -intercept, or where x = 0 x = 0 . f ( 0 ) f ( 0 ) would be 4 less than the first term listed in this case, which means that f ( 0 ) = − 7 f ( 0 ) = − 7 . The linear equation for this table would be f ( x ) = 4 x − 7 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.

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