The graph of an exponential function, f of x, is shown.

Short description, A graph.,Long description,
The graph shows an exponential function that opens up and to the left. The graphed curve passes through the points
(0, 2).
(1, 3).
(2, 5).
(3, 9).
(4, 17).
Question
Move numbers to the blanks to describe how f of x increases and grows over different intervals.
Response area with 5 blank spaces
The function f of x increases by
Blank space 11 empty
over the interval 0 to 1, by
Blank space 12 empty
over the interval of 1 to 2, by
Blank space 13 empty
over the interval of 2 to 3, and by
Blank space 14 empty
over the interval of 3 to 4. Therefore, this exponential function grows by a factor of
Blank space 10 empty
over every interval of 1.

1 answer

To describe how the function \( f(x) \) increases over the specified intervals, we can calculate the changes in the values of \( f(x) \) at the given points.

From the points you provided:

  • \( f(0) = 2 \)
  • \( f(1) = 3 \)
  • \( f(2) = 5 \)
  • \( f(3) = 9 \)
  • \( f(4) = 17 \)

Now, let's calculate the increases:

  1. From \( x = 0 \) to \( x = 1 \): \[ f(1) - f(0) = 3 - 2 = 1 \]

  2. From \( x = 1 \) to \( x = 2 \): \[ f(2) - f(1) = 5 - 3 = 2 \]

  3. From \( x = 2 \) to \( x = 3 \): \[ f(3) - f(2) = 9 - 5 = 4 \]

  4. From \( x = 3 \) to \( x = 4 \): \[ f(4) - f(3) = 17 - 9 = 8 \]

    Now, we can summarize the findings:

  • The function \( f(x) \) increases by 1 over the interval from \( 0 \) to \( 1 \).
  • The function \( f(x) \) increases by 2 over the interval from \( 1 \) to \( 2 \).
  • The function \( f(x) \) increases by 4 over the interval from \( 2 \) to \( 3 \).
  • The function \( f(x) \) increases by 8 over the interval from \( 3 \) to \( 4 \).

Next, to find the growth factor, we can see the increments are related. The increase from \( 1 \) to \( 2 \), from \( 2 \) to \( 3 \), and from \( 3 \) to \( 4 \) show a doubling pattern:

1 → 2 (factor of 2), 2 → 4 (factor of 2), 4 → 8 (factor of 2).

Thus, the growth factor is 2.

Therefore, the answers you are looking for in the blanks are:

  • 10: 2 (growth factor)
  • 11: 1 (from 0 to 1)
  • 12: 2 (from 1 to 2)
  • 13: 4 (from 2 to 3)
  • 14: 8 (from 3 to 4)

So the filled response area is:

The function \( f(x) \) increases by 1 over the interval from \( 0 \) to \( 1 \), by 2 over the interval of \( 1 \) to \( 2 \), by 4 over the interval of \( 2 \) to \( 3 \), and by 8 over the interval of \( 3 \) to \( 4 \). Therefore, this exponential function grows by a factor of 2 over every interval of 1.