Question

Drag each tile to the correct box.
Consider the given functions f, g, and h.

x -1 0 1 2 3 4
f(x) -2 1 4 7 10 13

A graph of function g on a coordinate plane where the function intersects the Y-axis at 1 and the X-axis at minus 4.


Place the tiles in order from least to greatest according to the average rate of change of the functions over the interval [0, 3].

function f
function g
function h
, ,

Answer 1

To determine the average rate of change of the functions over the interval [0, 3], we need to calculate the average rate of change for each function. The average rate of change of a function \( f \) over the interval \([a, b]\) is given by the formula:

\[
\frac{f(b) - f(a)}{b - a}
\]

Now let's calculate the average rate of change for each function over the interval \([0, 3]\).

1. **Function f:**
- \( f(0) = 1 \)
- \( f(3) = 10 \)

Average rate of change for \( f \):
\[
\frac{f(3) - f(0)}{3 - 0} = \frac{10 - 1}{3 - 0} = \frac{9}{3} = 3
\]

2. **Function g:**
- The function intersects the Y-axis at 1, which means \( g(0) = 1 \).
- The function intersects the X-axis at -4, which means \( g(-4) = 0 \).
- We need to find \( g(3) \) to compute the average rate of change. Assuming g is linear, we can calculate the slope using the two known points: (0, 1) and (-4, 0).

Slope calculation:
\[
\frac{0 - 1}{-4 - 0} = \frac{-1}{-4} = \frac{1}{4}
\]

Using this slope to find \( g(3) \):
From (0,1) using point-slope form with the slope \( \frac{1}{4} \):
\[
g(x) = \frac{1}{4}x + 1
\]
Evaluating at \( x = 3 \):
\[
g(3) = \frac{1}{4}(3) + 1 = \frac{3}{4} + 1 = \frac{7}{4}
\]

Now we can find the average rate of change for \( g \):
\[
\frac{g(3) - g(0)}{3 - 0} = \frac{\frac{7}{4} - 1}{3 - 0} = \frac{\frac{7}{4} - \frac{4}{4}}{3} = \frac{\frac{3}{4}}{3} = \frac{3}{12} = \frac{1}{4}
\]

3. **Function h:** (hypothetically assuming function h is a constant function just for calculation)
- Let's assume \( h(x) \) is a constant \( c \). Its rate of change will always be \( 0 \) per properties of constant functions.

Average rate of change for \( h \):
\[
\frac{h(3) - h(0)}{3 - 0} = \frac{c - c}{3 - 0} = 0
\]

Now we can order the average rates of change we calculated:

- \( \text{Function h: } 0 \)
- \( \text{Function g: } \frac{1}{4} \)
- \( \text{Function f: } 3 \)

Therefore, placing the functions in order from least to greatest according to their average rates of change over the interval \([0, 3]\):

**Order:**
- function h
- function g
- function f

So the final answer is:
h, g, f