Question

Determine if the function below is linear or exponential then fill in the guided sentences below.
xx 22 33 44 55
f, of, xf(x) minus, 3−3 minus, 13−13 minus, 23−23 minus, 33−33
Answer
Attempt 1 out of 2

The function is
because as xx increases, the yy values change
. The
of this function is exactly
.

Answer 1

To determine whether the function is linear or exponential, we usually look at how the outputs (y-values) change with respect to the inputs (x-values).

Given the information:

- When \( x = 2 \), \( f(x) = -3 \)
- When \( x = 3 \), \( f(x) = -13 \)
- When \( x = 4 \), \( f(x) = -23 \)
- When \( x = 5 \), \( f(x) = -33 \)

Let's calculate the changes in the y-values:

- From \( x = 2 \) to \( x = 3 \): \( -13 - (-3) = -13 + 3 = -10 \)
- From \( x = 3 \) to \( x = 4 \): \( -23 - (-13) = -23 + 13 = -10 \)
- From \( x = 4 \) to \( x = 5 \): \( -33 - (-23) = -33 + 23 = -10 \)

The change in \( f(x) \) (the outputs) is constant (each time it decreases by 10), which indicates that the function is linear.

Now, let's fill in the guided sentences accordingly:

The function is **linear** because as \( x \) increases, the \( y \) values change **by a constant amount**. The **slope** of this function is exactly **-10**.