Question

the function is linear because as x increases, the y values change of this function is exactly -10. (with a table of x and f(x) values: x=2, f(x)=-3; x=3, f(x)=-13; x=4, f(x)=-23; x=5, f(x)=-33)

Explanation:

Step1: Identify the type of function

To determine if the function is linear, we check the rate of change (slope) between consecutive points. For a linear function, the slope should be constant.

Step2: Calculate the slope between first two points

Using the formula for slope \( m=\frac{y_2 - y_1}{x_2 - x_1} \), for \( (x_1,y_1)=(2, - 3) \) and \( (x_2,y_2)=(3,-13) \), we have \( m=\frac{-13-(-3)}{3 - 2}=\frac{-10}{1}=-10 \).

Step3: Verify slope with other points

Check between \( (3,-13) \) and \( (4,-23) \): \( m=\frac{-23-(-13)}{4 - 3}=\frac{-10}{1}=-10 \). And between \( (4,-23) \) and \( (5,-33) \): \( m=\frac{-33-(-23)}{5 - 4}=\frac{-10}{1}=-10 \). Since the slope is constant (\(-10\)), the function is linear, and the common difference/slope is \(-10\).

Answer:

The function is linear because as \( x \) increases, the \( y \)-values change with a common difference/slope of \(-10\).