Question

The options are A,B,C,D can we make this quick please I am in a rush to turn this in!! thank you so much.

Answer 1

The function we have is:

$y=-x+4$

First, we need to find the rate of change of this function and then we can compare it with the rate of change of each option.

To find the rate of change, we compare the given equation with the general slope-intercept equation:

$y=mx+b$

Where m is the slope, also called the rate of change and b is the y-intercept.

By comparing the two equations, we find that the rate of change is:

$m=-1$

So now we will analyze the given options to see in which of them we find a rate of change of -1.

Option A:

In this option (and in option B) we have a table of values for x and y.

We calculate the rate of change by taking two (x,y) points from the table,

Here, we will take the first two (x,y) values and label them as follows:

$\begin{gathered} x_1=-4 \\ y_1=1 \\ x_2=-2 \\ y_2=2 \end{gathered}$

And we calculate the rate of change "m" using the slope formula:

$\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ \end{gathered}$

Substituting the values we get:

$m=\frac{2-1}{-2-(-4)}$

Solving the operations:

$\begin{gathered} m=\frac{1}{-2+4} \\ m=-\frac{1}{2} \end{gathered}$

The rate of change if NOT -1, this option is not correct.

Option B. We do the same as in the first option.

Label the first two (x,y) values as follows:

$\begin{gathered} x_1=4 \\ y_1=5 \\ x_2=8 \\ y_2=8 \end{gathered}$

And use the slope formula:

$m=\frac{y_2-y_1}{x_2-x_1}$

Substituting the values:

$\begin{gathered} m=\frac{8-5}{8-4} \\ m=\frac{3}{4} \end{gathered}$

Again, the slope or rate of change is NOT -1, this is also not the option we are looking for,

Option C. In options, C and D we have a graph. To find the rate of change from the graph of a line, we take two points where the line passes, and find the rate of change as follows:

$m=\frac{\text{change in y}}{change\text{ in x}}$

For the graph in C, we will take the following red points

Drawing a triangle between the points we can find the change in y and the change in x:

$\begin{gathered} \text{change in y=-1} \\ \text{change in x=1} \end{gathered}$

Thus, the rate of change is:

$\begin{gathered} m=-\frac{1}{1} \\ m=-1 \end{gathered}$

C is the correct option.