Q4. - Algebra 1

Question

Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper.

If the graph of a line has a positive slope and a negative $y$-intercept, what happens to the $x$–intercept if the slope and the $y$–intercept are doubled?

F The $x$–intercept becomes four times as great.

G The $x$–intercept becomes twice as great.

H The $x$–intercept becomes one-fourth as great.

J The $x$–intercept remains the same.

Step-by-Step Solution

Answer

The $x$-intercept remains the same. Therefore, the option J is correct.

Step 1. Write the slope-intercept form of a line.

The equation of line in slope-intercept form is:

$y=mx+b$

Where $m$ is the slope of the line and $b$ is the $y$-intercept of the line.

The $x$-intercept of the line having equation $y=mx+b$ is given by:

$$\begin{gathered} 0=mx+b \\ -b=mx \\ \frac{-b}{m}=x \\ -\frac{y-\text{intercept}}{slope}=x \end{gathered}$$

Therefore, the $x$-intercept of the line having equation $y=mx+b$ is given by $-\frac{y-\text{intercept}}{slope}$.

Step 2. Determine what happens to the x –intercept if the slope and the y –intercept are doubled.

Let the initial slope and $y$-intercept of the line be $m$ and $b$.

Therefore, the initial $x$-intercept of the line is:

$$\begin{gathered} \text{initial }x-\text{intercept}=-\frac{y-\text{intercept}}{slope} \\ =-\frac{b}{m} \end{gathered}$$

It is given that the slope and $y$-intercept of the line have been doubled.

Therefore, the final slope and $y$-intercept of the line are $2m$ and $2b$.

Therefore, the final $x$-intercept of the line is:

$$\begin{gathered} \text{final }x-\text{intercept}=-\frac{y-\text{intercept}}{slope} \\ =-\frac{2b}{2m} \\ =-\frac{b}{m} \end{gathered}$$

Therefore, it can be noticed that the initial $x$-intercept of the line is equal to the final $x$-intercept of the line.

Therefore, the $x$-intercept remains the same.

Therefore, the option J is correct.