Question

when yoshi graphed the lines $y = 2x + 3$ and $y = 2x - 2$, she got the graph shown at right.
a. one of the lines above matches the equation $y = 2x + 3$, and the other matches and $y = 2x - 2$. which line matches which equation?
b. yoshi wants to add the line and $y = 2x + 1$ to her graph. predict where it would lie and sketch a graph to show its position. justify your prediction.
c. where would the line $y = -2x + 1$ lie? again, justify your prediction and add the graph of this line to your graph from part (b).

Explanation:

Step1: Identify slope-intercept form

The line equation form is $y=mx+b$, where $m$=slope, $b$=y-intercept.

Step2: Match lines to equations (part a)

For $y=2x+3$, $b=3$ (crosses y-axis at $(0,3)$): this is line $b$.
For $y=2x-2$, $b=-2$ (crosses y-axis at $(0,-2)$): this is line $a$.

Step3: Predict $y=2x+1$ position (part b)

$y=2x+1$ has $m=2$ (parallel to given lines) and $b=1$. Its y-intercept $(0,1)$ is between $(0,-2)$ and $(0,3)$, so it lies between lines $a$ and $b$.

Step4: Predict $y=-2x+1$ position (part c)

$y=-2x+1$ has $m=-2$ (negative slope, downward left to right) and $b=1$ (crosses y-axis at $(0,1)$). It will cross the y-axis at $(0,1)$ and slope downward.

Answer:

a. Line $b$ matches $y=2x+3$; Line $a$ matches $y=2x-2$
b. The line $y=2x+1$ will lie between lines $a$ and $b$. It is parallel to the other two lines (same slope $2$) and has a y-intercept of $1$, which is between the y-intercepts $-2$ and $3$ of the given lines.
c. The line $y=-2x+1$ will cross the y-axis at $(0,1)$ and slope downward from left to right (negative slope $-2$). It will intersect $y=2x+1$ at the point $(0,1)$.