Question

1. graph the equation $y = x + 3$ and label the line with the equation.

a. predict how the graph of $y = x + 1$ will compare to the graph of $y = x + 3$.

b. predict how the graph of $y = x - 3$ will compare to the graph of $y = x + 3$.

c. graph the following equations on the same grid and label each line with its equation.
$y = x + 1$
$y = x - 3$

d. were your predictions correct? why or why not?

e. what is the relationship between the lines $y = x + 3$, $y = x + 1$, and $y = x - 3$?

f. write a different equation that would be parallel to the equations in this problem.

g. describe the movement of a line when $b$ is increased or decreased while $m$ is held constant.

Explanation:

Step1: Graph $y=x+3$

Find intercepts:

• When $x=0$, $y=0+3=3$ → $(0,3)$
• When $y=0$, $0=x+3$ → $x=-3$ → $(-3,0)$

Plot and connect these points, label the line $y=x+3$.

Step2: Predict $y=x+1$ vs $y=x+3$

Both have slope $m=1$. $y=x+1$ has a y-intercept of $1$, which is $2$ units less than $3$.
Prediction: Parallel, shifted down 2 units.

Step3: Predict $y=x-3$ vs $y=x+3$

Both have slope $m=1$. $y=x-3$ has a y-intercept of $-3$, which is $6$ units less than $3$.
Prediction: Parallel, shifted down 6 units.

Step4: Graph $y=x+1$ and $y=x-3$

For $y=x+1$:

• Intercepts: $(0,1)$, $(-1,0)$; plot, connect, label.

For $y=x-3$:

• Intercepts: $(0,-3)$, $(3,0)$; plot, connect, label.

Step5: Verify predictions

All lines have equal slope ($m=1$), so they are parallel. The vertical shifts match predictions:

• $y=x+1$ is 2 units below $y=x+3$
• $y=x-3$ is 6 units below $y=x+3$

Predictions are correct.

Step6: Identify line relationship

All lines share the same slope $m=1$, so they are parallel.

Step7: Write parallel equation

Any line with $m=1$ works, e.g., $y=x+2$.

Step8: Describe $b$ change effect

In $y=mx+b$, fixed $m$:

• Increasing $b$ shifts line up.
• Decreasing $b$ shifts line down.

Answer:

1. (Graph: Line through $(-3,0)$ and $(0,3)$, labeled $y=x+3$)

a. The graph of $y=x+1$ is parallel to $y=x+3$, shifted 2 units downward.
b. The graph of $y=x-3$ is parallel to $y=x+3$, shifted 6 units downward.
c. (Graph: $y=x+1$ through $(-1,0)$ and $(0,1)$; $y=x-3$ through $(3,0)$ and $(0,-3)$, each labeled)
d. Yes, the predictions were correct. All lines have the same slope ($m=1$), so they are parallel, and the vertical shift matches the difference in y-intercepts.
e. The three lines are parallel to each other (they have identical slopes of 1).
f. $\boldsymbol{y=x+2}$ (any equation of the form $y=x+b$ where $b$ is a real number works, e.g., $y=x-1$)
g. When the slope $m$ is constant, increasing the y-intercept $b$ shifts the line upward, and decreasing $b$ shifts the line downward.