Question

1. graph the equation $y = 2x - 4$ and label the line with the equation.

a. predict how the graph of $y = x - 4$ will compare to the graph of $y = 2x - 4$.
b. predict how the graph of $y = \frac{1}{2}x - 4$ will compare to the graph of $y = 2x - 4$.
c. predict how the graph of $y = -2x - 4$ will compare to the graph of $y = 2x - 4$.
d. graph the following equations and label each line with its equation.
$y = x - 4$
$y = \frac{1}{2}x - 4$
$y = -2x - 4$
e. were your predictions correct? why or why not?
f. describe the movement of a line when the slope is increased or decreased while the y-intercept is held constant.
g. describe the movement of a line when $m$ is changed to $-m$.
h. write the equation of a line that would be steeper than all of the equations in this problem.

Explanation:

Step1: Graph $y=2x-4$

First, identify intercepts:

• y-intercept: when $x=0$, $y=-4$, so point $(0,-4)$
• x-intercept: when $y=0$, $0=2x-4 \implies x=2$, so point $(2,0)$

Plot and connect these points, label the line $y=2x-4$.

Step2: Predict $y=x-4$ vs $y=2x-4$

Both have y-intercept $-4$. Slope of $y=x-4$ is $1$, which is less than $2$.
Conclusion: Same y-intercept, less steep, increasing.

Step3: Predict $y=\frac{1}{2}x-4$ vs $y=2x-4$

Both have y-intercept $-4$. Slope of $y=\frac{1}{2}x-4$ is $\frac{1}{2}$, less than $2$.
Conclusion: Same y-intercept, even less steep, increasing.

Step4: Predict $y=-2x-4$ vs $y=2x-4$

Both have y-intercept $-4$. Slope of $y=-2x-4$ is $-2$, negative opposite of $2$.
Conclusion: Same y-intercept, same steepness, decreasing (falls left to right).

Step5: Graph $y=x-4$, $y=\frac{1}{2}x-4$, $y=-2x-4$

• For $y=x-4$: y-intercept $(0,-4)$, x-intercept $(4,0)$; plot and label.
• For $y=\frac{1}{2}x-4$: y-intercept $(0,-4)$, x-intercept $(8,0)$; plot and label.
• For $y=-2x-4$: y-intercept $(0,-4)$, x-intercept $(-2,0)$; plot and label.

Step6: Verify predictions

All predictions are correct. The slope $m$ controls steepness and direction; same y-intercept means lines share $(0,-4)$, slope magnitude controls steepness, sign controls direction.

Step7: Slope change with fixed y-intercept

When slope magnitude increases, line becomes steeper; when magnitude decreases, line becomes less steep. Positive slope: line rises left to right; negative slope: line falls left to right.

Step8: Slope sign flip ($m$ to $-m$)

The line is reflected across the horizontal line through its y-intercept (or across the y-axis if y-intercept is 0); it maintains the same steepness but reverses direction (rises instead of falls, or vice versa).

Step9: Steeper line equation

Choose slope with magnitude greater than 2, e.g., $m=3$, same y-intercept: $y=3x-4$.

Answer:

1. (Graph of $y=2x-4$: passes through $(0,-4)$ and $(2,0)$, labeled appropriately)
2. a. It shares the y-intercept $(0,-4)$, is less steep, and rises left to right.

b. It shares the y-intercept $(0,-4)$, is even less steep, and rises left to right.
c. It shares the y-intercept $(0,-4)$, has the same steepness, but falls left to right.

1. d. (Graphs:

• $y=x-4$: passes through $(0,-4)$ and $(4,0)$, labeled
• $y=\frac{1}{2}x-4$: passes through $(0,-4)$ and $(8,0)$, labeled
• $y=-2x-4$: passes through $(0,-4)$ and $(-2,0)$, labeled)

1. e. Yes, the predictions are correct. All lines share the y-intercept $-4$, and the slope controls steepness and direction as predicted.
2. f. When the slope magnitude increases, the line becomes steeper; when the magnitude decreases, the line becomes less steep. Positive slopes make the line rise left to right, negative slopes make it fall.
3. g. The line is reflected across the horizontal line through its y-intercept, reversing its direction (rising ↔ falling) while keeping the same steepness.
4. h. $y=3x-4$ (any equation with $|m|>2$ and same y-intercept, e.g., $y=-3x-4$, is valid)