Question

problem 1
suppose you wanted to graph the equation $y = -4x - 1$.
a. describe the steps you would take to draw the graph.
b. how would you check that the graph you drew is correct?

Explanation:

For part a:

1. Identify the slope-intercept form $y=mx+b$, where $m$ is the slope and $b$ is the y-intercept. For $y=-4x-1$, $m=-4$ and $b=-1$.
2. Plot the y-intercept first: this is the point $(0, -1)$ on the y-axis.
3. Use the slope to find a second point: the slope $-4$ means $\frac{\text{change in } y}{\text{change in } x} = \frac{-4}{1}$. From $(0,-1)$, move 1 unit right and 4 units down to get the point $(1, -5)$.
4. Draw a straight line through the two plotted points, extending it across the graph.

For part b:

1. Select an arbitrary $x$-value not used to plot the original points (e.g., $x=-1$).
2. Substitute this value into the equation to calculate the corresponding $y$-value: $y=-4(-1)-1=4-1=3$, so the point is $(-1, 3)$.
3. Check if this calculated point lies on the drawn line. If it does, the graph is likely correct; repeat with another $x$-value for confirmation.

Answer:

a. 1. Identify $m=-4$, $b=-1$ from $y=mx+b$.

1. Plot y-intercept $(0, -1)$.
2. Use slope to plot $(1, -5)$.
3. Draw line through both points.

b. 1. Pick an unused $x$-value (e.g., $x=-1$).

1. Calculate $y$ using the equation: $y=-4(-1)-1=3$.
2. Verify the point $(-1, 3)$ lies on the line; repeat if needed.