Question

which steps should be used to graph the equation below?
$y - 4 = \frac{1}{3}(x + 2)$
\bigcirc 1. plot the point (2, 4).
\quad \\ 2. from that point, count left 3 units and down 1 unit and plot a second point.
\quad \\ 3. draw a line through the two points.
\bigcirc 1. plot the point (2, 4).
\quad \\ 2. from that point, count left 1 unit and down 3 units and plot a second point.
\quad \\ 3. draw a line through the two points.
\bigcirc 1. plot the point (-2,4).
\quad \\ 2. from that point, count left 3 units and down 1 unit and plot a second point.
\quad \\ 3. draw a line through the two points.
\bigcirc 1. plot the point (-2,4).

Explanation:

The equation \( y - 4=\frac{1}{3}(x + 2) \) is in point - slope form \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(-2,4) \) and the slope \( m = \frac{1}{3}=\frac{\text{rise}}{\text{run}} \). A slope of \( \frac{1}{3} \) means for a run (change in \( x \)) of 3 units (left or right), the rise (change in \( y \)) is 1 unit (up or down). If we start at the point \( (-2,4) \), to find another point, we can use the slope. The slope \( \frac{1}{3} \) can also be thought of as moving left 3 units (since moving left is a negative change in \( x \)) and down 1 unit (negative change in \( y \)) (because \( \frac{- 1}{-3}=\frac{1}{3} \)). Let's analyze each option:

• Option 1: The point \( (2,4) \) is incorrect as the point from the point - slope form is \( (-2,4) \), so this option is wrong.
• Option 2: The point \( (2,4) \) is incorrect, and also the slope interpretation (left 1, down 3) is wrong for a slope of \( \frac{1}{3} \), so this option is wrong.
• Option 3: The point \( (-2,4) \) is correct. Using the slope \( \frac{1}{3} \), moving left 3 units (change in \( x=-3 \)) and down 1 unit (change in \( y = - 1 \)) gives a new point, and then drawing a line through the two points is the correct way to graph the line.
• The last option (the fourth one, though not fully shown) likely has an incorrect slope movement or point, but from the first three, option 3 is correct.

Answer:

1. Plot the point \((-2,4)\).
2. From that point, count left 3 units and down 1 unit and plot a second point.
3. Draw a line through the two points. (This is the third option in the given choices)