The image shows four graphs representing linear functions with their equations:

1. $f(x) = -2x - 5$
2. $f(x) = \frac{2}{5}x + 6$
3. $f(x) = -4x + 2$
4. $f(x) = \frac{4}{3}x - 3$

Each graph illustrates the slope and y-intercept from the corresponding equation. Let's break this down:

Key Details for Each Equation:

3. $f(x) = -2x - 5$:

• Slope ($m$): $-2$ (line descends steeply).
• Y-Intercept ($b$): $-5$ (crosses y-axis at $-5$).
• The graph correctly starts at $(0, -5)$ and decreases by 2 units vertically for every 1 unit horizontally.

4. $f(x) = \frac{2}{5}x + 6$:

• Slope ($m$): $\frac{2}{5}$ (line ascends gradually).
• Y-Intercept ($b$): $6$ (crosses y-axis at $6$).
• The graph begins at $(0, 6)$ and rises 2 units vertically for every 5 units horizontally.

5. $f(x) = -4x + 2$:

• Slope ($m$): $-4$ (steeply descending line).
• Y-Intercept ($b$): $2$ (crosses y-axis at $2$).
• The graph starts at $(0, 2)$ and decreases by 4 units vertically for every 1 unit horizontally.

6. $f(x) = \frac{4}{3}x - 3$:

• Slope ($m$): $\frac{4}{3}$ (moderately ascending line).
• Y-Intercept ($b$): $-3$ (crosses y-axis at $-3$).
• The graph begins at $(0, -3)$ and rises 4 units vertically for every 3 units horizontally.

Questions to Expand Understanding:

1. How can you use the slope-intercept form to determine a line’s direction?
2. What is the impact of the slope's sign (positive or negative) on a graph?
3. How can you identify the y-intercept from a given graph without the equation?
4. How does the steepness of the slope compare between $m = \frac{4}{3}$ and $m = -4$?
5. Can you rewrite these equations in standard form $Ax + By = C$, and why is this useful?

Tip:

To quickly sketch a line, start at the y-intercept, then use the slope to plot subsequent points (rise/run). For example, $m = -2$ means go down 2 and right 1.