SOLUTION: Key features of linear functions - Studypool - Free Printable

Let's solve each of the four linear equations step by step. For each equation, we will:

1. Identify the slope ($ m $) and y-intercept ($ b $).
2. Find the x-intercept (where $ y = 0 $).
3. Determine the zero of the function (same as x-intercept).
4. Identify the starting point (usually the y-intercept, since that’s where $ x = 0 $).
5. Compute $ f(0) $ (which is the y-intercept).
6. Solve $ f(x) = 0 $ to find the x-value where the graph crosses the x-axis.

We'll also briefly explain how to graph each line on the coordinate plane.

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1) $ y = \frac{1}{2}x - 4 $

- Slope ($ m $): $ \frac{1}{2} $
- y-intercept ($ b $): $ -4 $ → So the line passes through $ (0, -4) $
- x-intercept: Set $ y = 0 $:
$$
0 = \frac{1}{2}x - 4 \Rightarrow \frac{1}{2}x = 4 \Rightarrow x = 8
$$
So x-intercept is $ (8, 0) $
- Zero: The value of $ x $ when $ y = 0 $ → $ x = 8 $
- Starting Point: Usually the y-intercept → $ (0, -4) $
- $ f(0) = \frac{1}{2}(0) - 4 = -4 $
- $ f(x) = 0 $ → $ x = 8 $

✔ Graphing: Start at $ (0, -4) $. Use slope $ \frac{1}{2} $: go up 1, right 2 to get next point.

| Slope | $ m = \frac{1}{2} $ |
|-------|------------------------|
| y-intercept | $ b = -4 $ |
| x-intercept | $ x = 8 $ |
| Zero | $ x = 8 $ |
| Starting Point | $ (0, -4) $ |
| $ f(0) $ | $ y = -4 $ |
| $ f(x) = 0 $ | $ x = 8 $ |

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2) $ y = \frac{1}{2}x $

- Slope ($ m $): $ \frac{1}{2} $
- y-intercept ($ b $): $ 0 $ → Passes through origin $ (0, 0) $
- x-intercept: Set $ y = 0 $:
$$
0 = \frac{1}{2}x \Rightarrow x = 0
$$
So x-intercept is $ (0, 0) $
- Zero: $ x = 0 $
- Starting Point: $ (0, 0) $
- $ f(0) = \frac{1}{2}(0) = 0 $
- $ f(x) = 0 $ → $ x = 0 $

✔ Graphing: Starts at origin. Use slope $ \frac{1}{2} $: up 1, right 2.

| Slope | $ m = \frac{1}{2} $ |
|-------|------------------------|
| y-intercept | $ b = 0 $ |
| x-intercept | $ x = 0 $ |
| Zero | $ x = 0 $ |
| Starting Point | $ (0, 0) $ |
| $ f(0) $ | $ y = 0 $ |
| $ f(x) = 0 $ | $ x = 0 $ |

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3) $ y = 2x + 3 $

- Slope ($ m $): $ 2 $
- y-intercept ($ b $): $ 3 $ → Point: $ (0, 3) $
- x-intercept: Set $ y = 0 $:
$$
0 = 2x + 3 \Rightarrow 2x = -3 \Rightarrow x = -\frac{3}{2} = -1.5
$$
So x-intercept is $ (-1.5, 0) $
- Zero: $ x = -1.5 $
- Starting Point: $ (0, 3) $
- $ f(0) = 2(0) + 3 = 3 $
- $ f(x) = 0 $ → $ x = -1.5 $

✔ Graphing: Start at $ (0, 3) $. Use slope $ 2 $: up 2, right 1.

| Slope | $ m = 2 $ |
|-------|-------------|
| y-intercept | $ b = 3 $ |
| x-intercept | $ x = -1.5 $ |
| Zero | $ x = -1.5 $ |
| Starting Point | $ (0, 3) $ |
| $ f(0) $ | $ y = 3 $ |
| $ f(x) = 0 $ | $ x = -1.5 $ |

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4) $ y = -\frac{5}{4}x + 2 $

- Slope ($ m $): $ -\frac{5}{4} $
- y-intercept ($ b $): $ 2 $ → Point: $ (0, 2) $
- x-intercept: Set $ y = 0 $:
$$
0 = -\frac{5}{4}x + 2 \Rightarrow \frac{5}{4}x = 2 \Rightarrow x = 2 \cdot \frac{4}{5} = \frac{8}{5} = 1.6
$$
So x-intercept is $ (1.6, 0) $
- Zero: $ x = \frac{8}{5} $ or $ 1.6 $
- Starting Point: $ (0, 2) $
- $ f(0) = -\frac{5}{4}(0) + 2 = 2 $
- $ f(x) = 0 $ → $ x = \frac{8}{5} $

✔ Graphing: Start at $ (0, 2) $. Use slope $ -\frac{5}{4} $: down 5, right 4 (or up 5, left 4).

| Slope | $ m = -\frac{5}{4} $ |
|-------|------------------------|
| y-intercept | $ b = 2 $ |
| x-intercept | $ x = \frac{8}{5} $ or $ 1.6 $ |
| Zero | $ x = \frac{8}{5} $ |
| Starting Point | $ (0, 2) $ |
| $ f(0) $ | $ y = 2 $ |
| $ f(x) = 0 $ | $ x = \frac{8}{5} $ |

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✔ Summary Table

| Problem | Equation | Slope (m) | y-int (b) | x-int | Zero | f(0) | f(x)=0 |
|--------|----------|-----------|-----------|--------|------|--------|---------|
| 1 | $ y = \frac{1}{2}x - 4 $ | $ \frac{1}{2} $ | -4 | 8 | 8 | -4 | 8 |
| 2 | $ y = \frac{1}{2}x $ | $ \frac{1}{2} $ | 0 | 0 | 0 | 0 | 0 |
| 3 | $ y = 2x + 3 $ | 2 | 3 | -1.5 | -1.5 | 3 | -1.5 |
| 4 | $ y = -\frac{5}{4}x + 2 $ | $ -\frac{5}{4} $ | 2 | $ \frac{8}{5} $ | $ \frac{8}{5} $ | 2 | $ \frac{8}{5} $ |

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📌 How to Graph Each Line:

For each:
1. Plot the y-intercept.
2. Use the slope to find another point.
3. Draw a straight line through them.

Example for #1:
- Start at $ (0, -4) $
- Rise 1, run 2 → to $ (2, -3) $
- Connect the dots.

You can do this similarly for others.

Let me know if you'd like me to draw the graphs or provide coordinates for plotting!