imath - more exercises, re. sketching linear graphs. - Free Printable

Let's solve each part of the problem step by step.

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Problem 1: Calculate the intercepts and sketch graphs

We are given linear functions in the form $ y = mx + c $. For each, we'll find:
- x-intercept: Set $ y = 0 $ and solve for $ x $
- y-intercept: Set $ x = 0 $ and solve for $ y $

Then we'll sketch a rough graph based on these points.

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#### 1.1 $ y = 2x + 3 $

- y-intercept: Let $ x = 0 $
$$
y = 2(0) + 3 = 3 \Rightarrow (0, 3)
$$
- x-intercept: Let $ y = 0 $
$$
0 = 2x + 3 \Rightarrow 2x = -3 \Rightarrow x = -\frac{3}{2} \Rightarrow \left(-\frac{3}{2}, 0\right)
$$

✔ Intercepts:
- x-intercept: $ \left(-\frac{3}{2}, 0\right) $
- y-intercept: $ (0, 3) $

📌 Graph: Line with positive slope (increasing), crossing y-axis at 3 and x-axis at $-1.5$

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#### 1.2 $ y = 3x - 5 $

- y-intercept: $ x = 0 $
$$
y = 3(0) - 5 = -5 \Rightarrow (0, -5)
$$
- x-intercept: $ y = 0 $
$$
0 = 3x - 5 \Rightarrow 3x = 5 \Rightarrow x = \frac{5}{3} \Rightarrow \left(\frac{5}{3}, 0\right)
$$

✔ Intercepts:
- x-intercept: $ \left(\frac{5}{3}, 0\right) $
- y-intercept: $ (0, -5) $

📌 Graph: Increasing line, crosses y-axis below origin, x-axis at ~1.67

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#### 1.3 $ y = -4x + 3 $

- y-intercept: $ x = 0 $
$$
y = -4(0) + 3 = 3 \Rightarrow (0, 3)
$$
- x-intercept: $ y = 0 $
$$
0 = -4x + 3 \Rightarrow 4x = 3 \Rightarrow x = \frac{3}{4} \Rightarrow \left(\frac{3}{4}, 0\right)
$$

✔ Intercepts:
- x-intercept: $ \left(\frac{3}{4}, 0\right) $
- y-intercept: $ (0, 3) $

📌 Graph: Decreasing line, crosses y-axis at 3, x-axis at 0.75

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#### 1.4 $ y = -3x - 1 $

- y-intercept: $ x = 0 $
$$
y = -3(0) - 1 = -1 \Rightarrow (0, -1)
$$
- x-intercept: $ y = 0 $
$$
0 = -3x - 1 \Rightarrow 3x = -1 \Rightarrow x = -\frac{1}{3} \Rightarrow \left(-\frac{1}{3}, 0\right)
$$

✔ Intercepts:
- x-intercept: $ \left(-\frac{1}{3}, 0\right) $
- y-intercept: $ (0, -1) $

📌 Graph: Decreasing line, crosses y-axis at -1, x-axis at $-0.33$

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> ✔ Summary Table for Problem 1:

| Function | x-intercept | y-intercept |
|--------|-------------|-------------|
| $ y = 2x + 3 $ | $ (-1.5, 0) $ | $ (0, 3) $ |
| $ y = 3x - 5 $ | $ (1.67, 0) $ | $ (0, -5) $ |
| $ y = -4x + 3 $ | $ (0.75, 0) $ | $ (0, 3) $ |
| $ y = -3x - 1 $ | $ (-0.33, 0) $ | $ (0, -1) $ |

Sketches would be straight lines passing through these two points for each function.

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Problem 2: Determine equations from graphs

Given graphs of linear functions $ f(x) = mx + c $, determine $ m $ (slope) and $ c $ (y-intercept), then write equation.

Use formula for slope:
$$
m = \frac{y_2 - y_1}{x_2 - x_1}
$$

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#### 2.1

Points: $ (-2, 0) $ and $ (0, 8) $

- $ c = $ y-intercept = 8 → $ c = 8 $
- $ m = \frac{8 - 0}{0 - (-2)} = \frac{8}{2} = 4 $

✔ Equation: $ y = 4x + 8 $

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#### 2.2

Points: $ (0, 6) $ and $ (3, 0) $

- $ c = 6 $
- $ m = \frac{0 - 6}{3 - 0} = \frac{-6}{3} = -2 $

✔ Equation: $ y = -2x + 6 $

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#### 2.3

Points: A = $ \left(-\frac{2}{3}, 0\right) $, B = $ (0, 2) $

- $ c = 2 $
- $ m = \frac{2 - 0}{0 - (-\frac{2}{3})} = \frac{2}{\frac{2}{3}} = 2 \cdot \frac{3}{2} = 3 $

✔ Equation: $ y = 3x + 2 $

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#### 2.4

Points: A = $ \left(\frac{4}{3}, 0\right) $, B = $ (0, 4) $

- $ c = 4 $
- $ m = \frac{4 - 0}{0 - \frac{4}{3}} = \frac{4}{-\frac{4}{3}} = -3 $

✔ Equation: $ y = -3x + 4 $

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> ✔ Summary Table for Problem 2:

| Graph | Points Used | Slope (m) | Intercept (c) | Equation |
|-------|-------------|-----------|---------------|----------|
| 2.1 | (-2,0), (0,8) | 4 | 8 | $ y = 4x + 8 $ |
| 2.2 | (0,6), (3,0) | -2 | 6 | $ y = -2x + 6 $ |
| 2.3 | (-2/3,0), (0,2) | 3 | 2 | $ y = 3x + 2 $ |
| 2.4 | (4/3,0), (0,4) | -3 | 4 | $ y = -3x + 4 $ |

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✔ Final Answers

#### Problem 1: Intercepts

| Function | x-intercept | y-intercept |
|--------|-------------|-------------|
| $ y = 2x + 3 $ | $ \left(-\frac{3}{2}, 0\right) $ | $ (0, 3) $ |
| $ y = 3x - 5 $ | $ \left(\frac{5}{3}, 0\right) $ | $ (0, -5) $ |
| $ y = -4x + 3 $ | $ \left(\frac{3}{4}, 0\right) $ | $ (0, 3) $ |
| $ y = -3x - 1 $ | $ \left(-\frac{1}{3}, 0\right) $ | $ (0, -1) $ |

#### Problem 2: Equations

| Graph | Equation |
|-------|----------|
| 2.1 | $ y = 4x + 8 $ |
| 2.2 | $ y = -2x + 6 $ |
| 2.3 | $ y = 3x + 2 $ |
| 2.4 | $ y = -3x + 4 $ |

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📌 How to Sketch the Graphs

For each function:
1. Plot the y-intercept (where $ x = 0 $).
2. Plot the x-intercept (where $ y = 0 $).
3. Draw a straight line through both points.
4. Use the sign of $ m $ to know direction:
- Positive $ m $: upward slope
- Negative $ m $: downward slope

For example:
- $ y = 2x + 3 $: goes from bottom-left to top-right, passes through $ (0,3) $ and $ (-1.5,0) $
- $ y = -4x + 3 $: goes from top-left to bottom-right, passes through $ (0,3) $ and $ (0.75,0) $

You can now draw all four graphs accordingly.

Let me know if you'd like ASCII sketches or help plotting them!