Point Slope Form Practice Worksheet | PDF | Mathematical Objects ... - Free Printable

Let's solve each part of this Point-Slope Form Practice Worksheet step by step. The point-slope form of a line is:

$$
y - y_1 = m(x - x_1)
$$

Where:
- $ (x_1, y_1) $ is a point on the line,
- $ m $ is the slope.

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Part 1: Write an equation in point-slope form given a point and slope

We'll plug the values directly into the formula.

#### ① $ (2, 7); m = -4 $
$$
y - 7 = -4(x - 2)
$$

#### ② $ (12, 5); m = -3 $
$$
y - 5 = -3(x - 12)
$$

#### ③ $ (4, -5); m = 6 $
$$
y + 5 = 6(x - 4)
$$

#### ④ $ (-6, -2); m = 3 $
$$
y + 2 = 3(x + 6)
$$

#### ⑤ $ (7, -6); m = \frac{1}{3} $
$$
y + 6 = \frac{1}{3}(x - 7)
$$

#### ⑥ $ (-8, 2); m = -\frac{3}{4} $
$$
y - 2 = -\frac{3}{4}(x + 8)
$$

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Part 2: Graph the equations below

These are already in point-slope form. We'll identify a point and use the slope to graph.

#### ⑦ $ y + 3 = -2(x - 2) $

Rewrite: $ y - (-3) = -2(x - 2) $ → Point: $ (2, -3) $, Slope: $ -2 $

- Start at $ (2, -3) $
- Use slope $ -2 = \frac{-2}{1} $: go down 2, right 1

#### ⑧ $ y + 3 = -2(x - 2) $ → Wait! This is same as ⑦? Let’s check.

Actually, it's listed twice — probably a typo. But assuming:

⑧ $ y + 3 = -2(x - 2) $ → Same as ⑦.

But let's assume it's different. Looking at the list:

Wait — actually:

⑦ $ y + 3 = -2(x - 2) $
⑧ $ y - 1 = 3(x + 6) $
⑨ $ y + 4 = -\frac{5}{3}(x - 3) $

So correct:

#### ⑦ $ y + 3 = -2(x - 2) $
- Point: $ (2, -3) $
- Slope: $ -2 $
- Plot $ (2, -3) $, then go down 2, right 1

#### ⑧ $ y - 1 = 3(x + 6) $
- Point: $ (-6, 1) $
- Slope: $ 3 $
- Plot $ (-6, 1) $, then up 3, right 1

#### ⑨ $ y + 4 = -\frac{5}{3}(x - 3) $
- Point: $ (3, -4) $
- Slope: $ -\frac{5}{3} $
- Plot $ (3, -4) $, then down 5, right 3

#### ⑩ $ y + 4 = -\frac{5}{3}(x - 3) $ → Same as ⑨?

Wait — likely a duplication. But looking at worksheet, maybe just ⑦–⑩ are four separate graphs.

Assuming:

⑩ $ y + 4 = -\frac{5}{3}(x - 3) $ → Same as ⑨? Possibly typo.

But let’s proceed with what’s written.

So for graphing:
- Use the point from the equation.
- Use the slope to plot another point.
- Draw the line.

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Part 3: Write an equation in point-slope form from the graph (use the right-hand point)

We need to determine the slope and use the right-hand point.

Let’s analyze each graph.

#### ⑪ Graph: Line going down from left to right

From grid:
- Right-hand point: appears to be $ (3, 1) $
- Left-hand point: $ (0, 4) $
- So change in $ y = 1 - 4 = -3 $, change in $ x = 3 - 0 = 3 $
- Slope $ m = \frac{-3}{3} = -1 $

Equation using $ (3, 1) $:
$$
y - 1 = -1(x - 3)
$$

#### ⑫ Graph: Line going steeply up

Points:
- Right-hand point: $ (2, 4) $
- Left-hand point: $ (0, -2) $
- $ \Delta y = 4 - (-2) = 6 $, $ \Delta x = 2 - 0 = 2 $
- Slope $ m = \frac{6}{2} = 3 $

Equation:
$$
y - 4 = 3(x - 2)
$$

#### ⑬ Graph: Line going up gently

Right-hand point: $ (4, 2) $
Left-hand point: $ (0, 0) $
$ \Delta y = 2 - 0 = 2 $, $ \Delta x = 4 - 0 = 4 $
Slope $ m = \frac{2}{4} = \frac{1}{2} $

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

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Part 4: Write an equation in point-slope form through two points (use first point)

We need to find slope first, then use the first point.

#### ⑭ $ (4, 7) $ and $ (5, 1) $

Slope:
$$
m = \frac{1 - 7}{5 - 4} = \frac{-6}{1} = -6
$$

Use $ (4, 7) $:
$$
y - 7 = -6(x - 4)
$$

#### ⑮ $ (9, -2) $ and $ (-3, 2) $

Slope:
$$
m = \frac{2 - (-2)}{-3 - 9} = \frac{4}{-12} = -\frac{1}{3}
$$

Use $ (9, -2) $:
$$
y + 2 = -\frac{1}{3}(x - 9)
$$

#### ⑯ $ (3, -8) $ and $ (7, -2) $

Slope:
$$
m = \frac{-2 - (-8)}{7 - 3} = \frac{6}{4} = \frac{3}{2}
$$

Use $ (3, -8) $:
$$
y + 8 = \frac{3}{2}(x - 3)
$$

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✔ Final Answers:

#### Part 1:
1. $ y - 7 = -4(x - 2) $
2. $ y - 5 = -3(x - 12) $
3. $ y + 5 = 6(x - 4) $
4. $ y + 2 = 3(x + 6) $
5. $ y + 6 = \frac{1}{3}(x - 7) $
6. $ y - 2 = -\frac{3}{4}(x + 8) $

#### Part 2: Graphs (instructions):
- Use point and slope to draw lines. For example:
- ⑦: Point $ (2, -3) $, slope $ -2 $
- ⑧: Point $ (-6, 1) $, slope $ 3 $
- ⑨: Point $ (3, -4) $, slope $ -\frac{5}{3} $
- ⑩: Same as ⑨? Possibly typo.

#### Part 3: From graphs:
11. $ y - 1 = -1(x - 3) $
12. $ y - 4 = 3(x - 2) $
13. $ y - 2 = \frac{1}{2}(x - 4) $

#### Part 4: Two points:
14. $ y - 7 = -6(x - 4) $
15. $ y + 2 = -\frac{1}{3}(x - 9) $
16. $ y + 8 = \frac{3}{2}(x - 3) $

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Let me know if you'd like the graphs sketched or explained further!