Question

Review of the Coordinate Plane & Slope Practice

Answer 1

Sure! Here's a brief review of the coordinate plane, slope, and some practice problems related to both topics.

### Coordinate Plane

1. **Definition**: The coordinate plane is a two-dimensional surface formed by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point on the plane is represented by an ordered pair (x, y), where 'x' is the horizontal position and 'y' is the vertical position.

2. **Quadrants**: The coordinate plane is divided into four quadrants:
- **Quadrant I**: (x > 0, y > 0)
- **Quadrant II**: (x < 0, y > 0)
- **Quadrant III**: (x < 0, y < 0)
- **Quadrant IV**: (x > 0, y < 0)

3. **Points**: Points on the coordinate plane are usually denoted as (x, y).

### Slope

1. **Definition**: The slope of a line represents the rate of change of y with respect to x. It is a measure of the steepness or incline of the line.

2. **Formula**: The slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

3. **Interpretation**:
- A positive slope indicates the line rises from left to right.
- A negative slope indicates the line falls from left to right.
- A slope of 0 indicates a horizontal line.
- An undefined slope (when x2 = x1) indicates a vertical line.

### Practice Problems

Here are some practice problems you can try:

1. **Plotting Points**: Plot the following points on the coordinate plane:
- A(2, 3)
- B(-4, 5)
- C(-2, -3)
- D(1, -1)

2. **Finding Slope**: Calculate the slope of the line passing through the points:
- (1, 2) and (3, 6)
- (-2, -1) and (2, 3)

3. **Identifying Quadrants**: Determine which quadrant each of the following points is in:
- P(5, -2)
- Q(-3, 4)
- R(-1, -5)
- S(0, 2)

4. **Line Equation from Slope**: Given a slope of 2 and a y-intercept of -3, write the equation of the line in slope-intercept form.

5. **Finding Points on a Line**: If a line has a slope of -1 and passes through the point (3, 2), what is the equation of the line?

### Solutions

1. **Plotting Points**: This involves plotting on graph paper or a digital graphing tool.

2. **Finding Slope**:
- For (1, 2) and (3, 6):
\[
m = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2
\]
- For (-2, -1) and (2, 3):
\[
m = \frac{3 - (-1)}{2 - (-2)} = \frac{4}{4} = 1
\]

3. **Identifying Quadrants**:
- P(5, -2) - Quadrant IV
- Q(-3, 4) - Quadrant II
- R(-1, -5) - Quadrant III
- S(0, 2) - On the y-axis (not in any quadrant)

4. **Line Equation from Slope**: The slope-intercept form is \(y = mx + b\).
\[
y = 2x - 3
\]

5. **Finding Points on a Line**: Using point-slope form, the equation is:
\[
y - 2 = -1(x - 3)
\]
Simplifying gives:
\[
y = -x + 5
\]

Feel free to reach out if you need more explanations or additional problems!