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!