2.3+Linear+Functions+and+Slope

Section 2.3 Linear Functions and Slope

Objectives:

  • Calculate a line’s slope.

  • Write the point-slope form of the equation of a line.

  • Write and graph the slope-intercept form of the equation of a line.

  • Graph horizontal or vertical lines.

  • Recognize and use the general form of a line’s equation.

  • Use intercepts to graph the general form of a line’s equation.

  • Model data with linear functions and make predictions.

Definition of Slope

The slope of a line quantifies its steepness and direction. It is calculated by taking two distinct points on a line:

  • Point 1: (x1, y1)

  • Point 2: (x2, y2)

Slope Calculation:

  • Formula: Slope = rise/run = (y2 - y1) / (x2 - x1)

  • Example: For points (1, 1) and (2, 2), the slope calculation is (2 - 1) / (2 - 1) = 1.

  • Standard notation for slope is denoted as m, which represents the steepness of a line.

Possibilities for a Line’s Slope

A line can have different types of slopes:

  • Positive slope: Rises from left to right, indicating that as x increases, y also increases.

  • Negative slope: Falls from left to right, indicating that as x increases, y decreases.

  • Zero slope: A horizontal line where y does not change as x changes, represented by the equation y = c.

  • Undefined slope: A vertical line where x does not change as y changes, represented by the equation x = c.

Exercise 1

Task: Compute the slope for points (2, 3) and (4, -12).

Slope-Intercept Form

The slope-intercept form of a line is utilized to express equations conveniently:

  • Formula: y = mx + b

    • Where:

      • m = slope

      • b = y-intercept (the point where the line crosses the y-axis)

Exercise 2

Task: Determine the slope for the equation 4 + 6 - 7 = 11.

Point-Slope Form

The point-slope form is particularly useful for writing equations of lines that pass through a specific point:

  • Formula: y - y1 = m(x - x1)

    • Where:

      • m = slope

      • (x1, y1) = coordinates of a specific point on the line.

Exercise 3

Task: Write the equation in slope-intercept form for a line with a slope of 2 passing through the point (10, -9).

Exercise 4

Task: Write the slope-intercept form for a line passing through points (6,-9) and (-6,-5).

Graphing y = mx + b

To graph a linear equation in slope-intercept form:

  1. Plot the y-intercept point (0, b).

  2. Use the slope (expressed as rise/run) to obtain a second point starting from the y-intercept.

  3. Plot this second point.

  4. Use a straightedge to draw the line, adding arrowheads to show that it extends indefinitely.

Exercise 5

Task: Graph the linear function (3/15)x + 1.

Exercise 6

Task: Use the slope-intercept form to graph 4 - 5 = -7 - 2 + 6.

Horizontal and Vertical Lines

  • Horizontal lines have:

    • Form: y = c

    • Slope: 0

  • Vertical lines have:

    • Form: x = c

    • Slope: undefined.

Exercise 7

Task: Graph the horizontal line y = 3 in the rectangular coordinate system.

General Form of the Equation of a Line

Every line can be expressed in the general form:

  • Equation: Ax + By = C, where A, B, and C are real numbers, and A and B are not both zero.

  • Example: An example of a general form equation is 3x + 4y = 12.

Using Intercepts to Graph

To graph a line from its general form:

  1. Find the x-intercept by letting y = 0 and solving for x; plot (x, 0).

  2. Find the y-intercept by letting x = 0 and solving for y; plot (0, y).

  3. Connect the intercept points with a straightedge, indicating the line extends infinitely.

Exercise 8

Task: Find the x- and y-intercepts for the equation 4x + 3y = 24 and graph it.

Exercise 9

Task: Find the x- and y-intercepts for the equation 2(-x) + 1 = 6 - 15 and graph it.