Linear Functions and Graphs: Slope, Equations, and Applications

Linear Functions and Graphs

Introduction

This section introduces the foundational concepts of linear functions, their graphical representations, and the various forms of linear equations. Understanding these concepts is essential for modeling real-world phenomena and solving problems in precalculus.

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

Understanding Slope

The slope of a line measures its steepness and direction. It is defined as the ratio of the vertical change to the horizontal change between two distinct points on the line.

• Formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

• Vertical change (rise): $y_2 - y_1$

• Horizontal change (run): $x_2 - x_1$

Example: Find the slope of the line passing through the points (3, 2) and (–1, 5):

$m = \frac{5 - 2}{-1 - 3} = \frac{3}{-4} = -\frac{3}{4}$

Point-Slope Form of the Equation of a Line

Definition and Application

The point-slope form is useful for writing the equation of a line when you know its slope and a point it passes through.

• Formula:

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

• m: slope of the line

• (x1, y1): a point on the line

Example: Write the equation in point-slope form for the line with slope 6 passing through (2, –5):

$y - (-5) = 6(x - 2)$ $y + 5 = 6(x - 2)$ $y = 6x - 12 - 5$ $y = 6x - 17$

Slope-Intercept Form of the Equation of a Line

Definition and Usage

The slope-intercept form is the most common way to express the equation of a line, especially for graphing.

• Formula:

$y = mx + b$

• m: slope of the line

• b: y-intercept (the value of y when x = 0)

Graphing Using the Slope and y-Intercept

Step-by-Step Procedure

• Step 1: Plot the point containing the y-intercept on the y-axis. This is the point (0, b).

• Step 2: Obtain a second point using the slope, m. Write m as a fraction and use "rise over run" starting at the y-intercept.

• Step 3: Use a straightedge to draw a line through the two points. Draw arrowheads at the ends to indicate the line continues indefinitely.

Example: Graph the linear function $y = \frac{3}{5}x + 1$:

• Plot the y-intercept: (0, 1)

• Slope $m = \frac{3}{5}$: From (0, 1), move up 3 units (rise) and right 5 units (run) to plot (5, 4)

• Draw the line through (0, 1) and (5, 4)

Horizontal and Vertical Lines

Special Cases

• Horizontal Line: Equation is $y = b$ where b is the y-intercept. Slope is zero.

• Vertical Line: Equation is $x = a$ where a is the x-intercept. Slope is undefined.

Example: The line $y = 3$ is horizontal and passes through all points with y-coordinate 3.

Example: The line $x = -2$ is vertical and passes through all points with x-coordinate -2.

General Form of the Equation of a Line

Definition and Properties

The general form of a line is useful for algebraic manipulation and finding intercepts.

• Formula:

$Ax + By + C = 0$

• A, B, and C are real numbers, with A and B not both zero.

Finding Slope and Intercepts from General Form

Procedure

• To find the x-intercept: Set $y = 0$ and solve for $x$.

• To find the y-intercept: Set $x = 0$ and solve for $y$.

• To find the slope: Rearrange to slope-intercept form $y = mx + b$ and identify $m$.

Example: For $2x - 3y = 6$:

• x-intercept: $y = 0 \Rightarrow 2x = 6 \Rightarrow x = 3$

• y-intercept: $x = 0 \Rightarrow -3y = 6 \Rightarrow y = -2$

• Slope: Rearranged, $y = \frac{2}{3}x - 2$ so $m = \frac{2}{3}$

Graphing Using Intercepts

Step-by-Step Method

• Find the x-intercept and plot it on the x-axis.

• Find the y-intercept and plot it on the y-axis.

• Draw a line through both intercepts, extending in both directions.

Example: For $x - 3y = 6$:

• x-intercept: $y = 0 \Rightarrow x = 6$

• y-intercept: $x = 0 \Rightarrow -3y = 6 \Rightarrow y = -2$

• Plot (6, 0) and (0, -2), then draw the line.

Modeling Data with Linear Functions

Application Example

Linear functions can be used to model relationships between variables, such as predicting temperature based on carbon dioxide concentration.

• Given data points: (317, 57.04) and (354, 57.64)

• Find the slope:

$m = \frac{57.64 - 57.04}{354 - 317} = \frac{0.60}{37} \approx 0.016$

• Find the y-intercept (b): Use one point and solve for b in $y = mx + b$

$57.04 = 0.016 \times 317 + b$ $b = 57.04 - (0.016 \times 317)$ $b \approx 57.04 - 5.072 = 51.97$

• Model: $f(x) = 0.016x + 51.97$

This function models average global temperature as a function of atmospheric carbon dioxide concentration.

Summary Table: Forms of Linear Equations

Form	Equation	Key Features
Slope-Intercept	$y = mx + b$	Slope (m), y-intercept (b)
Point-Slope	$y - y_1 = m(x - x_1)$	Slope (m), point $(x_1, y_1)$
General Form	$Ax + By + C = 0$	Intercepts, can be rearranged
Horizontal Line	$y = b$	Slope = 0
Vertical Line	$x = a$	Slope undefined

Additional info: Some steps and examples were expanded for clarity and completeness, based on standard precalculus curriculum.