Linear Functions and Equations of Lines: Precalculus Study Notes

Linear Functions and Their Applications

Introduction to Linear Equations

Linear equations are fundamental in mathematics and are widely used in various fields such as statistics, economics, and science. In precalculus, understanding linear functions is essential for analyzing relationships between variables and modeling real-world scenarios.

• Linear regression models: Used to predict outcomes and analyze trends.

• Calculating rates of change: Slope represents the rate at which one variable changes with respect to another.

• Maximizing profits: Linear models can help optimize business decisions.

• Other applications: Linear functions appear in physics, engineering, and everyday problem-solving.

Definition and Structure of Linear Functions

General Form of a Linear Function

A linear function is a function that can be written in the form:

$f(x) = mx + b$

• x: Input variable (independent variable)

• f(x): Output variable (dependent variable)

• m: Slope of the line

• b: y-intercept of the line

Example: $f(x) = 2x - 1$

• m = 2 (slope)

• b = -1 (y-intercept)

Graphing Linear Functions

Graph of a Linear Function

To graph a linear function, plot points $(x, y)$ that satisfy the equation $y = mx + b$ on an xy-coordinate plane. All such pairs form a continuous straight line. The domain and range of a linear function are typically $(- , 1e)$ unless restricted by context.

Example: The graph of $y = 2x - 1$ is a straight line passing through points that satisfy the equation.

Tables and Linear Functions

Using Tables to Represent Linear Functions

Tables can be used to display input-output pairs for a linear function. Consistent increments in $x$ are common, but not required. Missing values can be found if the table represents a linear function.

x	-2	-1	0	1	2	2.5	6
f(x)	-5	-3	-1	1	3	4	11

Example: Find missing values in a table by asserting the linear relationship and using the formula $f(x) = mx + b$.

Slope of a Line

Definition and Calculation of Slope

The slope ($m$) of a line measures its steepness and direction. It is calculated as the ratio of the change in $y$ to the change in $x$ between two points.

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

Alternatively, $m = \frac{\text{change in } y}{\text{change in } x} = \frac{\Delta y}{\Delta x} = \frac{\text{rise}}{\text{run}}$

• Always match $x_1$ with $y_1$ and $x_2$ with $y_2$.

Example: Given points $(2, 0)$ and $(4, 4)$, $m = \frac{4 - 0}{4 - 2} = 2$.

Finding the y-Intercept

Methods for Determining the y-Intercept

The y-intercept ($b$) is the value of $y$ when $x = 0$. It can be found by:

• Setting $x = 0$ in the equation and solving for $y$.

• Using a known point $(x_1, y_1)$ and the slope $m$:

$b = y_1 - m x_1$

Example: Given slope $m = 4$ and point $(2, 0)$, $b = 0 - 4 d 2 = -8$.

Forms of Linear Equations

Slope-Intercept Form

The slope-intercept form is:

$y = mx + b$

• Variables: $x$ and $y$

• Parameters: $m$ (slope), $b$ (y-intercept)

Point-Slope Form

The point-slope form is:

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

• Variables: $x$ and $y$

• Parameters: $m$ (slope), $(x_1, y_1)$ (a point on the line)

Example: Find the slope-intercept form for the line with slope $-5$ passing through $(-2, 7)$:

Start with point-slope form: $y - 7 = -5(x + 2)$ Expand: $y = -5x - 10 + 7 = -5x - 3$

Parallel and Perpendicular Lines

Parallel Lines

Two lines are parallel if they have the same slope.

Example: $y = 3x + 4$ and $y = 3x - 2$ are parallel because both have slope $3$.

Perpendicular Lines

Two lines are perpendicular if their slopes are opposite reciprocals.

• The opposite of positive is negative; the opposite of negative is positive.

• Reciprocal means "flip the fraction".

Original Slope (m)	Opposite Reciprocal
4	-1/4
-3	1/3
1/5	-5
-3/4	4/3

Example: A line perpendicular to $y = 2x + 7$ has slope $-1/2$.

Horizontal and Vertical Lines

Horizontal Lines

A line is horizontal if and only if its slope is $0$.

Equation: $y = b$

• Example: $y = 0$ (means $b = 0$)

• Example: $y = -10$ (means $b = -10$)

Vertical Lines

A line is vertical if its slope is undefined. The equation is $x = a$, where $a$ is the x-coordinate of all points on the line.

• Example: $x = 2$

• Example: $x = 99$

Practice Problems

Finding Equations of Lines

For each scenario, find the equation of the line:

• The line contains the points $(0, 3)$ and $(1, -1)$.

• The line contains the points $(-1, 2)$ and $(1, 6)$.

• The line contains the point $(0, 3)$ and is perpendicular to the line given by $x + 2y = 7$.

• The vertical line containing the point $(1, 3)$.

• The horizontal line containing the point $(1, 3)$.

Additional info: To solve these, use the slope formula, point-slope form, and knowledge of perpendicular/parallel relationships as described above.