Essential Concepts
Graphs of Linear Functions
A linear function is a function with a constant rate of change, represented by a polynomial of degree 1. Its graph is always a straight line.
Key Features of Linear Graphs
- Slope (m): The rate of change, showing how steep the line is
- Positive slope: line rises from left to right (increasing function)
- Negative slope: line falls from left to right (decreasing function)
- Zero slope: horizontal line (constant function)
- Undefined slope: vertical line
- y-intercept (b): The point where the line crosses the y-axis, written as [latex](0, b)[/latex]
- Find by setting [latex]x = 0[/latex] in the equation
- Represents the initial value or starting point
- x-intercept: The point where the line crosses the x-axis, written as [latex](a, 0)[/latex]
- Find by setting [latex]y = 0[/latex] and solving for [latex]x[/latex]
- Not all linear functions have an x-intercept (horizontal lines like [latex]y = 5[/latex] do not)
Slope measures how one quantity changes in relation to another:
[latex]m = \dfrac{\text{change in output}}{\text{change in input}} = \dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 - y_1}{x_2 - x_1}[/latex]
The units of slope are “output units per input unit” (like miles per hour or dollars per year).
Graphing by Plotting Points
- Choose at least two input values (three is better to check for errors)
- Evaluate the function at each input
- Plot the coordinate pairs
- Draw a line through the points
Graphing Using Slope and y-intercept
- Find the y-intercept and plot that point
- Use the slope [latex]\frac{\text{rise}}{\text{run}}[/latex] to find additional points
- Draw a line through the points
Linear Functions
Three Forms of Linear Equations
Slope-Intercept Form: [latex]y = mx + b[/latex]
- [latex]m[/latex] is the slope
- [latex]b[/latex] is the y-intercept
- Most common form for graphing
Point-Slope Form: [latex]y - y_1 = m(x - x_1)[/latex]
- [latex]m[/latex] is the slope
- [latex](x_1, y_1)[/latex] is a point on the line
- Useful when you know the slope and one point
Standard Form: [latex]Ax + By = C[/latex]
- [latex]A[/latex], [latex]B[/latex], and [latex]C[/latex] are integers
- [latex]A[/latex] and [latex]B[/latex] are not both zero
- Useful for certain applications
Writing Equations of Lines
Given two points:
- Calculate slope: [latex]m = \frac{y_2 - y_1}{x_2 - x_1}[/latex]
- Use point-slope form with either point
- Simplify to slope-intercept form
Given a graph:
- Identify two clear points on the line
- Calculate the slope
- Identify the y-intercept (or use point-slope form)
- Write the equation
Horizontal and Vertical Lines
- Horizontal line: [latex]y = c[/latex] (slope = 0)
- Vertical line: [latex]x = c[/latex] (slope is undefined)
Parallel and Perpendicular Lines
- Parallel lines: Have the same slope but different y-intercepts
- If line 1 has slope [latex]m[/latex], a parallel line also has slope [latex]m[/latex]
- Perpendicular lines: Have slopes that are negative reciprocals
- If line 1 has slope [latex]m[/latex], a perpendicular line has slope [latex]-\frac{1}{m}[/latex]
- Exception: Horizontal and vertical lines are perpendicular to each other
Parallel and Perpendicular to Horizontal or Vertical Lines:
- A line parallel to [latex]y = c[/latex] is [latex]y = d[/latex] (another horizontal line)
- A line parallel to [latex]x = c[/latex] is [latex]x = d[/latex] (another vertical line)
- A line perpendicular to [latex]y = c[/latex] is [latex]x = d[/latex] (vertical)
- A line perpendicular to [latex]x = c[/latex] is [latex]y = d[/latex] (horizontal)
Linear Models
Building Linear Models from Word Problems
- Identify changing quantities and define variables
- Determine the initial value (y-intercept)
- Determine the rate of change (slope)
- Write the equation using [latex]y = mx + b[/latex]
- Use the model to make predictions or solve problems
Scatter Plots and Lines of Best Fit
- Scatter plot: A graph of data points that may show a relationship between two variables
- Linear relationship: Points form a pattern resembling a straight line
- Line of best fit: A line that best represents the trend in the data
- Can be estimated by sketching
- Can be calculated precisely using linear regression
Interpolation vs. Extrapolation
- Interpolation: Predicting a value inside the domain and range of the data
- Generally more reliable
- Extrapolation: Predicting a value outside the domain and range of the data
- Less reliable; may lead to inaccurate predictions
Absolute Value Functions
Absolute value measures distance from zero on a number line, always non-negative: [latex]|x| = \begin{cases} x & \text{if } x \geq 0 \ -x & \text{if } x < 0 \end{cases}[/latex]
Graphing Absolute Value Functions
The basic absolute value function [latex]f(x) = |x|[/latex] has a V-shape with a corner point at the origin. It can be transformed using:
- Vertical/horizontal shifts
- Vertical/horizontal stretches or compressions
- Reflections across the x-axis or y-axis
Solving Absolute Value Equations
For [latex]|A| = B[/latex] where [latex]B \geq 0[/latex]:
- Set up two equations: [latex]A = B[/latex] or [latex]A = -B[/latex]
- Solve each equation
- Check solutions in the original equation
Special cases:
- If [latex]B < 0[/latex]: No solution (absolute value cannot be negative)
- If [latex]B = 0[/latex]: One solution
- If [latex]B > 0[/latex]: Two solutions (usually)
Solving Absolute Value Inequalities
For [latex]|X| < k[/latex] where [latex]k > 0[/latex]:
- Equivalent to [latex]-k < X < k[/latex]
- Interval notation: [latex](-k, k)[/latex]
For [latex]|X| > k[/latex] where [latex]k > 0[/latex]:
- Equivalent to [latex]X < -k[/latex] or [latex]X > k[/latex]
- Interval notation: [latex](-\infty, -k) \cup (k, \infty)[/latex]
Same logic applies for [latex]\leq[/latex] and [latex]\geq[/latex]:
- [latex]|X| \leq k[/latex] gives [latex][-k, k][/latex]
- [latex]|X| \geq k[/latex] gives [latex](-\infty, -k] \cup [k, \infty)[/latex]
Key Equations
| Slope formula | [latex]m = \frac{y_2 - y_1}{x_2 - x_1}[/latex] |
| Slope-intercept form of a line | [latex]y = mx + b[/latex] |
| Point-slope form of a line | [latex]y - y_1 = m(x - x_1)[/latex] |
| Standard form of a line | [latex]Ax + By = C[/latex] |
Glossary
absolute value
the distance of a number from zero on the number line, always non-negative; denoted [latex]|x|[/latex]
absolute value equation
an equation of the form [latex]|A| = B[/latex] with [latex]B \geq 0[/latex]; has solutions when [latex]A = B[/latex] or [latex]A = -B[/latex]
absolute value inequality
a relationship in the form [latex]|A| < B[/latex], [latex]|A| \leq B[/latex], [latex]|A| > B[/latex], or [latex]|A| \geq B[/latex]
decreasing linear function
a linear function with a negative slope; as input increases, output decreases
extrapolation
predicting a value outside the domain and range of observed data
horizontal line
a line defined by [latex]y = b[/latex] where [latex]b[/latex] is a constant; has slope of 0
increasing linear function
a linear function with a positive slope; as input increases, output also increases
interpolation
predicting a value inside the domain and range of observed data
linear function
a function with a constant rate of change, represented as a polynomial of degree 1; graph is a straight line
line of best fit
a line that best represents the trend in a set of data points; can be found using linear regression
parallel lines
two or more lines with the same slope
perpendicular lines
two lines that intersect at right angles and have slopes that are negative reciprocals of each other
point-slope form
the equation of a line in the form [latex]y - y_1 = m(x - x_1)[/latex] where [latex]m[/latex] is the slope and [latex](x_1, y_1)[/latex] is a point on the line
scatter plot
a graph of plotted points that may show a relationship between two sets of data
slope
the ratio of the change in output to the change in input; measures the steepness and direction of a line; calculated as [latex]m = \frac{y_2 - y_1}{x_2 - x_1}[/latex]
slope-intercept form
the equation of a line in the form [latex]y = mx + b[/latex] where [latex]m[/latex] is the slope and [latex]b[/latex] is the y-intercept
standard form
the equation of a line in the form [latex]Ax + By = C[/latex] where [latex]A[/latex], [latex]B[/latex], and [latex]C[/latex] are integers
vertical line
a line defined by [latex]x = a[/latex] where [latex]a[/latex] is a constant; has undefined slope
x-intercept
the point where a graph crosses the x-axis; has coordinates [latex](a, 0)[/latex]
y-intercept
the point where a graph crosses the y-axis; has coordinates [latex](0, b)[/latex]; also called the initial value