Linear Functions and Graphing: Study Guide for Intermediate Algebra

Linear Functions and Graphing

Linear vs. Nonlinear Equations

Understanding the distinction between linear and nonlinear equations is fundamental in algebra. Linear equations produce straight lines when graphed, while nonlinear equations result in curves or other shapes.

• Linear Equations: Can be written in the form $y = mx + b$.

• Graph: Always a straight line.

• Examples: $y = 3x - 2$, $f(x) = -5x + 1$, $2x + 3y = 12$

• Nonlinear Equations: Contain powers (e.g., $x^2$), absolute values, or other operations.

• Graph: Not a straight line; may be a curve or V-shape.

• Examples: $y = x^2$, $f(x) = |x|$

Slope (m)

The slope measures the steepness and direction of a line. It is calculated using two points on the line.

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

• Positive slope: Line rises from left to right.

• Negative slope: Line falls from left to right.

• Slope = 0: Horizontal line.

• Undefined slope: Vertical line.

Slope-Intercept Form

The slope-intercept form is a common way to express linear equations, highlighting the slope and y-intercept.

• Formula: $y = mx + b$

• m: Slope

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

• Example: If $f(x) = 5x + 3$, then slope $= 5$, y-intercept $= (0, 3)$

Finding an Equation of a Line Given Slope and a Point

To write the equation of a line when you know the slope and a point, use the slope-intercept form and solve for the y-intercept.

1. Start with $y = mx + b$

2. Plug in the given point $(x, y)$

3. Solve for $b$

• Example: Slope $= 3$, point $(5, 3)$

  • $3 = 3 \times 5 + b$

  • $3 = 15 + b$

  • $b = -12$

  • Equation: $y = 3x - 12$

Writing Equations from Two Points

When two points are given, first find the slope, then use one point to solve for the y-intercept.

1. Find the slope using $m = \frac{y_2 - y_1}{x_2 - x_1}$

2. Use $y = mx + b$

3. Plug in one point to find $b$

• Example: Points $(2, 1)$ and $(5, -8)$

  • $m = \frac{-8 - 1}{5 - 2} = \frac{-9}{3} = -3$

  • Using $(2, 1)$: $1 = -3 \times 2 + b \Rightarrow 1 = -6 + b \Rightarrow b = 7$

  • Equation: $f(x) = -3x + 7$

Standard Form of a Linear Equation

Linear equations can also be written in standard form, which is useful for certain types of analysis and graphing.

• Format: $Ax + By = C$

• No fractions: Coefficients should be integers.

• A is usually positive.

• Example: $y = -2x + 4 \rightarrow 2x + y = 4$

Graphing Lines

Lines can be graphed using the slope and y-intercept or by finding the x- and y-intercepts.

• Using slope & y-intercept:

  1. Plot the y-intercept.

  2. Use the slope (rise/run) to find another point.

  3. Draw the line through the points.

• Using intercepts:

  • x-intercept: Set $y = 0$

  • y-intercept: Set $x = 0$

• Example: $2x + 3y = 12$

  • x-intercept: $y = 0 \Rightarrow 2x = 12 \Rightarrow x = 6$ (point $(6, 0)$)

  • y-intercept: $x = 0 \Rightarrow 3y = 12 \Rightarrow y = 4$ (point $(0, 4)$)

Parallel & Perpendicular Lines

Parallel and perpendicular lines have specific relationships between their slopes.

• Parallel lines: Have the same slope.

• Perpendicular lines: Slopes are negative reciprocals (e.g., $2$and$-\frac{1}{2}$).

Functions & Function Notation

A function assigns each input (x-value) exactly one output (y-value). Function notation is used to express this relationship.

• Function notation: $f(x)$ means the output when $x$ is plugged in.

• Example: If $f(x) = -2x + 4$, then $f(3) = -2 \times 3 + 4 = -2$

• Ordered pair: $f(5) = 10 \rightarrow (5, 10)$

Domain & Range

The domain and range describe the set of possible input and output values for a function.

• Domain: All possible x-values (inputs).

• Range: All possible y-values (outputs).

• Note: The range is the set of all second components of ordered pairs.

Vertical & Horizontal Lines

Vertical and horizontal lines have unique equations and properties.

• Vertical line: $x = a$ (not a function; slope is undefined)

• Horizontal line: $y = b$ (slope $= 0$)

Checking Ordered Pairs

To determine if a point is a solution to an equation, substitute the values and check if the equation is satisfied.

1. Plug $x$ and $y$ into the equation.

2. If the equation is true, the point is a solution.

Absolute Value Graphs

Absolute value functions create V-shaped graphs with a vertex at the origin.

• Equation: $f(x) = |x|$

• Shape: V-shape

• Vertex: $(0, 0)$

Quadratic vs Linear (Conceptual)

Linear and quadratic equations differ in their graphs and rates of change.

• Linear: Straight line; constant rate of change.

• Quadratic: Parabola (curve); changing rate of change.

Test Tips

• Always find the slope first.

• Watch your signs (+ / −).

• Read carefully: parallel vs perpendicular.

• For graphs: intercepts save time.