Precalculus Study Guide: Lines, Functions, Symmetry, Transformations, and Equations

Lines and Their Equations

Finding the Equation of a Line

To determine the equation of a line, you need either two points on the line or one point and the slope.

• Point-Slope Form: The equation of a line with slope $m$ passing through $(x_1, y_1)$ is $y - y_1 = m(x - x_1)$.

• Slope-Intercept Form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Standard Form: $Ax + By = C$, where $A$, $B$, and $C$ are constants.

Example: Find the equation of the line through $(2,3)$ and $(4,3)$.

• First, calculate the slope: $m = \frac{3-3}{4-2} = 0$.

• Equation: $y = 3$.

Parallel and Perpendicular Lines

• Parallel Lines: Two lines are parallel if they have the same slope.

• Perpendicular Lines: Two lines are perpendicular if the product of their slopes is $-1$.

Example: Find the equation of the line through $(-1,6)$ parallel to $y = 2x + 9$.

• Slope is $2$ (same as given line).

• Use point-slope form: $y - 6 = 2(x + 1)$.

• Simplify: $y = 2x + 8$.

Example: Find the equation of the line through $(-1,6)$ perpendicular to $y = 2x + 9$.

• Perpendicular slope: $m = -\frac{1}{2}$.

• Equation: $y - 6 = -\frac{1}{2}(x + 1)$.

• Simplify: $y = -\frac{1}{2}x + \frac{11}{2}$.

Intercepts

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

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

Example: For $4x + 5y = 7$:

• x-intercept: $4x = 7 \implies x = \frac{7}{4}$

• y-intercept: $5y = 7 \implies y = \frac{7}{5}$

Functions and Their Properties

Average Rate of Change

The average rate of change of a function $f(x)$ from $x = a$ to $x = b$ is:

• $\frac{f(b) - f(a)}{b - a}$

Example: For $f(x) = x^3 + x + 1$ from $x = 2$ to $x = 5$:

• $f(5) = 125 + 5 + 1 = 131$

• $f(2) = 8 + 2 + 1 = 11$

• Average rate: $\frac{131 - 11}{5 - 2} = \frac{120}{3} = 40$

Solving for Variables

To solve for a variable, isolate it using algebraic operations.

• Example: $3h = 7c + 2w$; solve for $w$:

• $2w = 3h - 7c \implies w = \frac{3h - 7c}{2}$

Graphical Analysis

Continuity

A function $f(x)$ is continuous at $x = a$ if:

• $f(a)$ is defined

• $\lim_{x \to a} f(x)$ exists

• $\lim_{x \to a} f(x) = f(a)$

Example: Given a graph, check for holes, jumps, or asymptotes at the specified $x$-values.

Symmetry

• Even Function: $f(-x) = f(x)$ for all $x$ in the domain. Graph is symmetric about the y-axis.

• Odd Function: $f(-x) = -f(x)$ for all $x$ in the domain. Graph is symmetric about the origin.

Example: $f(x) = x^2$ is even; $f(x) = x^3$ is odd.

Transformations of Functions

Types of Transformations

• Vertical Shifts: $f(x) + c$ shifts up by $c$ units; $f(x) - c$ shifts down by $c$ units.

• Horizontal Shifts: $f(x + c)$ shifts left by $c$ units; $f(x - c)$ shifts right by $c$ units.

• Reflections: $-f(x)$ reflects over the x-axis; $f(-x)$ reflects over the y-axis.

• Vertical Stretch/Compression: $a f(x)$ stretches if $|a| > 1$, compresses if $0 < |a| < 1$.

• Horizontal Stretch/Compression: $f(bx)$ compresses horizontally if $|b| > 1$, stretches if $0 < |b| < 1$.

Example: $F(x) = f(x + 3)$ is a shift left by 3 units.

Solving Absolute Value Equations and Inequalities

Absolute Value Equations

• To solve $|A| = B$, set $A = B$ and $A = -B$.

• To solve $|A| < B$, $-B < A < B$.

• To solve $|A| > B$, $A < -B$ or $A > B$.

Example: Solve $|2x - 5| + 4 = 1$:

• $|2x - 5| = -3$ (no solution, since absolute value cannot be negative).

Summary Table: Types of Symmetry

Type	Algebraic Test	Graphical Symmetry
Even	$f(-x) = f(x)$	y-axis
Odd	$f(-x) = -f(x)$	Origin
Neither	Fails both tests	No symmetry

Summary Table: Function Transformations

Transformation	Equation	Description
Vertical Shift Up	$f(x) + c$	Up by $c$ units
Vertical Shift Down	$f(x) - c$	Down by $c$ units
Horizontal Shift Left	$f(x + c)$	Left by $c$ units
Horizontal Shift Right	$f(x - c)$	Right by $c$ units
Reflection x-axis	$-f(x)$	Over x-axis
Reflection y-axis	$f(-x)$	Over y-axis
Vertical Stretch	$a f(x)$, $|a| > 1$	Stretched vertically
Vertical Compression	$a f(x)$, $0 < |a| < 1$	Compressed vertically

Key Definitions

• Function: A relation where each input has exactly one output.

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (y-values).

• Intercept: The point where a graph crosses an axis.

• Continuous Function: A function with no breaks, holes, or jumps in its graph.

Sample Problems and Solutions

• Find the equation of the line through $(1,2)$ and $(2,7)$: Slope: $m = \frac{7-2}{2-1} = 5$ Equation: $y - 2 = 5(x - 1) \implies y = 5x - 3$

• Is $f(x) = x^2 - 4x - 2$ even, odd, or neither? $f(-x) = (-x)^2 - 4(-x) - 2 = x^2 + 4x - 2$ Not equal to $f(x)$ or $-f(x)$, so neither.

• Solve $|3x - 5| = 7$: $3x - 5 = 7 \implies x = 4$ $3x - 5 = -7 \implies x = -\frac{2}{3}$

Additional info: Some context and explanations have been expanded for clarity and completeness, as the original material was in exercise format.