Precalculus Study Guide: Functions and Graphs (Chapter 1)

Rectangular Coordinate System

Graphs & the Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian plane, is fundamental for graphing equations and visualizing relationships between variables in two dimensions.

• Axes: The x-axis (horizontal) and y-axis (vertical) are perpendicular and intersect at the origin (0,0).

• Ordered pairs: Points are represented as (x, y).

• Quadrants: The plane is divided into four quadrants, numbered counterclockwise starting from the upper right.

• Plotting points: Move right/left for x, up/down for y from the origin.

Example: Plot the points A (4, 3), B (–2, 2), C (–2, –3), D (3, –4), E (0, 5) on the graph.

Equations of Two Variables

Solving Two-Variable Equations

Many equations in precalculus involve two variables, typically x and y. Their solutions are ordered pairs (x, y) that satisfy the equation.

• Equations with one variable: Solutions are points on a line.

• Equations with two variables: Solutions are points on a plane.

Example: Determine if points (1, 7), (–1, 5), (4, 3) satisfy y = –2x + 9.

Graphing by Plotting Points

To graph an equation, calculate ordered pairs that make the equation true.

1. Isolate y (or x) if possible.

2. Choose values for x (or y) and solve for the other variable.

3. Plot the points and connect with a line or curve.

Example: Graph –2x + y = –1 by plotting points.

Intercepts

Graphing Intercepts

Intercepts are points where a graph crosses the axes.

• x-intercept: Where the graph crosses the x-axis (y = 0).

• y-intercept: Where the graph crosses the y-axis (x = 0).

Example: Find the x- and y-intercepts of y = 2x – 4.

Lines

Slopes of Lines

The slope of a line measures its steepness and direction.

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

• Positive slope: Line rises left to right.

• Negative slope: Line falls left to right.

• Zero slope: Horizontal line.

• Undefined slope: Vertical line.

Example: Find the slope of the line through (–1, 1) and (4, 3).

Slope-Intercept Form

The slope-intercept form of a line is $y = mx + b$, where m is the slope and b is the y-intercept.

• To graph, plot the y-intercept, then use the slope to find another point.

Example: Write the equation of a line with slope 2 and y-intercept –3.

Point-Slope Form

Used when given a point and a slope: $y - y_1 = m(x - x_1)$

• Convert to slope-intercept form if needed.

Example: Write the equation of a line with slope 3 passing through (2, –4).

Standard Form

The standard form of a line is $Ax + By = C$.

• To find intercepts, set x or y to zero and solve for the other variable.

Example: Find the slope and y-intercept of 4x + 2y = 6.

Parallel and Perpendicular Lines

• Parallel lines: Same slope, different y-intercepts.

• Perpendicular lines: Slopes are negative reciprocals ($m_1 \cdot m_2 = -1$).

Example: Write the equation of a line parallel to y = 2x – 4 through (–1, 5).

Intro to Functions & Their Graphs

Relations and Functions

A relation is a set of ordered pairs. A function is a relation where each input (x) has exactly one output (y).

• Vertical Line Test: If any vertical line crosses a graph more than once, it is not a function.

Example: Determine if the graph of y = x^2 is a function.

Domain and Range

The domain is the set of all possible input values (x), and the range is the set of all possible output values (y).

• Interval notation: Use brackets [ ] for included values, parentheses ( ) for excluded values.

• Set-builder notation: Describes the set using inequalities.

Example: Find the domain and range of $f(x) = \sqrt{x-2}$.

Finding Domain Algebraically

• For square roots: Set the radicand ≥ 0.

• For denominators: Set the denominator ≠ 0.

Example: Find the domain of $f(x) = \frac{1}{x-3}$.

Common Functions

Graphs of Common Functions

Several basic functions frequently appear in precalculus. Their graphs, domains, and ranges are important to recognize.

Function	Equation	Domain	Range
Constant	$f(x) = c$	All real x	$y = c$
Identity	$f(x) = x$	All real x	All real y
Square	$f(x) = x^2$	All real x	$y \geq 0$
Cube	$f(x) = x^3$	All real x	All real y
Square Root	$f(x) = \sqrt{x}$	$x \geq 0$	$y \geq 0$
Cube Root	$f(x) = \sqrt[3]{x}$	All real x	All real y

Transformations of Functions

Reflections

A reflection flips a graph over a specified axis.

• Over x-axis: $y = -f(x)$

• Over y-axis: $y = f(-x)$

Shifts

• Vertical shift: $y = f(x) + k$ (up if k > 0, down if k < 0)

• Horizontal shift: $y = f(x - h)$ (right if h > 0, left if h < 0)

Stretches and Shrinks

• Vertical stretch/shrink: $y = a f(x)$ (stretch if |a| > 1, shrink if 0 < |a| < 1)

• Horizontal stretch/shrink: $y = f(bx)$ (shrink if |b| > 1, stretch if 0 < |b| < 1)

Combining Transformations

Multiple transformations can be applied to a function in sequence.

Example: $y = -f(x-3) + 2$ reflects over x-axis, shifts right 3, up 2.

Domain & Range of Transformed Functions

Transformations affect the domain and range of functions. Shifts move the domain/range, while reflections may invert them.

Function Operations

Adding & Subtracting Functions

• $(f + g)(x) = f(x) + g(x)$

• $(f - g)(x) = f(x) - g(x)$

• Domain: Intersection of domains of f and g.

Multiplying & Dividing Functions

• $(f \cdot g)(x) = f(x) \cdot g(x)$

• $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

• Domain: Intersection of domains, and for division, exclude where $g(x) = 0$.

Function Composition

Composing Functions

Function composition involves substituting one function into another: $(f \circ g)(x) = f(g(x))$.

• Evaluate inside function first, then outside.

• Domain: Values of x for which g(x) is in the domain of f.

Example: If $f(x) = x^2 + 4$ and $g(x) = \sqrt{x}$, then $(f \circ g)(x) = f(g(x)) = (\sqrt{x})^2 + 4 = x + 4$.

Summary Table: Forms of Linear Equations

Form	Equation	When to Use
Slope-Intercept	$y = mx + b$	Given slope and y-intercept
Point-Slope	$y - y_1 = m(x - x_1)$	Given slope and a point
Standard	$Ax + By = C$	General form, easy for intercepts

Additional info: This guide covers the foundational concepts of Chapter 1 in Precalculus, focusing on functions, graphs, lines, and transformations. Mastery of these topics is essential for success in later chapters involving more advanced functions and analytic geometry.