College Algebra Study Guide: Graphs, Functions, and Linear & Quadratic Functions

Graphs and Functions

Ordered Pairs and the Coordinate Plane

The coordinate plane is a two-dimensional surface defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Points are represented as ordered pairs (x, y).

• Graphing Ordered Pairs: To plot (x, y), move x units along the x-axis and y units along the y-axis.

• Distance Formula: The distance between points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Midpoint Formula: The midpoint between $(x_1, y_1)$ and $(x_2, y_2)$ is: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Equations with x and y: These represent lines or curves on the plane. To graph, solve for y in terms of x or vice versa, and plot points.

Relations and Functions

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

• Determining Functions: A relation is a function if no x-value is paired with more than one y-value.

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

• Example: The set {(1,2), (2,3), (3,4)} is a function; {(1,2), (1,3)} is not.

Function Notation

Functions are often written as $f(x)$, which means "the output of function f for input x."

• Example: If $f(x) = x^2 + 1$, then $f(2) = 2^2 + 1 = 5$.

Domain and Range

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

• Finding Domain: Exclude x-values that cause division by zero or negative square roots.

• Finding Range: Consider the possible y-values as x varies over the domain.

• Example: For $f(x) = \sqrt{x}$, domain is $x \geq 0$, range is $y \geq 0$.

Increasing, Decreasing, and Constant Functions

A function is increasing if its output rises as x increases, decreasing if it falls, and constant if it stays the same.

• Even Functions: Satisfy $f(-x) = f(x)$ (symmetric about the y-axis).

• Odd Functions: Satisfy $f(-x) = -f(x)$ (symmetric about the origin).

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

Evaluating and Combining Functions

Functions can be evaluated and combined using arithmetic operations or composition.

• Addition: $(f + g)(x) = f(x) + g(x)$

• Subtraction: $(f - g)(x) = f(x) - g(x)$

• Multiplication: $(f \cdot g)(x) = f(x) \cdot g(x)$

• Division: $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

• Composition: $(f \circ g)(x) = f(g(x))$

• Example: If $f(x) = x + 1$, $g(x) = 2x$, then $(f \circ g)(x) = f(g(x)) = 2x + 1$

Linear Functions

Forms of Linear Equations

Linear functions graph as straight lines and can be written in several forms:

• Slope-Intercept Form: $y = mx + b$

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Standard Form: $Ax + By = C$

• Where: m = slope, b = y-intercept, $(x_1, y_1)$ is a point on the line.

Basic Concepts of Linear Functions

• Vertical Line: Equation $x = a$ (undefined slope)

• Horizontal Line: Equation $y = b$ (slope = 0)

• Positive Slope: Line rises left to right

• Negative Slope: Line falls left to right

Basic Function Graphs

Common Parent Functions

• Squared Function: $f(x) = x^2$ (parabola)

• Cubic Function: $f(x) = x^3$ (S-shaped curve)

• Absolute Value Function: $f(x) = |x|$ (V-shaped graph)

• Square Root Function: $f(x) = \sqrt{x}$ (starts at origin, increases slowly)

• Piecewise Function: Defined by different expressions for different intervals of x

Graphing Techniques: Transformations

Graphs can be transformed by shifting, stretching, shrinking, or reflecting.

• Vertical Shift: $f(x) + k$ shifts up by k units

• Horizontal Shift: $f(x - h)$ shifts right by h units

• Reflection: $-f(x)$ reflects over x-axis; $f(-x)$ reflects over y-axis

• Vertical Stretch/Shrink: $a f(x)$ stretches if $|a| > 1$, shrinks if $0 < |a| < 1$

Quadratic Functions

Properties of Quadratic Functions

A quadratic function has the form $f(x) = ax^2 + bx + c$.

• Vertex: The highest or lowest point on the graph (parabola)

• Axis of Symmetry: Vertical line through the vertex, $x = -\frac{b}{2a}$

• Direction: Opens upward if $a > 0$, downward if $a < 0$

• Graphing: Plot the vertex, axis of symmetry, and additional points to sketch the parabola

Example: Graphing a Quadratic Function

• Given $f(x) = 2x^2 - 4x + 1$:

• Vertex: $x = -\frac{-4}{2 \cdot 2} = 1$, $f(1) = 2(1)^2 - 4(1) + 1 = -1$

• Axis of symmetry: $x = 1$

• Opens upward ($a = 2 > 0$)

Additional info: Students should be familiar with identifying key features of graphs, using function notation, and applying transformations to parent functions. Practice with graphing and interpreting equations is essential for mastery.