Precalculus Study Guide: Functions, Graphs, and Transformations

1.1 Plot Points, Quadrants, and Linear Equations

Plotting Points and Identifying Quadrants

Understanding the coordinate plane is fundamental in precalculus. The plane is divided into four quadrants, and each point is represented by an ordered pair (x, y).

• Quadrants: The four regions created by the intersection of the x- and y-axes.

• Plotting Points: To plot (x, y), move x units along the x-axis and y units along the y-axis.

• Example: The point (3, -2) is in Quadrant IV.

Solving Linear Equations

• Linear Equation: An equation of the form $ax + b = 0$.

• Solution: Isolate x: $x = -\frac{b}{a}$.

• Example: Solve $2x + 3 = 7$; $x = 2$.

Using Graphing Calculators

• Viewing Intercepts: Use the calculator's graphing function to find where the graph crosses the axes.

• x-intercept: Where $y = 0$.

• y-intercept: Where $x = 0$.

Reading and Interpreting Graphs

• Key Features: Intercepts, slope, increasing/decreasing intervals.

• Application: Use graphs to estimate solutions and analyze function behavior.

1.2 Functions and Their Representations

Definition and Notation

A function is a relation where each input has exactly one output. Functions can be represented as equations, tables, graphs, or mappings.

• Function Notation: $f(x)$ denotes the output when x is the input.

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

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

Evaluating Functions

• To evaluate $f(a)$, substitute $a$ for $x$ in the function's formula.

• Example: If $f(x) = 2x + 1$, then $f(3) = 7$.

Independent and Dependent Variables

• Independent Variable: The input (usually x).

• Dependent Variable: The output (usually y or f(x)).

Interval Notation

• Used to describe domains and ranges.

• Example: $[a, b]$ includes endpoints; $(a, b)$ excludes endpoints.

Graphing Functions

• Plot points for various x-values and connect smoothly.

• Identify key features: intercepts, maxima, minima.

Writing Equations from Scenarios

• Translate word problems into function equations.

• Example: "A taxi charges $3 plus $2 per mile" becomes$C(m) = 3 + 2m$.

1.3 Function Behavior and Symmetry

Increasing, Decreasing, and Constant Functions

• Increasing: $f(x_2) > f(x_1)$ when $x_2 > x_1$.

• Decreasing: $f(x_2) < f(x_1)$ when $x_2 > x_1$.

• Constant: $f(x_2) = f(x_1)$ for all $x_2, x_1$.

Relative Maximum and Minimum

• Relative Maximum: Highest point in a local region.

• Relative Minimum: Lowest point in a local region.

Symmetry

• y-axis Symmetry: $f(-x) = f(x)$ (even function).

• Origin Symmetry: $f(-x) = -f(x)$ (odd function).

• x-axis Symmetry: Not a function (fails vertical line test).

Piecewise Functions

• Defined by different expressions over different intervals.

• Example: $f(x) = \begin{cases} x^2 & x < 0 \\ x & x \geq 0 \end{cases}$

Difference Quotient

• Measures average rate of change:

$\frac{f(x+h) - f(x)}{h}$

1.4 Linear Functions and Slope

Definition and Forms

• Linear Function: $f(x) = mx + b$

• Slope (m): $m = \frac{y_2 - y_1}{x_2 - x_1}$

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

Graphing Lines

• Use slope and y-intercept to plot.

• Find x- and y-intercepts by setting $y=0$ and $x=0$ respectively.

Parallel and Perpendicular Lines

• Parallel: Same slope, different y-intercepts.

• Perpendicular: Slopes are negative reciprocals: $m_1 \cdot m_2 = -1$

Average Rate of Change

• For $f(x)$ from $x=a$ to $x=b$:

$\frac{f(b) - f(a)}{b - a}$

1.6 Transformations of Functions

Vertical and Horizontal Shifts

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

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

Reflections

• Across x-axis: $-f(x)$

• Across y-axis: $f(-x)$

Stretching and Shrinking

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

• Horizontal Stretch/Shrink: $f(bx)$, $|b| > 1$ shrinks, $0 < |b| < 1$ stretches.

Order of Transformations

• Apply in this order: 1) Horizontal shifts, 2) Stretch/Shrink/Reflect, 3) Vertical shifts.

1.7 Domains and Operations on Functions

Domain of a Function

• Set of all x-values for which the function is defined.

• Example: For $f(x) = \frac{1}{x}$, domain is $x \neq 0$.

Algebra of Functions

• Sum: $(f+g)(x) = f(x) + g(x)$

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

• Product: $(fg)(x) = f(x)g(x)$

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

Composition of Functions

• $(f \circ g)(x) = f(g(x))$

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

1.8 Inverse Functions

Definition and Properties

• Inverse Function: $f^{-1}(x)$ reverses the effect of $f(x)$.

• One-to-One Function: Passes the horizontal line test; each output is from only one input.

• Finding the Inverse: Swap x and y, then solve for y.

• Example: $f(x) = 2x + 3$; inverse is $f^{-1}(x) = \frac{x-3}{2}$.

Horizontal Line Test

• If any horizontal line crosses the graph more than once, the function does not have an inverse.

Summary Table: Types of Function Transformations

Transformation	Equation	Effect
Vertical Shift	$f(x) + k$	Up/down by $k$ units
Horizontal Shift	$f(x - h)$	Right by $h$ units
Reflection x-axis	$-f(x)$	Reflect over x-axis
Reflection y-axis	$f(-x)$	Reflect over y-axis
Vertical Stretch/Shrink	$a f(x)$	Stretch if $|a| > 1$, shrink if $0 < |a| < 1$
Horizontal Stretch/Shrink	$f(bx)$	Shrink if $|b| > 1$, stretch if $0 < |b| < 1$