Precalculus Final Review Study Guide

Functions and Graphs

Finding the Slope Between Two Points

The slope of a line measures its steepness and is calculated as the ratio of the change in y-values to the change in x-values between two points.

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

• Example: For points (7, -2) and (4, 3): $m = \frac{3 - (-2)}{4 - 7} = \frac{5}{-3} = -\frac{5}{3}$

Equation of a Line Perpendicular to a Given Line

To find the equation of a line passing through a point and perpendicular to another line, use the negative reciprocal of the given line's slope.

• Given line: $y = 2x + 1$ (slope $m = 2$)

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

• Point-slope form: $y - y_1 = m(x - x_1)$

• Example: Through (5, -3): $y + 3 = -\frac{1}{2}(x - 5)$

Domain of Functions

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

• Rational functions: Exclude x-values that make the denominator zero.

• Logarithmic functions: Argument must be positive.

• Square root functions: Radicand must be non-negative.

• Examples:

  • $f(x) = \frac{2}{x^2 - 5}$: Domain is all real x except $x^2 - 5 = 0 \implies x = \pm \sqrt{5}$

  • $f(x) = \log_2(2x + 3)$: Domain is $2x + 3 > 0 \implies x > -1.5$

  • $f(x) = \sqrt{x^2 - 4}$: Domain is $x^2 - 4 \geq 0 \implies x \leq -2$ or $x \geq 2$

Function Operations and Difference Quotient

Functions can be added, subtracted, multiplied, divided, and composed. The difference quotient is used to find the average rate of change.

• Difference quotient: $\frac{f(x+h) - f(x)}{h}$

• Example: For $f(x) = x^2 - 2x + 4$, $f(x+h) = (x+h)^2 - 2(x+h) + 4$

Polynomial and Rational Functions

Graphing Rational Functions

Rational functions are quotients of polynomials. Their graphs may have asymptotes and holes.

• Horizontal Asymptote (H.A.): Determined by degrees of numerator and denominator.

• Vertical Asymptote (V.A.): Values of x that make the denominator zero (unless canceled).

• Slant Asymptote (S.A.): If degree of numerator is one more than denominator.

• Holes: Occur where factors cancel in numerator and denominator.

• Example: $f(x) = \frac{x^2 - x - 2}{2x^2 + x - 1}$

Graphing Polynomial Functions

Polynomial functions are continuous and smooth curves. Key features include degree, leading coefficient, zeros, y-intercept, and end behavior.

• Degree: Highest power of x.

• Leading Coefficient: Coefficient of the highest power.

• Zeros: Values of x where $f(x) = 0$ (with multiplicity).

• End Behavior: Determined by degree and leading coefficient.

• Example: $f(x) = x^2(x - 3)^2(x - 1)$

Exponential and Logarithmic Functions

Solving Exponential and Logarithmic Equations

Exponential equations can be solved by taking logarithms; logarithmic equations require the argument to be positive.

• Exponential equation: $3^x = 411$

• Logarithmic equation: $\log_6(x + 3) - \log_6(2x - 4) = \log_6(x - 2)$

• Change of base formula: $\log_b a = \frac{\ln a}{\ln b}$

Converting Between Exponential and Logarithmic Forms

Exponential and logarithmic equations are related:

• $a^x = b \iff x = \log_a b$

• Example: $2^x = 81 \implies x = \log_2 81$

Systems of Equations and Inequalities

Solving Linear Systems

Systems of equations can be solved by substitution, elimination, or graphing.

• Example:

  • $3x - 2y = 16$

  • $5x + y = 23$

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• Solution: Find values of x and y that satisfy both equations.

Sequences, Induction, and Probability

Compound Interest and Loan Formulas

Exponential functions model compound interest and loan payments.

• Compound interest formula: $A = P\left(1 + \frac{r}{n}\right)^{nt}$

• Loan payment formula: $A = P\left[\frac{\frac{r}{n}}{1 - \left(1 + \frac{r}{n}\right)^{-nt}}\right]$

• Example: Calculate future value or loan amount given rate, time, and payment.

Additional Topics

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals.

• Example:

  • $f(x) = \begin{cases} -2x - 1, & x \leq -3 \\ x^2, & -3 < x < 2 \\ x, & x \geq 2 \end{cases}$

• Evaluate at specific points by determining which interval the input falls into.

Quadratic Functions and Their Graphs

Quadratic functions have parabolic graphs. Key features include vertex, axis of symmetry, maximum/minimum, domain, range, and intercepts.

• Standard form: $f(x) = ax^2 + bx + c$

• Vertex: $x = -\frac{b}{2a}$

• Maximum/Minimum: Value at the vertex

• Domain: All real numbers

• Range: Depends on whether parabola opens up or down

Transformations of Functions

Transformations shift, stretch, compress, or reflect the graph of a function.

• Vertical shift: $f(x) + k$

• Horizontal shift: $f(x - h)$

• Reflection: $-f(x)$ or $f(-x)$

• Example: $f(x) = \left|\frac{4}{5}x\right| - 3$

Function Composition

Composition combines two functions: $f(g(x))$.

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

• Example: $f(x) = \sqrt{x - 2}$, $g(x) = x^2 - 1$; $f(g(x)) = \sqrt{x^2 - 3}$

Summary Table: Asymptotes of Rational Functions

Type	How to Find	Example
Horizontal Asymptote	Compare degrees of numerator and denominator	$y = 0$ if degree numerator < denominator
Vertical Asymptote	Set denominator = 0 (exclude holes)	$x = a$ where denominator is zero
Slant Asymptote	Degree numerator = degree denominator + 1	Divide numerator by denominator

Summary Table: Function Transformations

Transformation	Equation	Effect
Vertical Shift	$f(x) + k$	Up if $k > 0$, down if $k < 0$
Horizontal Shift	$f(x - h)$	Right if $h > 0$, left if $h < 0$
Reflection	$-f(x)$ or $f(-x)$	Over x-axis or y-axis

Additional info: These study notes cover key Precalculus topics including functions, graphing, domains, polynomial and rational functions, exponential and logarithmic equations, systems of equations, piecewise functions, and transformations, as presented in the final review questions.