Precalculus Final Review Study Guide

Final Review: Precalculus Concepts

Functions and Graphs

Understanding functions and their graphical representations is fundamental in precalculus. This section covers domains, ranges, piecewise functions, and transformations.

• Domain: The set of all possible input values (x-values) for which the function is defined.

• Range: The set of all possible output values (y-values) the function can produce.

• Piecewise Functions: Functions defined by different expressions over different intervals.

• Transformations: Shifts, stretches, compressions, and reflections applied to parent functions.

• Difference Quotient: Used to find the average rate of change of a function: $\frac{f(x+h) - f(x)}{h}$

Example: For $f(x) = x^2 - 2x + 4$, the difference quotient is $\frac{(x+h)^2 - 2(x+h) + 4 - [x^2 - 2x + 4]}{h}$.

Linear Equations and Slope

Linear equations describe straight lines. The slope measures the steepness and direction of a line.

• Slope Formula: For points $(x_1, y_1)$ and $(x_2, y_2)$: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Equation of a Line: Point-slope form: $y - y_1 = m(x - x_1)$

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

Example: Find the slope between $(-7, -2)$ and $(4, 3)$:

• $m = \frac{3 - (-2)}{4 - (-7)} = \frac{5}{11}$

Polynomial and Rational Functions

Polynomials and rational functions are key topics in precalculus, involving their graphs, zeros, and asymptotes.

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

• Degree: The highest power of x.

• Leading Coefficient: The coefficient of the term with the highest degree.

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

• End Behavior: Determined by the degree and leading coefficient.

• Rational Function: A function of the form $f(x) = \frac{P(x)}{Q(x)}$ where $P$ and $Q$ are polynomials.

• Asymptotes:

  • Horizontal Asymptote (H.A.): Describes end behavior as $x \to \infty$.

  • Vertical Asymptote (V.A.): Occurs where denominator is zero (and numerator is nonzero).

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

• Holes: Occur where both numerator and denominator are zero at the same x-value.

Example: For $f(x) = \frac{x^2 - x - 2}{2x^2 + x - 1}$, factor numerator and denominator to find zeros, asymptotes, and holes.

Exponential and Logarithmic Functions

Exponential and logarithmic functions are used to model growth, decay, and many real-world phenomena.

• Exponential Equation: $a^x = b$

• Logarithmic Equation: $\log_a b = x$

• Change of Base Formula: $\log_a b = \frac{\log_c b}{\log_c a}$

• Solving Exponential Equations: Take logarithms of both sides.

• Solving Logarithmic Equations: Combine logs and exponentiate both sides.

Example: Solve $3^x = 411$ by taking logarithms:

• $x = \log_3 411$

Compound Interest and Financial Applications

Compound interest and loan formulas are applications of exponential functions in finance.

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

  • $A$ = final amount

  • $P$ = principal

  • $r$ = annual interest rate

  • $n$ = number of compounding periods per year

  • $t$ = number of years

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

Example: Calculate the amount borrowed for a car loan with monthly payments and interest rate.

Systems of Equations

Systems of equations involve finding values that satisfy multiple equations simultaneously.

• Solving by Substitution or Elimination: Methods to find the values of variables.

• Example System:

  • $3x - 2y = 14$

  • $5x + y = 23$

Example: Solve for x and y using elimination or substitution.

Function Operations and Composition

Functions can be added, subtracted, multiplied, divided, and composed to form new 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: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$

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

Example: If $f(x) = x^2 + x - 2$ and $g(x) = 1/x + 5$, find $f(g(x))$ and its domain.

Quadratic Functions and Their Graphs

Quadratic functions have the form $f(x) = ax^2 + bx + c$ and their graphs are parabolas.

• Vertex: The maximum or minimum point of the parabola.

• Axis of Symmetry: $x = -\frac{b}{2a}$

• Maximum/Minimum Value: The y-value at the vertex.

• Domain: All real numbers, unless restricted.

• Range: Depends on whether the parabola opens up or down.

• x-intercepts: Solutions to $f(x) = 0$.

Example: For $f(x) = x^2 - 4x + 3$, vertex is at $(2, -1)$.

Graphing Rational and Polynomial Functions

Graphing involves identifying key features such as intercepts, asymptotes, holes, and end behavior.

• x-intercepts: Set numerator equal to zero.

• y-intercept: Evaluate $f(0)$.

• Vertical Asymptotes: Set denominator equal to zero.

• Horizontal/Slant Asymptotes: Compare degrees of numerator and denominator.

• Holes: Common factors in numerator and denominator.

Example: For $f(x) = \frac{x^2 - x - 2}{2x^2 + x - 1}$, factor and analyze for intercepts and asymptotes.

Solving Equations Involving Exponents and Logarithms

Techniques include rewriting equations, applying logarithms, and using properties of exponents and logs.

• Exponent Rules: $a^{x+y} = a^x a^y$

• Logarithm Rules: $\log_a(xy) = \log_a x + \log_a y$

• Solving: Isolate the variable and apply appropriate rules.

Example: Solve $5^{x+2} = 411$ by taking logarithms.

Interval Notation and Increasing/Decreasing Functions

Functions can be classified as increasing, decreasing, or constant over intervals.

• Increasing: $f(x_2) > f(x_1)$ for $x_2 > x_1$

• Decreasing: $f(x_2) < f(x_1)$ for $x_2 > x_1$

• Constant: $f(x_2) = f(x_1)$ for $x_2 > x_1$

Example: Use the graph to identify intervals where the function increases, decreases, or remains constant.

Summary Table: Types of Asymptotes in Rational Functions

Type	How to Find	Example
Horizontal Asymptote	Compare degrees of numerator and denominator	$f(x) = \frac{2x^2}{x^2 + 1}$, H.A.: $y=2$
Vertical Asymptote	Set denominator equal to zero	$f(x) = \frac{1}{x-3}$, V.A.: $x=3$
Slant Asymptote	Degree of numerator is one more than denominator	$f(x) = \frac{x^2}{x+1}$, S.A.: $y=x-1$

Additional info:

• Some questions require calculator use for logarithms and roots.

• Graphing prompts expect students to identify intercepts, asymptotes, and holes.

• Loan and compound interest formulas are standard in financial mathematics.