Precalculus Final Exam Review: Key Concepts and Practice Problems

Systems of Equations and Inequalities

Solving Systems of Linear Equations

Systems of equations involve finding values for variables that satisfy all given equations simultaneously.

• Substitution Method: Solve one equation for one variable and substitute into the other.

• Elimination Method: Add or subtract equations to eliminate a variable.

• Example: Solve the system:

  • $y = 4x - 3$

  • $2y + 10x = 48$

Solving Systems with Three Variables

Systems with three equations and three unknowns can be solved using substitution, elimination, or matrix methods.

• Example: Solve:

  • $3x + 5y = 2z - 21$

  • $2x - 4y - z = 12$

  • $2x - y + 4z = 2$

Functions and Graphs

Definition of a Function

A function is a relation in which each input (x-value) corresponds to exactly one output (y-value).

• Vertical Line Test: A graph represents a function if no vertical line intersects it more than once.

• Example: Determine if $6x + 5y = 13$ defines y as a function of x.

Domain and Range

The domain is the set of all possible input values; the range is the set of all possible output values.

• Example: For the relation {(4, 9), (2, 7), (4, -5), (5, -7)}, state the domain and range.

Even and Odd Functions

Functions can be classified as even, odd, or neither based on their symmetry.

• Even Function: $f(-x) = f(x)$ (symmetric about the y-axis)

• Odd Function: $f(-x) = -f(x)$ (symmetric about the origin)

• Example: $f(x) = x^3 - 4x$

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals.

• Example: $f(x) = \begin{cases} 4x - 4 & x < -2 \\ 2x + 1 & -2 \leq x < 3 \end{cases}$

Graphical Analysis of Functions

Using Graphs to Analyze Functions

Graphs provide visual information about domain, range, intercepts, intervals of increase/decrease, and extrema.

• Domain and Range: Identify from the graph.

• x-intercepts/y-intercepts: Points where the graph crosses the axes.

• Intervals of Increase/Decrease: Where the function rises or falls.

• Relative Minimum/Maximum: Lowest/highest points in a local region.

Rates of Change and Difference Quotients

Average Rate of Change

The average rate of change of $f(x)$ from $x_1$ to $x_2$ is:

• $\frac{f(x_2) - f(x_1)}{x_2 - x_1}$

• Example: $f(x) = \sqrt{x}$ from $x_1 = 2$ to $x_2 = 8$

Difference Quotient

The difference quotient is used to measure the average rate of change over an interval $h$:

• $\frac{f(x + h) - f(x)}{h}$

Inverse Functions

Finding the Inverse

The inverse function reverses the roles of x and y. To find the inverse:

• Replace $f(x)$ with $y$

• Swap x and y

• Solve for y

• Example: $f(x) = 5x + 1$

Analytic Geometry

Distance and Midpoint Formulas

Used to find the distance and midpoint between two points in the plane.

• Distance: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Midpoint: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Equations of Circles

The standard form of a circle with center $(h, k)$ and radius $r$:

• $(x - h)^2 + (y - k)^2 = r^2$

• Example: Center $(2, 9)$, radius $10$

Transformations of Functions

Quadratic and Absolute Value Functions

Transformations include shifts, stretches, compressions, and reflections.

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

• Horizontal Shift: $f(x - h)$ shifts right/left

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

• Example: $g(x) = |x + 2| + 4$

Polynomial and Rational Functions

Zeros and Multiplicity

Zeros are values of x where $f(x) = 0$. Multiplicity is the number of times a zero occurs.

• Graph Behavior: If multiplicity is odd, the graph crosses the axis; if even, it touches and turns around.

Synthetic Division and Remainder Theorem

Synthetic division is a shortcut for dividing polynomials by linear factors. The Remainder Theorem states that the remainder of $f(x)$ divided by $(x - a)$ is $f(a)$.

• Example: $f(x) = 4x^3 - 8x^2 - 3x + 2$ divided by $x - 2$

Rational Zero Theorem

Lists all possible rational zeros of a polynomial based on its coefficients.

• Possible zeros: $\pm$ (factors of constant term)/(factors of leading coefficient)

Asymptotes of Rational Functions

Vertical asymptotes occur where the denominator is zero; horizontal asymptotes depend on the degrees of numerator and denominator.

• Example: $h(x) = \frac{x^2 + 4}{x^2 - 16}$

Inequalities

Polynomial and Rational Inequalities

Solving inequalities involves finding intervals where the expression is positive or negative.

• Express solutions in interval notation.

• Example: $x^2 - 3x + 4 < 0$

Exponential and Logarithmic Functions

Exponential Growth and Compound Interest

Exponential functions model growth and decay. Compound interest formulas:

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

• $A = Pe^{rt}$

• Example: Find accumulated value for $P = 1500$, $r = 0.08$, $t = 2$

Exponential and Logarithmic Equations

Convert between exponential and logarithmic forms:

• $\log_b a = x \iff b^x = a$

• Example: $\log_4 256 = x$

Properties of Logarithms

Logarithms can be expanded or condensed using these properties:

• $\log_b (xy) = \log_b x + \log_b y$

• $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$

• $\log_b (x^r) = r \log_b x$

Tables: Population Growth Model

Population Growth Table

Population growth can be modeled exponentially. The table below compares initial and projected populations:

2005 Population (millions)	Projected 2041 Population (millions)	Projected Growth Rate (%)
13.4	44.8	Additional info: Use exponential growth formula to calculate

Summary

This review covers essential Precalculus topics including systems of equations, functions and their properties, graphing, analytic geometry, polynomial and rational functions, inequalities, exponential and logarithmic functions, and applications such as compound interest and population growth. Mastery of these concepts is foundational for success in calculus and further mathematical studies.