Precalculus Final Review Study Notes

Functions and Graphs

Finding the Slope Between Two Points

The slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is a measure of its steepness.

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

• Example: For points $(-7, -2)$ and $(4, 3)$, $m = \frac{3 - (-2)}{4 - (-7)} = \frac{5}{11}$

Equation of a Line Perpendicular to a Given Line

To find the equation of a line through a point and perpendicular to a given line:

• Find the slope of the given line. For $y = 2x + 1$, slope $m = 2$.

• The perpendicular slope is the negative reciprocal: $m = -\frac{1}{2}$.

• Use 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.

• For rational functions $f(x) = \frac{2}{x^2 - 9}$, exclude values where the denominator is zero: $x^2 - 9 = 0 \implies x = \pm3$

• For logarithmic functions $f(x) = \log_2(2x + 3)$, require $2x + 3 > 0 \implies x > -\frac{3}{2}$

• For square root functions $f(x) = \sqrt{x^2 - 4}$, require $x^2 - 4 \geq 0 \implies x \leq -2$ or $x \geq 2$

Polynomial and Rational Functions

Function Operations and Difference Quotient

Operations on functions include addition, subtraction, and composition. The difference quotient is used to find the average rate of change.

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

• Example: For $f(x) = x^2 - 2x + 4$, compute $f(x+h)$ and $f(x)$, then substitute into the formula.

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.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals.

• Evaluate by determining which interval the input falls into.

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

Quadratic Functions and Their Graphs

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

• Vertex: $(h, k)$ where $h = -\frac{b}{2a}$

• Maximum/Minimum: Value at the vertex, depending on the sign of $a$

• Domain: All real numbers

• Range: $y \geq k$ if $a > 0$, $y \leq k$ if $a < 0$

• x-intercepts: Solve $ax^2 + bx + c = 0$

Polynomial Graphs and Zeros

Polynomials can be graphed by identifying degree, leading coefficient, zeros (with multiplicity), y-intercept, and end behavior.

• Degree: Highest power of $x$

• Leading Coefficient: Coefficient of highest power

• End Behavior: Determined by degree and leading coefficient

• Zeros: Solutions to $f(x) = 0$

Exponential and Logarithmic Functions

Solving Exponential and Logarithmic Equations

Exponential equations have the form $a^x = b$. Logarithmic equations use properties of logarithms.

• Example: $3^x = 411$

• Take logarithms to solve: $x = \log_3 411$

• Use properties: $\log_a b = \frac{\ln b}{\ln a}$

Converting Between Exponential and Logarithmic Forms

Exponential and logarithmic forms are related:

• $a^x = b \iff x = \log_a b$

• Example: $5^x = 125 \implies x = \log_5 125$

Applications: Loans and Compound Interest

Exponential functions model financial growth and decay.

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

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

Systems of Equations and Inequalities

Solving Linear Systems

Systems of equations can be solved by substitution or elimination.

• Example: $3x - 2y = 8$, $5x + y = 23$

• Solve for one variable, substitute into the other equation.

Additional Topics

Transformations of Functions

Transformations include shifts, stretches, and reflections.

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

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

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

Intervals of Increase, Decrease, and Constancy

Analyze the graph to determine where the function is increasing, decreasing, or constant.

• 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 $x_2 > x_1$

Table: Types of Asymptotes in Rational Functions

Type	How to Find	Example
Horizontal Asymptote	Compare degrees of numerator and denominator	$f(x) = \frac{x^2}{2x^2 + x - 1}$, H.A.: $y = \frac{1}{2}$
Vertical Asymptote	Set denominator equal to zero	$2x^2 + x - 1 = 0$
Slant Asymptote	Degree numerator = degree denominator + 1	$f(x) = \frac{x^2 + 4x + 2}{x + 1}$

Table: Properties of Polynomial Functions

Property	Description
Degree	Highest exponent of $x$
Leading Coefficient	Coefficient of highest degree term
Zeros	Values of $x$ where $f(x) = 0$
End Behavior	Behavior as $x \to \pm\infty$

Additional info: These notes expand on the original review questions by providing definitions, formulas, and examples for each major Precalculus topic covered in the file.