Precalculus Final Exam Study Guide: Key Concepts and Formulas

Functions and Their Properties

Determining if a Relation is a Function

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

• Ordered Pairs: If any x-value repeats with different y-values, the relation is not a function.

• Graphical Test (Vertical Line Test): If a vertical line intersects the graph more than once, the relation is not a function.

• Example: (1,2), (2,3), (1,5) is not a function because x=1 maps to two different y-values.

Average Rate of Change

The average rate of change of a function f(x) from x=a to x=b is given by:

• $\frac{f(b)-f(a)}{b-a}$

• Example: For $f(x)=x^2$ from x=1 to x=3: $\frac{9-1}{3-1}=\frac{8}{2}=4$

Inverse Functions

An inverse function reverses the roles of inputs and outputs. To find the inverse:

1. Replace f(x) with y.

2. Swap x and y.

3. Solve for y.

4. Rename y as $f^{-1}(x)$.

• Example: $y=2x+3$ Swap: $x=2y+3$ Solve: $y=\frac{x-3}{2}$ Inverse: $f^{-1}(x)=\frac{x-3}{2}$

Polynomial and Rational Functions

Polynomial Functions

• Degree: The highest exponent of x. Example: $x^4-3x^2+1$ has degree 4.

• Domain: All real numbers $(-\infty, \infty)$.

• Range: Depends on the graph's behavior.

• Zeros: Solve $f(x)=0$. Example: $x^2-4=0$ gives $x=2, -2$.

• Local Maximum/Minimum: Turning points (peaks and valleys).

• Increasing/Decreasing Intervals: Where the graph rises or falls from left to right.

• Continuity: Polynomials are always continuous (no breaks or holes).

Graphing with Transformations

• Parent Function: $y=x^2$

• Transformations:

  • $(x-h)$: shift right h units

  • $(x+h)$: shift left h units

  • +$k$: shift up k units

  • −$k$: shift down k units

  • Negative outside: reflect over x-axis

• Example: $y=(x-2)^2+3$ is shifted right 2 and up 3. Vertex: (2,3)

Piecewise Functions

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

• Example: $f(x)=\begin{cases} x+1 & x<0 \\ x^2 & x\geq0 \end{cases}$

• Check domain, range, and whether endpoints are included (open/closed circles).

Graphing Polynomial Functions

• Zeros and Multiplicity: The number of times a zero occurs affects the graph's behavior at that point.

  • Even multiplicity: graph bounces

  • Odd multiplicity: graph crosses

• End Behavior: Determined by degree and leading coefficient.

  • Even degree, positive leading coefficient: up on both sides

Rational Functions

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

• x-intercepts: Set numerator = 0 ($x=-1$)

• y-intercept: Plug in $x=0$ ($y=-1/2$)

• Vertical Asymptotes: Set denominator = 0 ($x=2$)

• Horizontal Asymptote: If degrees are equal, ratio of leading coefficients ($y=1$)

• Holes: Factor and cancel common factors

Rational Zero Theorem

• Possible rational zeros: $\pm\frac{\text{factors of constant}}{\text{factors of leading coefficient}}$

• Test candidates using synthetic division

Exponential and Logarithmic Functions

Exponential Equations

• General Form: $y=a\cdot b^x$

• Growth: $b>1$

• Decay: $0

• Horizontal Asymptote: $y=0$

• Solving Equations: If bases are equal, set exponents equal. Example: $2^{x+1}=2^5 \implies x+1=5 \implies x=4$

Logarithmic Equations and Properties

• Product Rule: $\log_a(MN)=\log_aM+\log_aN$

• Quotient Rule: $\log_a(M/N)=\log_aM-\log_aN$

• Power Rule: $\log_a(M^p)=p\log_aM$

• Example: $\log(x-1)=2 \implies x-1=10^2 \implies x=101$

Trigonometric Functions and Applications

Basic Trigonometric Ratios (SOH CAH TOA)

• $\sin\theta=\frac{\text{opp}}{\text{hyp}}$

• $\cos\theta=\frac{\text{adj}}{\text{hyp}}$

• $\tan\theta=\frac{\text{opp}}{\text{adj}}$

Graphing Trigonometric Functions

• Parent Sine Function: $y=\sin x$

• Period: $2\pi$

• Cosine: Starts at maximum value

Trigonometric Identities

• Pythagorean Identity: $\sin^2x+\cos^2x=1$

Polar Coordinates and Vectors

Polar and Rectangular Conversion

• Polar to Rectangular:

  • $x=r\cos\theta$

  • $y=r\sin\theta$

• Rectangular to Polar:

  • $r=\sqrt{x^2+y^2}$

  • $\tan\theta=\frac{y}{x}$

Vectors

• Magnitude: $|\vec v|=\sqrt{x^2+y^2}$

• Dot Product: $\vec a\cdot\vec b=a_1b_1+a_2b_2$

• Cross Product: Defined only in 3D

Analytic Geometry (Conic Sections)

Parabolas

• Standard Form: $y=a(x-h)^2+k$

• Vertex: (h, k)

• Direction: a>0 opens up, a<0 opens down

• Axis of Symmetry: $x=h$

Ellipses

• Center: (h, k)

• Major/Minor Axes: Longest/shortest diameters

Hyperbolas

• Standard Form: $\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$

• Asymptotes: $y-k=\pm\frac{b}{a}(x-h)$

Systems of Equations and Matrices

Solving Systems of Equations

• Methods: Substitution, elimination

Matrices

• Operations: Addition, subtraction, multiplication

• Multiplication Rule: Inside dimensions must match

Sequences and Series

Arithmetic Sequences

• Formula: $a_n=a_1+(n-1)d$

Geometric Sequences

• Formula: $a_n=a_1r^{n-1}$

Infinite Geometric Series

• Convergence: $|r|<1$

• Sum: $S=\frac{a}{1-r}$

Limits and Applications

Limits

• Definition: The value a function approaches as x approaches a specific value.

• Example: $\lim_{x\to2}(x^2+1)=5$

Compound Interest

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

• P = principal, r = annual rate, n = number of times compounded per year, t = years

Triangles and Trigonometric Laws

Triangle Cases

• SSS, SAS, ASA, AAS, SSA (ambiguous case)

Law of Sines

• $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

Law of Cosines

• $c^2=a^2+b^2-2ab\cos C$

Final Exam Checklist

• Identify functions

• Graph transformations

• Solve logarithmic and exponential equations

• Graph polynomial and rational functions

• Find asymptotes and holes

• Use synthetic division

• Graph trigonometric functions

• Prove trigonometric identities

• Convert between polar and rectangular coordinates

• Solve vector operations

• Graph conic sections

• Solve systems of equations

• Work with matrices

• Solve sequences and series

• Evaluate limits

• Solve triangle problems

• Use compound interest formulas