Precalculus Review Problems: Lines, Functions, Graphs, and Logarithms

Lines and Linear Equations

Finding the Equation of a Line

To find the equation of a line passing through two points, use the point-slope form or slope-intercept form.

• Point-Slope Form: $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

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

• Example: Find the equation of the line through (2,5) and (4,7): $m = \frac{7-5}{4-2} = 1$ So, $y - 5 = 1(x - 2) \implies y = x + 3$

Solving Equations and Inequalities

Solving Linear Equations

To solve equations like $\frac{x-1}{2} = 3$, isolate $x$ by performing inverse operations.

• Multiply both sides by 2: $x-1 = 6$

• Add 1 to both sides: $x = 7$

Distance Formula and Applications

Distance Between Two Points

The distance between points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

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

• Example: Distance from (2,5) to (10,4): $d = \sqrt{(10-2)^2 + (4-5)^2} = \sqrt{64 + 1} = \sqrt{65}$

Functions and Their Graphs

Identifying and Sketching Functions

Functions can be represented algebraically and graphically. Key types include linear, quadratic, and piecewise functions.

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

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

• Graphing: Plot each piece over its domain, ensuring continuity or noting any jumps.

Transformations of Functions

• Vertical and Horizontal Shifts: $f(x) + c$ shifts up, $f(x-c)$ shifts right.

• Reflections: $-f(x)$ reflects over the x-axis.

• Stretching/Compressing: $af(x)$ stretches vertically if $|a|>1$.

Quadratic and Polynomial Functions

Graphing Quadratic Functions

• Standard Form: $y = ax^2 + bx + c$

• Vertex: $x = -\frac{b}{2a}$

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

• Example: $y = 2(x+2)^2 - 3$ is a parabola opening upward, vertex at $(-2, -3)$.

Exponential and Logarithmic Functions

Exponential Growth and Decay

• General Form: $y = ab^{x}$

• Applications: Population growth, radioactive decay, compound interest.

Logarithms

• Definition: $\log_b a = c$ means $b^c = a$

• Properties:

  • $\log_b (xy) = \log_b x + \log_b y$

  • $\log_b (x/y) = \log_b x - \log_b y$

  • $\log_b (x^r) = r \log_b x$

• Change of Base Formula: $\log_b a = \frac{\log_k a}{\log_k b}$

• Example: $\log_{10} 100 = 2$ because $10^2 = 100$

Applications and Word Problems

Optimization Problems

• Set up equations based on the problem statement.

• Use algebraic methods or calculus (if appropriate) to find maximum or minimum values.

• Example: Maximizing area with a fixed perimeter, or minimizing cost.

Interpreting Graphs and Functions

• Match equations to their graphs by identifying key features: intercepts, asymptotes, vertex, etc.

• Analyze transformations and their effects on the graph.

Sample Table: Logarithm Properties

Property	Equation
Product	$\log_b(xy) = \log_b x + \log_b y$
Quotient	$\log_b(x/y) = \log_b x - \log_b y$
Power	$\log_b(x^r) = r \log_b x$
Change of Base	$\log_b a = \frac{\log_k a}{\log_k b}$

Sequences and Series

Arithmetic and Geometric Sequences

• Arithmetic Sequence: $a_n = a_1 + (n-1)d$

• Geometric Sequence: $a_n = a_1 r^{n-1}$

• Sum of Arithmetic Series: $S_n = \frac{n}{2}(a_1 + a_n)$

• Sum of Geometric Series: $S_n = a_1 \frac{1 - r^n}{1 - r}$ (for $r \neq 1$)

Summary

• Review of lines, equations, and graphing techniques

• Practice with function transformations and matching equations to graphs

• Application of logarithmic properties and solving exponential/logarithmic equations

• Optimization and word problems involving algebraic modeling

• Sequences and series basics