Comprehensive Study Guide: College Algebra Final Exam Topics (Units 1-3)

Graphs, Functions, and Models

Distance and Midpoint Formulas

The distance formula and midpoint formula are essential tools for analyzing points in the coordinate plane.

• Distance Formula: Calculates the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$. $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Midpoint Formula: Finds the midpoint between two points $(x_1, y_1)$ and $(x_2, y_2)$. $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Example: For points (2, 3) and (6, 7):

  • Distance: $d = \sqrt{(6-2)^2 + (7-3)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}$

  • Midpoint: $M = (\frac{2+6}{2}, \frac{3+7}{2}) = (4, 5)$

Solving Equations Algebraically and Graphically

Equations can be solved using algebraic manipulation or by interpreting their graphs.

• Linear Equations: Equations of the form $ax + b = 0$; solution is $x = -\frac{b}{a}$.

• Quadratic Equations: Equations of the form $ax^2 + bx + c = 0$; solutions found using factoring, completing the square, or the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Exponential Equations: Equations involving terms like $a^x$; often solved using logarithms.

• Logarithmic Equations: Equations involving $\log_b(x)$; solved by exponentiating both sides.

• Graphical Solutions: The solution to $f(x) = 0$ is the x-intercept of the graph of $f(x)$.

• Example: Solve $2x + 3 = 7$ algebraically: $x = 2$; graphically, find where the line $y = 2x + 3$ crosses $y = 7$.

Solving Inequalities

Inequalities express a range of possible solutions and can be solved algebraically or graphically.

• Linear Inequalities: $ax + b > 0$ or $ax + b < 0$; solve like equations, then graph solution on a number line.

• Quadratic Inequalities: $ax^2 + bx + c > 0$; solve by finding roots and testing intervals.

• Example: Solve $x^2 - 4 < 0$; roots at $x = -2, 2$; solution is $-2 < x < 2$.

More on Functions

Identifying and Graphing Functions

Functions can be linear, quadratic, exponential, or logarithmic. Their graphs have distinct shapes and properties.

• Linear Functions: $f(x) = mx + b$; graph is a straight line.

• Quadratic Functions: $f(x) = ax^2 + bx + c$; graph is a parabola.

• Exponential Functions: $f(x) = a b^x$; graph increases or decreases rapidly.

• Logarithmic Functions: $f(x) = a \log_b(x)$; graph increases slowly, undefined for $x \leq 0$.

• Example: Graph $f(x) = 2^x$; passes through (0,1), (1,2), (2,4).

Domain and Range of Functions

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

• From Graph: Identify the x-values (domain) and y-values (range) covered by the graph.

• From Equation: Analyze restrictions (e.g., denominator cannot be zero, argument of square root must be non-negative).

• Example: $f(x) = \sqrt{x-1}$; domain: $x \geq 1$, range: $y \geq 0$.

x- and y-Intercepts of a Function

Intercepts are points where the graph crosses the axes.

• x-intercept: Set $y = 0$ and solve for $x$.

• y-intercept: Set $x = 0$ and solve for $y$.

• Example: For $f(x) = x^2 - 4$, x-intercepts at $x = -2, 2$; y-intercept at $y = -4$.

Even, Odd, and Neither Functions

Functions can be classified 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.

• Neither: Does not satisfy either property.

• Example: $f(x) = x^2$ is even; $f(x) = x^3$ is odd.

Transformations of Graphs and Equations

Transformations alter the position and shape of a graph.

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

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

• Vertical Compression/Stretch: $a f(x)$ compresses if $0 < a < 1$, stretches if $a > 1$.

• Horizontal Compression/Stretch: $f(bx)$ compresses if $b > 1$, stretches if $0 < b < 1$.

• Example: $f(x) = (x-2)^2 + 3$ is a parabola shifted right 2 units and up 3 units.

Quadratic, Exponential, and Logarithmic Functions

Linear Models

Linear models describe relationships with constant rates of change.

• Equation: $y = mx + b$

• Application: Used in economics, physics, and statistics to model trends.

• Example: Predicting cost based on quantity: $C = 5q + 20$

Composition of Functions

Composition combines two functions into one by substituting one into the other.

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

• Example: If $f(x) = x^2$, $g(x) = x + 1$, then $(f \circ g)(x) = f(x+1) = (x+1)^2$

Inverse Functions

An inverse function reverses the effect of the original function.

• Definition: $f^{-1}(f(x)) = x$ for all $x$ in the domain of $f$.

• Finding the Inverse: Swap $x$ and $y$ in $y = f(x)$, then solve for $y$.

• Example: $f(x) = 2x + 3$; inverse: $f^{-1}(x) = \frac{x-3}{2}$

Equations for Inverse of One-to-One Functions

One-to-one functions have unique inverses.

• Process: Set $y = f(x)$, solve for $x$ in terms of $y$, then write $f^{-1}(y)$.

• Example: $f(x) = 3x - 5$; $y = 3x - 5 \Rightarrow x = \frac{y+5}{3}$; $f^{-1}(y) = \frac{y+5}{3}$

Solving Logarithmic Expressions

Logarithmic expressions can be simplified using properties of logarithms.

• Product Rule: $\log_b(xy) = \log_b(x) + \log_b(y)$

• Quotient Rule: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$

• Power Rule: $\log_b(x^k) = k \log_b(x)$

• Example: $\log_2(8) = 3$ because $2^3 = 8$

Systems of Equations and Matrices

Solving Systems of Equations

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

• Methods: Substitution, elimination, and matrix methods.

• Example: Solve:

  • $x + y = 5$

  • $x - y = 1$

  Solution: Add equations: $2x = 6 \Rightarrow x = 3$; then $y = 2$.

Function Type	General Form	Graph Shape	Domain	Range
Linear	$f(x) = mx + b$	Straight line	All real numbers	All real numbers
Quadratic	$f(x) = ax^2 + bx + c$	Parabola	All real numbers	$y \geq$ vertex (if $a > 0$)
Exponential	$f(x) = a b^x$	Curve, rapid increase/decrease	All real numbers	$y > 0$ (if $a > 0$)
Logarithmic	$f(x) = a \log_b(x)$	Slowly increasing curve	$x > 0$	All real numbers

Additional info: Some context and examples have been inferred to ensure completeness and clarity for exam preparation.