College Algebra Exam 1 Study Guide: Graphs, Functions, and Linear Equations

Graphs, Functions, and Models

Distance and Midpoint Between Two Points

Understanding the geometric relationship between two points in the coordinate plane is fundamental in algebra. The distance formula and midpoint formula are essential tools for analyzing points and line segments.

• Distance Formula: The distance between points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Midpoint Formula: The midpoint $M$ of the segment joining $(x_1, y_1)$ and $(x_2, y_2)$ is: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

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

Equations of Circles

The equation of a circle in the coordinate plane is based on its center and radius.

• Standard Form: For a circle with center $(h, k)$ and radius $r$: $(x - h)^2 + (y - k)^2 = r^2$

• Finding Center and Radius: Given an equation in standard form, the center is $(h, k)$ and the radius is $r$.

• Example: $(x - 3)^2 + (y + 2)^2 = 16$ has center $(3, -2)$ and radius $4$.

More on Functions

Relations and Functions

A relation is a set of ordered pairs. A function is a relation in which each input (domain value) corresponds to exactly one output (range value).

• Determining Functions: A relation is a function if no input value is paired with more than one output.

• Domain and Range: The domain is the set of all possible input values; the range is the set of all possible output values.

• Example: The relation $\{(1,2), (2,3), (3,4)\}$ is a function. The domain is $\{1,2,3\}$; the range is $\{2,3,4\}$.

Evaluating Functions and Determining Domain

Given a function $f(x)$, you can evaluate it at specific values and determine its domain.

• Evaluating: Substitute the given value into the function.

• Domain of Rational Functions: Exclude values that make the denominator zero.

• Domain of Square Root Functions: The expression under the square root must be non-negative.

• Example: For $f(x) = \frac{1}{x-2}$, the domain is all real numbers except $x=2$.

Linear Equations and Their Graphs

Slope and Equations of Lines

The slope of a line measures its steepness, and the equation of a line can be written in several forms.

• Slope Formula: For points $(x_1, y_1)$ and $(x_2, y_2)$: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Slope-Intercept Form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Standard Form: $Ax + By = C$

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Parallel Lines: Have the same slope.

• Perpendicular Lines: Slopes are negative reciprocals: $m_1 \cdot m_2 = -1$

• Example: Find the equation of a line through $(1,2)$ with slope $3$:$y - 2 = 3(x - 1) \implies y = 3x - 1$

Linear Models and Regression

Linear regression is used to model the relationship between two variables using a straight line.

• Linear Model: $y = mx + b$ is fitted to data points to predict future values.

• Application: Use a calculator or software to find the best-fit line for a table of values.

Solving Linear Equations and Inequalities

Solving equations and inequalities is a core skill in algebra.

• Linear Equations: Solve for the variable using inverse operations.

• Linear Inequalities: Solve similarly to equations, but reverse the inequality sign when multiplying or dividing by a negative number.

• Interval Notation: Express solution sets using intervals, e.g., $(-\infty, 2]$.

• Example: Solve $2x - 3 < 7$: $2x < 10 \implies x < 5$; solution: $(-\infty, 5)$

Word Problems: Perimeter and Area

Algebraic equations can be used to solve geometric word problems involving perimeter and area.

• Perimeter of a Rectangle: $P = 2l + 2w$

• Area of a Rectangle: $A = lw$

• Distance, Rate, Time: $d = rt$

• Example: A rectangle has length $5$and width $3$. Perimeter: $P = 2(5) + 2(3) = 16$; Area: $A = 5 \times 3 = 15$

Graphing and Analyzing Functions

Graphing Functions and Identifying Intervals

Graphing functions helps visualize their behavior, including where they increase, decrease, or remain constant.

• Increasing Interval: Where the function rises as $x$ increases.

• Decreasing Interval: Where the function falls as $x$ increases.

• Constant Interval: Where the function remains unchanged as $x$ increases.

• Relative Maxima/Minima: Points where the function reaches a local highest or lowest value.

Piecewise-Defined Functions

A piecewise-defined function is defined by different expressions over different intervals of the domain.

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

• Graph each piece over its specified interval.

Operations and Compositions of Functions

Operations on Functions

Functions can be combined using addition, subtraction, multiplication, and division.

• Sum: $(f+g)(x) = f(x) + g(x)$

• Difference: $(f-g)(x) = f(x) - g(x)$

• Product: $(fg)(x) = f(x) \cdot g(x)$

• Quotient: $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

• Domain: The domain of the combined function is the intersection of the domains of $f$ and $g$, except for division, where $g(x) \neq 0$.

Composition of Functions

The composition of two functions $f$ and $g$ is written as $(f \circ g)(x) = f(g(x))$.

• Order Matters: $(f \circ g)(x) \neq (g \circ f)(x)$ in general.

• Domain: The domain of $f \circ g$ consists of all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.

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

Summary Table: Key Formulas and Concepts

Concept	Formula	Description
Distance between points	$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$	Length of segment between $(x_1, y_1)$ and $(x_2, y_2)$
Midpoint	$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$	Point halfway between $(x_1, y_1)$ and $(x_2, y_2)$
Circle (standard form)	$(x - h)^2 + (y - k)^2 = r^2$	Circle with center $(h, k)$ and radius $r$
Slope	$m = \frac{y_2 - y_1}{x_2 - x_1}$	Steepness of a line
Slope-intercept form	$y = mx + b$	Line with slope $m$ and y-intercept $b$
Point-slope form	$y - y_1 = m(x - x_1)$	Line through $(x_1, y_1)$ with slope $m$
Perimeter of rectangle	$P = 2l + 2w$	Sum of all sides
Area of rectangle	$A = lw$	Product of length and width
Distance, rate, time	$d = rt$	Distance equals rate times time

Additional info: This guide covers foundational topics from the first chapters of a college algebra course, including coordinate geometry, functions, linear equations, and basic modeling. For further practice, refer to the textbook pages and problem numbers listed in your review sheet.