College Algebra Exam 1 Review: Study Notes and Key Concepts

Review of Basic Algebra and Linear Equations

Solving Linear Equations and Substitution

Linear equations are equations of the first degree, meaning the variable(s) are only to the first power. Substitution is a method used to determine if a given ordered pair is a solution to an equation.

• Key Point: Substitute the x and y values from the ordered pair into the equation and check if the equation is satisfied.

• Example: For the equation $3x - 2y = 5$, check if (1, -2) is a solution by substituting x = 1 and y = -2.

Intercepts and Graphing Lines

The intercepts of a line are the points where the line crosses the x-axis and y-axis.

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

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

• Graphing: Plot the intercepts and draw a straight line through them.

• Example: For $3x + 2y = 12$, x-intercept: $x = 4$, y-intercept: $y = 6$.

Graphing Linear Equations

To graph a linear equation, plot at least two points that satisfy the equation and connect them with a straight line.

• Example: Graph $y = -2x + 4$ by finding points such as (0, 4) and (2, 0).

Functions and Their Properties

Definition of a Function

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

• Vertical Line Test: A graph represents a function if no vertical line intersects the graph at more than one point.

• Example: The set {(1,2), (2,3), (3,4)} is a function, but {(1,2), (1,3)} is not.

Domain and Range

The domain of a function is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values).

• Example: For $f(x) = \sqrt{x-1}$, the domain is $x \geq 1$.

Evaluating Functions

To evaluate a function, substitute the given value for the variable.

• Example: If $f(x) = 2x^2 - 3x + 1$, then $f(2) = 2(2)^2 - 3(2) + 1 = 8 - 6 + 1 = 3$.

Circles and Distance

Equation of a Circle

The standard form of the equation of a circle with center $(h, k)$ and radius $r$ is:

• $(x - h)^2 + (y - k)^2 = r^2$

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

Distance Formula

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

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

• Application: Used to find the length of a segment or the diameter/radius of a circle.

Midpoint Formula

The midpoint of a segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is:

• $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Graphing and Analyzing Functions

Graphing Quadratic and Other Functions

Quadratic functions have the form $f(x) = ax^2 + bx + c$ and graph as parabolas. Other functions may include square roots, absolute values, or piecewise definitions.

• Example: $f(x) = \sqrt{x-2}$ is defined for $x \geq 2$ and its graph starts at (2,0).

Piecewise Functions

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

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

Transformations of Functions

Transformations include shifting, stretching, compressing, and reflecting graphs.

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

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

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

• Vertical Stretch/Shrink: $a f(x)$ stretches if $|a| > 1$, shrinks if $0 < |a| < 1$.

Properties of Lines

Slope and y-Intercept

The slope of a line measures its steepness, and the y-intercept is where the line crosses the y-axis.

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

• Slope-Intercept Form: $y = mx + b$

Parallel and Perpendicular Lines

• Parallel Lines: Have the same slope ($m_1 = m_2$).

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

Function Operations and Composition

Operations on Functions

Functions can be added, subtracted, multiplied, or divided (where defined).

• Sum: $(f+g)(x) = f(x) + g(x)$

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

• Product: $(fg)(x) = f(x)g(x)$

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

Composition of Functions

The composition of $f$ and $g$ is $(f \circ g)(x) = f(g(x))$.

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

Analyzing Graphs and Function Behavior

Increasing, Decreasing, and Constant Intervals

A function is increasing where its graph rises as you move left to right, decreasing where it falls, and constant where it remains flat.

• Example: For a function $f(x)$, if $f(x_1) < f(x_2)$ for $x_1 < x_2$, then $f$ is increasing on that interval.

Relative Maxima and Minima

A relative maximum is a point where the function value is higher than all nearby points; a relative minimum is lower than all nearby points.

• Identified visually on the graph as peaks (maxima) and valleys (minima).

Application Problems

Modeling with Linear Equations

Linear equations can be used to model real-world situations, such as cost, revenue, or salary over time.

• Example: $C = 20x + 20000$ models the cost $C$ to produce $x$ units, where 20 is the cost per unit and 20000 is the fixed cost.

Solving Word Problems

Translate the problem into an equation, solve for the unknown, and interpret the result in context.

• Example: If the endpoints of a diameter are given, use the midpoint formula to find the center of the circle.

Tables: Function Correspondence and Data Modeling

Function Correspondence Table

Tables can be used to determine if a correspondence is a function (each input has only one output).

x	y
1	11
2	12
3	17
4	14

Main Purpose: To check if each x-value is paired with only one y-value.

Salary Data Table

Used for modeling salary as a function of years of experience.

Years	Salary
1	$32,000
2	$34,000
3	$36,000
4	$38,000

Main Purpose: To model the relationship between years and salary using a linear equation.

Additional info: These notes cover topics from Ch. R, Ch. 1, and Ch. 2 of a typical College Algebra course, including equations, graphing, functions, and applications.