College Algebra Study Guide: Coordinates, Functions, Graphs, and Operations

Rectangular Coordinates and Graphs

Distance and Midpoint Formulas

The rectangular coordinate system is fundamental for graphing equations and analyzing geometric relationships. The distance formula calculates the distance between two points P(x_1, y_1) and Q(x_2, y_2) in the plane:

• Distance Formula: $d(P, Q) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Midpoint Formula: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Example: Find the distance and midpoint between $P(-\sqrt{7}, 8\sqrt{3})$ and $Q(5\sqrt{7}, -\sqrt{13})$.

Right Triangles and Collinearity

To determine if three points form a right triangle, use the Pythagorean Theorem. Points are collinear if the sum of the distances between two pairs equals the distance between the remaining pair.

• Right Triangle: $a^2 + b^2 = c^2$ for sides $a$, $b$, and hypotenuse $c$.

• Collinearity: Three points are collinear if they lie on a straight line.

Graphing by Point Plotting

Graphing equations by plotting points involves finding x- and y-intercepts and plotting solutions.

• X-intercept: Solve for $x$ when $y = 0$.

• Y-intercept: Solve for $y$ when $x = 0$.

• Example Equations: $y = 4x - 1$, $y = \sqrt{x - 2}$, $y = |x + 4|$, $y = -x^3$, etc.

Circles

Equation of a Circle

The equation of a circle in the coordinate plane can be written in center-radius form or general form.

• Center-Radius Form: $(x - h)^2 + (y - k)^2 = r^2$ where $(h, k)$ is the center and $r$ is the radius.

• General Form: $x^2 + y^2 + Dx + Ey + F = 0$

• Special Cases:

  • $r > 0$: Circle

  • $r = 0$: Single point

  • $r < 0$: Nonexistent

Identifying Circles from Equations

Given equations, determine if they represent circles and find their centers and radii.

• Example: $4x^2 + 9y^2 + 4x - 16y - 19 = 0$

• Convert to center-radius form to identify circle properties.

Functions

Definition and Properties

A function is a relation in which each input (first component) has exactly one output (second component). The domain is the set of all possible input values, and the range is the set of all possible output values.

• Ordered Pair: $(x, y)$

• Independent Variable: $x$

• Dependent Variable: $y$

• No two ordered pairs in a function can have the same first component with different second components.

Domain and Range from Relations and Graphs

Determine the domain and range for given relations and graphs, and decide if the relation is a function.

• Example: Relation $ig\\{(-4, -2), (-1, 0), (3, 5)\\ig\\}$

• Use mapping diagrams and tables to analyze functions.

Vertical Line Test

The vertical line test determines if a graph represents a function: if any vertical line intersects the graph at more than one point, it is not a function.

Function Classification and Domain/Range

Decide whether a relation defines a function and give its domain and range.

• Example: $y = x + 4$ (Function, domain $(-\infty, \infty)$, range $(-\infty, \infty)$)

• For $x^2 = 12x$, not a function.

Function Evaluation

Evaluate functions for given values and analyze mappings.

• Example: $g(x) = -x^2 + 4x + 1$, find $g(-4)$, $g(2)$, $g(-x)$

Linear Functions

Definition and Standard Form

A linear function is of the form $f(x) = ax + b$. The standard form of a linear equation is $Ax + By = C$, where $A$, $B$, and $C$ are integers and $A > 0$.

• Domain and Range: If $a \neq 0$, domain and range are $(-\infty, \infty)$.

• Example: $f(x) = \frac{2}{3}x + 4$, $f(x) = 2$, $x = -3$

Slope of a Line

The slope $m$ of a line through $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

• $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Positive Slope: Line rises left to right

• Negative Slope: Line falls left to right

• Horizontal Line: $y = a$, $m = 0$

• Vertical Line: $x = a$, slope undefined

Finding and Graphing Slopes

Find the slope for given conditions and graph lines with specified slopes.

• Example: Through $(3, -1)$ and $(-3, 5)$, horizontal through $(3, 5)$, vertical through $(-8, 5)$

• Graph lines passing through points with indicated slopes.

Equations of Lines and Linear Models

Forms of Linear Equations

Linear equations can be written in several forms:

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

• Slope-Intercept Form: $y = mx + b$

• Standard Form: $Ax + By = C$

• Horizontal Line: $y = b$, $m = 0$

• Vertical Line: $x = a$, slope undefined

Writing Equations for Lines

Write equations for lines given points, slopes, and intercepts in both standard and slope-intercept forms.

• Example: Through $(-3, 2)$ and $(-1, -4)$, through $(5, 1)$ with undefined slope, through $(5, 1)$ with $m = 0$, x-intercept $(-4, 0)$, y-intercept $(0, 3)$

Parallel and Perpendicular Lines

Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals.

• Parallel: $m_1 = m_2$

• Perpendicular: $m_1 \cdot m_2 = -1$

• Write equations for lines parallel or perpendicular to given lines.

Graphs of Basic Functions

Continuity

A function is continuous on an interval if its graph can be drawn without lifting the pencil. Discontinuity occurs at points where the graph is broken.

• Continuous over entire domain: $(-\infty, \infty)$

• Point of discontinuity: At $x = 3$, continuous over $(-\infty, 3)$ and $(3, \infty)$

Basic Function Types

Common functions include identity, squaring, cubing, square root, cube root, absolute value, and greatest integer functions.

• Identity Function: $f(x) = x$, domain and range $(-\infty, \infty)$

• Squaring Function: $f(x) = x^2$, domain $(-\infty, \infty)$, range $[0, \infty)$

• Cubing Function: $f(x) = x^3$, domain and range $(-\infty, \infty)$

• Square Root Function: $f(x) = \sqrt{x}$, domain and range $[0, \infty)$

• Cube Root Function: $f(x) = \sqrt[3]{x}$, domain and range $(-\infty, \infty)$

• Absolute Value Function: $f(x) = |x|$, domain $(-\infty, \infty)$, range $[0, \infty)$

• Greatest Integer Function: $f(x) = \lfloor x \rfloor$, domain $(-\infty, \infty)$, range is all integers

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals.

• Example: $f(x) = \begin{cases} -3x & x < -3 \\ 3x - 1 & -3 \leq x \leq 2 \\ -4x & x > 2 \end{cases}$

• Evaluate and graph piecewise functions.

Even and Odd Functions

A function is even if $f(-x) = f(x)$ for all $x$ in its domain, and odd if $f(-x) = -f(x)$ for all $x$.

• Example: $f(x) = 8x^4 - 3x^2 + 1$ (even), $f(x) = 6x^3 - 9x$ (odd), $f(x) = 3x^2 + 5x$ (neither)

Graphing Techniques

Transformations

Graph transformations include translations, stretches, shrinks, and reflections.

• Vertical Translation: $y = f(x) + K$ (up), $y = f(x) - K$ (down)

• Horizontal Translation: $y = f(x + h)$ (left), $y = f(x - h)$ (right)

• Horizontal Stretch/Shrink: $y = f(ax)$ ($a > 1$ shrink, $0 < a < 1$ stretch)

• Vertical Stretch/Shrink: $y = af(x)$ ($a > 1$ stretch, $0 < a < 1$ shrink)

• Reflection: $y = -f(x)$ (x-axis), $y = f(-x)$ (y-axis)

Graphing Functions with Transformations

Apply transformations to graph functions such as $g(x) = 2|x|$, $h(x) = \frac{1}{2}|x|$, $K(x) = |x|$, $f(x) = -\sqrt{x}$, $h(x) = \sqrt{-x}$, $s(x) = |x - 4|$, $f(x) = -|x + 1| + 1$, $h(x) = |2x - 4|$.

Symmetry

Types of Symmetry

Graphs may exhibit symmetry with respect to the x-axis, y-axis, or origin.

• Y-axis Symmetry: Replace $x$ with $-x$

• X-axis Symmetry: Replace $y$ with $-y$

• Origin Symmetry: Replace both $x$ and $y$ with $-x$ and $-y$

Function Operations and Composition

Operations on Functions

Functions can be added, subtracted, multiplied, and divided.

• Sum: $(f + g)(x) = f(x) + g(x)$

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

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

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

Domains of Function Operations

The domain of a function operation is the intersection of the domains of the individual functions, with additional restrictions for division.

• Example: $f(x) = 8x - 9$, $g(x) = \sqrt{x - 1}$

• Domain for $g(x)$: $[1, \infty)$

• Domain for $\frac{f}{g}$: Intersection of domains, $g(x) \neq 0$

Difference Quotient

The difference quotient is used to analyze rates of change:

• $\frac{f(x + h) - f(x)}{h}$, $h \neq 0$

• Find the difference quotient for various functions.

Composition of Functions

The composition of functions $f$ and $g$ is defined as $(f \circ g)(x) = f(g(x))$. The domain is determined by the domains of $f$ and $g$.

• Example: $f(x) = \frac{6}{x - 3}$, $g(x) = \frac{1}{x}$

• Find $(f \circ g)(x)$ and $(g \circ f)(x)$ with domains.