Precalculus Review: Distance, Circles, Functions, Lines, and Inequalities

Distance and Circles

Distance Between Two Points

The distance between two points in the coordinate plane can be found using the distance formula, which is derived from the Pythagorean Theorem.

• Distance Formula: For points $(x_1, y_1)$ and $(x_2, y_2)$, the distance $d$ is given by:

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

• Example: Find the distance between $(4, -5)$ and $(7, 7)$.

• Apply the formula: $d = \sqrt{(7-4)^2 + (7-(-5))^2} = \sqrt{3^2 + 12^2} = \sqrt{9 + 144} = \sqrt{153}$

Equations of Circles

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

$(x - h)^2 + (y - k)^2 = r^2$

• Center: $(h, k)$

• Radius: $r$

• Example: A circle with center $(2, -2)$ and radius $\sqrt{5}$ has the equation $(x - 2)^2 + (y + 2)^2 = 5$.

Finding Center and Radius from Equation

• Given an equation in the form $(x - h)^2 + (y - k)^2 = r^2$, the center is $(h, k)$ and the radius is $\sqrt{r^2}$.

• Example: $(x + 6)^2 + (y - 2)^2 = 9$ has center $(-6, 2)$ and radius $3$.

Standard Form of a Circle

• To write the equation of a circle in standard form, complete the square if necessary.

• Example: $(x - 2)^2 + (y + 3)^2 = 25$ is already in standard form.

Functions and Relations

Definition of a Function

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

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

Domain and Range

• Domain: The set of all possible input values (usually $x$ values).

• Range: The set of all possible output values (usually $y$ values).

• Example: For $f(x) = \sqrt{x - 6}$, the domain is $x \geq 6$.

Evaluating Functions

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

• Example: If $f(x) = x^2 + 2x$, then $f(3) = 3^2 + 2 \cdot 3 = 9 + 6 = 15$.

Linear Equations and Slope

Slope of a Line

• Slope Formula: For points $(x_1, y_1)$ and $(x_2, y_2)$:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

• Example: The slope between $(7, -4)$ and $(1, 7)$ is $m = \frac{7 - (-4)}{1 - 7} = \frac{11}{-6} = -\frac{11}{6}$.

Slope-Intercept Form

• The slope-intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Example: A line with slope $2$ passing through $(3, 5)$: $y - 5 = 2(x - 3) \implies y = 2x - 1$.

Parallel and Perpendicular Lines

• Parallel lines have equal slopes.

• Perpendicular lines have slopes that are negative reciprocals: $m_1 \cdot m_2 = -1$.

• Example: $y = 4x - 6$ and $y = 4x + 6$ are parallel because both have slope $4$.

Solving Equations and Inequalities

Solving Linear Equations

• To solve $ax + b = cx + d$, isolate $x$ on one side.

• Example: $2x + 5 = 4(x + 5)$

• Expand: $2x + 5 = 4x + 20$

• Rearrange: $2x - 4x = 20 - 5 \implies -2x = 15 \implies x = -\frac{15}{2}$

Solving and Graphing Inequalities

• To solve inequalities, use similar steps as equations, but reverse the inequality sign when multiplying or dividing by a negative.

• Example: $-2(x + 2) < -9(x - 5)$

• Expand: $-2x - 4 < -9x + 45$

• Rearrange: $7x < 49 \implies x < 7$

• Graph the solution on a number line, using open or closed circles as appropriate.

Interval Notation

• Interval notation is used to describe sets of numbers between two endpoints.

• Example: $x \in (a, b]$ means $a < x \leq b$.

Domain and Range from Graphs

• To find the domain and range from a graph, look at the extent of the graph along the $x$-axis (domain) and $y$-axis (range).

• Example: If a graph extends from $x = -3$ to $x = 2$, the domain is $[-3, 2]$.

• If the graph extends from $y = -2$ to $y = 3$, the range is $[-2, 3]$.

Summary Table: Key Formulas and Concepts

Concept	Formula/Definition	Example
Distance between points	$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$	$(4, -5)$ and $(7, 7)$: $d = \sqrt{153}$
Equation of a circle	$(x - h)^2 + (y - k)^2 = r^2$	Center $(2, -2)$, $r = \sqrt{5}$: $(x - 2)^2 + (y + 2)^2 = 5$
Slope of a line	$m = \frac{y_2 - y_1}{x_2 - x_1}$	$(7, -4)$ and $(1, 7)$: $m = -\frac{11}{6}$
Slope-intercept form	$y = mx + b$	$m = 2$, $b = -1$: $y = 2x - 1$
Function	Each input has one output	$f(x) = x^2$ is a function
Domain	All possible $x$ values	$f(x) = \sqrt{x-6}$: $x \geq 6$
Range	All possible $y$ values	Graph from $y = -2$ to $y = 3$: $[-2, 3]$
Parallel lines	Equal slopes	$y = 4x - 6$ and $y = 4x + 6$
Perpendicular lines	Slopes multiply to $-1$	$m_1 = 2$, $m_2 = -\frac{1}{2}$

Additional info:

• This review covers foundational Precalculus topics including analytic geometry, functions, linear equations, and inequalities, all of which are essential for further study in mathematics.