The Timpson Lyceum · College Algebra

Lecture 12 (College Algebra)

Overview

This lecture transitions from a brief trigonometry review into the core algebra of the course: graphs, relations, functions, and lines. You will learn what it means to graph an equation (the graph as a solution set), how to recognize a linear equation, and why every linear equation graphs as a straight line. Along the way the vocabulary of relations and functions is made precise — including function notation, domain, and range — and functions are combined with arithmetic. The lecture closes with the family of equations for a line (two-point, point-slope, intercept, slope-intercept forms), a proof that linear equations and lines correspond one-to-one, and the conditions for lines to be parallel, perpendicular, or coincident.

Key Concepts

Radian (trig recap). A radian is the angle subtended by an arc whose length equals the radius. Since the circumference is $C = 2\pi r$, it takes $2\pi$ radii to wrap all the way around a circle, so a full turn is $2\pi$ radians. Hence $360^\circ = 2\pi$ rad, $180^\circ = \pi$ rad. The definition of $\pi$ itself is $\pi = \dfrac{C}{d}$ (circumference over diameter).

Right-triangle trig (recap). For a right triangle with hypotenuse $r$ and reference angle $\theta$:

• horizontal leg: $x = r\cos\theta$
• vertical leg: $y = r\sin\theta$
• slope of the hypotenuse: $\tan\theta = \dfrac{y}{x} = \dfrac{\sin\theta}{\cos\theta}$

There are six trig functions ($\sin,\cos,\tan$ and their reciprocals $\csc,\sec,\cot$); standard values such as $\cos\frac{\pi}{3} = \frac12$ come from the unit circle.

Graph (locus) of an equation. The graph — or locus — of an equation is the set of all points $(x, y)$ whose coordinates satisfy the equation. Graphing an equation means plotting its entire solution set; a point lies on the graph exactly when substituting it makes the equation true.

Linear equation in two variables. An equation that can be written as $Ax + By + C = 0,$ where $A, B, C$ are constants and both $x$ and $y$ appear only to the first power. For example $x = 2$ qualifies (take $A=1,\ B=0,\ C=-2$). An equation with a variable raised to any power other than one (e.g. $y = x^2 + 2x + 1$) is not linear — its graph is a curve such as a parabola.

Relation. A set of ordered pairs (each element is a pair $(a,b)$). The empty set is vacuously a relation.

Function. A relation in which each left-hand element (input) is paired with exactly one right-hand element (output) — "every input has a unique output." A repeated output is fine; a repeated input with different outputs is not. Note that $\{(1,2),(1,2)\}$ is just $\{(1,2)\}$, so it is a function. The vertical line test is the geometric version of this check.

Function notation. Writing $f(x) = x^2$ is shorthand for the set $\{(x,y) : y = x^2\}$. The statement $f(a) = b$ means exactly that the ordered pair $(a,b)$ belongs to $f$. Only functions support this notation. $x$ is the independent variable; $y$ or $f(x)$ is the dependent variable.

Domain and range. The domain $D_f$ is the set of all inputs (left-hand elements); the range (or image) is the set of all outputs (right-hand elements). They need not be the same size. For $f(x)=x^2$ over the reals, the domain is all real numbers and the range is $[0,\infty)$ — a square bracket includes the endpoint $0$. Unless stated otherwise, all numbers are real.

Arithmetic of functions. Functions combine pointwise. For $f(x)=x^2+1$ and $g(x)=2x-5$:

• $(f+g)(x) = x^2 + 2x - 4$
• $(f\cdot g)(x) = (x^2+1)(2x-5) = 2x^3 - 5x^2 + 2x - 5$
• $\left(\dfrac{f}{g}\right)(x) = \dfrac{x^2+1}{2x-5}$

Solution function. Given an expression $h(x,y) = 0$, solving for $y$ produces a solution function $y = f(x)$ such that the original equation holds whenever $y = f(x)$. Some equations yield more than one solution function: $x^2 + y^2 = 25$ solves to $y = \pm\sqrt{25 - x^2}$ (the upper and lower halves of a circle of radius $5$), each branch a separate function.

Equations of a line. All come from the fact that collinear points share one slope:

• Two-point form: $\dfrac{y - y_1}{x - x_1} = \dfrac{y_2 - y_1}{x_2 - x_1}$
• Point-slope form: $y - y_1 = m(x - x_1)$
• Intercept form (intercepts $a$ on the $x$-axis, $b$ on the $y$-axis): $\dfrac{x}{a} + \dfrac{y}{b} = 1$
• Slope-intercept form: $y = mx + b$
• Vertical line: has no slope; its equation is $x = a$.

Linear equations ↔ lines (one-to-one). Every linear equation graphs as a line, and every line is described by a linear equation.

Parallel, perpendicular, coincident. Two lines are parallel iff they have equal slopes. They are coincident (the same line) iff they are parallel and share the same $y$-intercept. Perpendicular lines have slopes that are negative reciprocals: if one slope is $m$, the perpendicular slope is $-\dfrac1m$ (e.g. slope $\frac34 \Rightarrow$ perpendicular slope $-\frac43$).

Worked Example

Find the equation of the line through $(-2, 3)$ that is perpendicular to $3x - 4y = 5$.

First find the slope of the given line by writing it in slope-intercept form, then take the negative reciprocal for the perpendicular slope, and finally apply point-slope form:

The perpendicular line is $y = -\dfrac{4}{3}x + \dfrac{1}{3}$.

Summary

• The graph (locus) of an equation is its solution set: all points $(x,y)$ that satisfy it.
• A linear equation has the form $Ax + By + C = 0$ with $x, y$ to the first power; it always graphs as a line, and conversely every line has such an equation.
• A relation is any set of ordered pairs; a function is a relation in which each input has exactly one output (vertical line test). $f(a)=b$ means $(a,b)\in f$.
• The domain is the set of inputs and the range the set of outputs; for $f(x)=x^2$ the domain is all reals and the range is $[0,\infty)$.
• A line can be written in two-point, point-slope ($y - y_1 = m(x-x_1)$), intercept ($\frac{x}{a}+\frac{y}{b}=1$), or slope-intercept ($y = mx + b$) form; choose whichever fits the given information.
• Lines are parallel when slopes are equal, coincident when slopes and $y$-intercepts both match, and perpendicular when slopes are negative reciprocals.