Precalculus Study Notes: Functions, Graphs, and Equations of Lines

1. Introduction to Functions and Their Graphs

1.1 Ordered Pairs and the Coordinate Plane

An ordered pair is a pair of numbers written as (x, y), where the first number is the x-coordinate and the second is the y-coordinate. Ordered pairs are used to locate points on the coordinate plane.

• Plotting Points: To plot (x, y), move x units along the horizontal axis and y units along the vertical axis.

• Example: The point (3, -2) is 3 units right and 2 units down from the origin.

1.2 The Equation of a Graph

The graph of an equation in two variables is the set of all points (x, y) that satisfy the equation. Each solution corresponds to a point on the graph.

• x-intercept: The point where the graph crosses the x-axis (y = 0).

• y-intercept: The point where the graph crosses the y-axis (x = 0).

• Example: For the equation $y = 2x + 1$, the y-intercept is (0, 1).

2. Basics of Functions and Their Graphs

2.1 Definition of a Function

A function is a rule that assigns to each element x in a set called the domain exactly one element y in a set called the range. The function is often written as $y = f(x)$.

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

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

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

• Example: $f(x) = x^2$ is a function; $x = y^2$ is not a function.

2.2 Graphs of Functions

The graph of a function is the set of all ordered pairs (x, f(x)).

• Intercepts: Points where the graph crosses the axes.

• Domain and Range from Graph: The domain is the set of all x-values on the graph; the range is the set of all y-values.

3. Equations of Lines and Circles

3.1 Slope of a Line

The slope (m) of a line measures its steepness and is calculated as:

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

• Interpretation: The change in y divided by the change in x between two points $(x_1, y_1)$ and $(x_2, y_2)$.

3.2 Point-Slope and Slope-Intercept Forms

• Point-Slope Form: $y - y_1 = m(x - x_1)$ Used to find the equation of a line given a point and the slope.

• Slope-Intercept Form: $y = mx + b$ Used to find y-values or graph the line; b is the y-intercept.

3.3 Distance Formula

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

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

3.4 Equation of a Circle

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

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

4. More on Slopes

• Parallel Lines: Have equal slopes.

• Perpendicular Lines: Have slopes that are negative reciprocals.

• Example: If one line has slope 2, a perpendicular line has slope $-\frac{1}{2}$.

5. Transformations of Functions

Transformations change the position or shape of a graph. Common transformations include:

• Vertical Shifts: $f(x) + c$ shifts the graph up by c units.

• Horizontal Shifts: $f(x - c)$ shifts the graph right by c units.

• Reflections: $-f(x)$ reflects the graph over the x-axis; $f(-x)$ reflects over the y-axis.

• Vertical Stretch/Compression: $a f(x)$ stretches (if $|a| > 1$) or compresses (if $0 < |a| < 1$) the graph vertically.

6. Combinations and Inverse Functions

6.1 Combinations of Functions

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

• $(f + g)(x) = f(x) + g(x)$

• $(f - g)(x) = f(x) - g(x)$

• $(f \cdot g)(x) = f(x) \cdot g(x)$

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

6.2 Composite Functions

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

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

6.3 Inverse Functions

The inverse of a function $f$ is denoted $f^{-1}$ and satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ for all x in the domain of $f^{-1}$.

• Finding the Inverse: Solve $y = f(x)$ for x in terms of y, then interchange x and y.

• Example: If $f(x) = 3x + 2$, then $y = 3x + 2 \implies x = \frac{y - 2}{3}$, so $f^{-1}(x) = \frac{x - 2}{3}$.

7. Summary Table: Forms and Properties of Lines

Form	Equation	Description
Slope Formula	$m = \frac{y_2 - y_1}{x_2 - x_1}$	Slope between two points $(x_1, y_1)$ and $(x_2, y_2)$
Point-Slope Form	$y - y_1 = m(x - x_1)$	Line with slope m through $(x_1, y_1)$
Slope-Intercept Form	$y = mx + b$	Line with slope m and y-intercept b
Distance Formula	$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$	Distance between two points
Circle Equation	$(x - h)^2 + (y - k)^2 = r^2$	Circle with center $(h, k)$ and radius r

Additional info: You do NOT need to memorize the basic graphs at the beginning of section 1.6 (e.g., $y = x^2$, $y = 2$, etc.).