Precalculus Study Guide: Graphs, Lines, Circles, and Functions (Sections 1.1–2.3)

Graphs and the Rectangular Coordinate System

Plotting Points and Identifying Coordinates

The rectangular (Cartesian) coordinate system allows us to represent points in a plane using ordered pairs (x, y). Each point corresponds to a unique location defined by its horizontal (x) and vertical (y) distances from the origin (0, 0).

• Plotting Points: To plot a point, move x units along the x-axis and y units along the y-axis.

• Identifying Coordinates: Given a point on the plane, its coordinates are written as (x, y).

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

Distance and Midpoint Formulas

To analyze the relationship between two points, we use the distance and midpoint formulas.

• Distance Formula: The distance between points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

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

• Example: For points (1, 4) and (5, 2): Distance: $\sqrt{(5-1)^2 + (2-4)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}$ Midpoint: $\left( \frac{1+5}{2}, \frac{4+2}{2} \right) = (3, 3)$

Application Problems

• Word problems may involve finding distances, midpoints, or interpreting coordinates in real-world contexts (e.g., navigation, geometry).

Graphing Equations and Analyzing Graphs

Intercepts and Graphing Using Intercepts

Intercepts are points where a graph crosses the axes.

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

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

• Graphing: Plot the intercepts and additional points to sketch the curve.

• Example: For $y = 2x - 4$, x-intercept: $0 = 2x - 4 \implies x = 2$; y-intercept: $y = 2(0) - 4 = -4$.

Symmetry of Graphs

• y-axis symmetry: Replace x with -x; if the equation is unchanged, the graph is symmetric about the y-axis.

• x-axis symmetry: Replace y with -y; if unchanged, symmetric about the x-axis.

• Origin symmetry: Replace x with -x and y with -y; if unchanged, symmetric about the origin.

• Example: $y = x^2$ is symmetric about the y-axis.

Calculator Skills: ZERO and INTERSECT Functions

• ZERO Function: Used to find x-intercepts (roots) of a function on a graphing calculator.

• INTERSECT Function: Used to approximate solutions to equations by finding where two graphs intersect.

• Application: Useful for solving equations graphically and checking solutions.

Linear Equations and Their Graphs

Slope of a Line

The slope measures the steepness of a line.

• Formula: For points $(x_1, y_1)$ and $(x_2, y_2)$: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Interpretation: Positive slope rises left to right; negative slope falls left to right.

• Example: Points (2, 3) and (5, 11): $m = \frac{11-3}{5-2} = \frac{8}{3}$

Equations of Lines

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

• Slope-Intercept Form: $y = mx + b$

• General Form: $Ax + By + C = 0$

• Finding the Equation: Use given information (point, slope, or two points) to write the equation in the desired form.

• Example: Through (1, 2) with slope 3: $y - 2 = 3(x - 1) \implies y = 3x - 1$

Graphing Lines

• Using Slope and a Point: Plot the point, use the slope to find another point, draw the line.

• Using Intercepts: Find x- and y-intercepts, plot both, draw the line.

Parallel and Perpendicular Lines

• Parallel Lines: Have equal slopes ($m_1 = m_2$).

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

• Example: $y = 2x + 1$ is parallel to $y = 2x - 3$; perpendicular to $y = -\frac{1}{2}x + 4$.

Word Problems Involving Lines

• Translate real-world situations into linear equations and solve for unknowns.

Circles in the Coordinate Plane

Center and Radius of a Circle

A circle is the set of all points equidistant from a fixed point (the center).

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

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

• Finding Center and Radius: Complete the square to convert general form to standard form.

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

Completing the Square

• Used to rewrite the general form of a circle into standard form.

• Steps: Group x and y terms, complete the square for each variable, adjust the equation accordingly.

• Example: $x^2 + y^2 - 4x + 6y = 12$ $x^2 - 4x + y^2 + 6y = 12$ Complete the square: $x^2 - 4x + 4 + y^2 + 6y + 9 = 12 + 4 + 9$ $(x - 2)^2 + (y + 3)^2 = 25$ Center: (2, -3), Radius: 5

Word Problems Involving Circles

• Apply circle equations to real-world contexts, such as navigation, design, or geometry.

Functions and Their Properties

Relations and Functions

A relation is a set of ordered pairs. A function is a relation in which each input (x) has exactly one output (y).

• Vertical Line Test: If any vertical line crosses the graph more than once, it is not a function.

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

Determining if an Equation is a Function

• For each x, there must be only one y.

• Check by solving for y in terms of x and seeing if multiple y-values result for a single x.

Evaluating Functions and Finding Domains

• Evaluating: Substitute the input value into the function.

• Domain: The set of all permissible input values (x-values) for which the function is defined.

• Example: For $f(x) = \frac{1}{x-2}$, domain is all real numbers except $x = 2$.

Operations with Functions

• Addition: $(f+g)(x) = f(x) + g(x)$

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

• Multiplication: $(fg)(x) = f(x) \cdot g(x)$

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

• Domain of Combined Functions: Intersection of the domains of the original functions, with additional restrictions for division.

Difference Quotient

• The difference quotient is a formula used to compute the average rate of change of a function over an interval and is foundational for calculus.

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

• Example: For $f(x) = x^2$, $\frac{(x+h)^2 - x^2}{h} = \frac{x^2 + 2xh + h^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h$

Word Problems Involving Functions

• Apply function concepts to real-world scenarios, such as cost, revenue, or motion problems.

Analyzing Graphs of Functions

Graph Information: Domain, Range, Intercepts

• Domain: All x-values for which the function is defined.

• Range: All y-values the function attains.

• Intercepts: Points where the graph crosses the axes.

Even and Odd Functions

• Even Function: $f(-x) = f(x)$ for all x in the domain; symmetric about the y-axis.

• Odd Function: $f(-x) = -f(x)$ for all x in the domain; symmetric about the origin.

• Neither: If neither condition holds.

• Example: $f(x) = x^2$ is even; $f(x) = x^3$ is odd; $f(x) = x + 1$ is neither.

Relative/Local Maxima and Minima; Increasing/Decreasing Intervals

• Relative Maximum: A point where the function changes from increasing to decreasing.

• Relative Minimum: A point where the function changes from decreasing to increasing.

• Increasing Interval: Where the function rises as x increases.

• Decreasing Interval: Where the function falls as x increases.

• Graphing Utility: Use calculators to approximate these values and intervals.

Average Rate of Change

• The average rate of change of $f(x)$ from $x = a$ to $x = b$ is: $\frac{f(b) - f(a)}{b - a}$

• Example: For $f(x) = x^2$ from $x = 1$ to $x = 3$: $\frac{9 - 1}{3 - 1} = \frac{8}{2} = 4$

Secant Line

• The secant line passes through two points on the graph of a function.

• Equation: Use the two points $(x_1, f(x_1))$ and $(x_2, f(x_2))$ to find the slope and write the equation in point-slope form.

Evaluating Difference Quotients

• Plug values into the difference quotient formula to compute the result.