Precalculus Study Guide test 1: Graphs, Lines, Circles, and Functions (Sections 1.1–1.4, B3/B4, 2.1–2.3)

Graphs and the Rectangular Coordinate System

Plotting Points and Identifying Coordinates

The rectangular (Cartesian) coordinate system is used to graph points, lines, and curves in two dimensions. Each point is represented by an ordered pair (x, y).

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

• Identifying Coordinates: The coordinates of a point are written as (x, y), where x is the horizontal value and y is the vertical value.

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

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, 2) and (5, 6): Distance: $d = \sqrt{(5-1)^2 + (6-2)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}$ Midpoint: $M = (\frac{1+5}{2}, \frac{2+6}{2}) = (3, 4)$

Graphing Equations and Analyzing Graphs

Intercepts and Symmetry

Intercepts and symmetry help in sketching and understanding graphs.

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

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

• Symmetry:

  • 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: The graph of $y = x^2$ 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 the intersection points of two graphs.

• Application: These functions help solve equations graphically and check solutions.

Linear Equations and Their Graphs

Slope and Forms of Linear Equations

The slope of a line measures its steepness and direction.

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

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

• Slope-Intercept Form: $y = mx + b$, where m is the slope and b is the y-intercept.

• General Form: $Ax + By + C = 0$

• Example: Find the equation of a line through (2, 3) with slope 4: $y - 3 = 4(x - 2) \implies y = 4x - 5$

Graphing Lines and Analyzing Relationships

• Graphing Using Intercepts: Set x = 0 for y-intercept, y = 0 for x-intercept, plot both, and draw the line.

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

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

• Example: Are the lines $y = 2x + 1$ and $y = -\frac{1}{2}x + 3$ perpendicular? Yes, since $2 \times (-\frac{1}{2}) = -1$.

Circles in the Coordinate Plane

Equations and Properties of Circles

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 from general to standard form.

• Example: Given $x^2 + y^2 - 4x + 6y - 12 = 0$, complete the square: $x^2 - 4x + y^2 + 6y = 12$ $(x - 2)^2 - 4 + (y + 3)^2 - 9 = 12$ $(x - 2)^2 + (y + 3)^2 = 25$ Center: (2, -3), Radius: 5

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-value) corresponds to exactly one output (y-value).

• 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 (fails the vertical line test).

Evaluating and Operating on Functions

• Function Evaluation: Substitute the input value into the function.

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

• Operations: For functions f and g: $(f + g)(x) = f(x) + g(x)$ $(f - g)(x) = f(x) - g(x)$ $(f \cdot g)(x) = f(x) \cdot g(x)$ $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

• Domain of Operations: The intersection of the domains of f and g, except for division where $g(x) \neq 0$.

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

Difference Quotient

The difference quotient is a fundamental concept for understanding rates of change and the foundation of calculus.

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

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

Analyzing Graphs of Functions

Graph Features: 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.

• Example: For $f(x) = \sqrt{x}$, domain is $x \geq 0$, range is $y \geq 0$.

Even and Odd Functions

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

• Odd Function: $f(-x) = -f(x)$ for all x in the domain. Graph is 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 Extrema and Intervals of Increase/Decrease

• 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 values rise as x increases.

• Decreasing Interval: Where the function values fall as x increases.

• Example: For $f(x) = -x^2 + 4$, the vertex at (0, 4) is a maximum.

Average Rate of Change and Secant Lines

• Average Rate of Change: For $f(x)$ from $x = a$ to $x = b$: $\frac{f(b) - f(a)}{b - a}$

• Secant Line: The line passing through $(a, f(a))$ and $(b, f(b))$; its slope is the average rate of change.

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

Summary Table: Key Formulas and Concepts

Concept	Formula/Definition	Example
Distance between points	$\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$	Between (1,2) and (4,6): $\sqrt{9 + 16} = 5$
Midpoint	$\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$	Between (1,2) and (4,6): (2.5, 4)
Slope	$\frac{y_2 - y_1}{x_2 - x_1}$	Between (1,2) and (4,6): $\frac{6-2}{4-1} = \frac{4}{3}$
Standard form of circle	$(x - h)^2 + (y - k)^2 = r^2$	Center (2, -3), radius 5: $(x-2)^2 + (y+3)^2 = 25$
Difference quotient	$\frac{f(x+h) - f(x)}{h}$	If $f(x) = x^2$, result is $2x + h$
Average rate of change	$\frac{f(b) - f(a)}{b - a}$	$f(x) = x^2$, $a=1$, $b=3$: $4$