Precalculus Study Notes: Graphing Equations, Linear and Rational Equations

Section 1.1 – Introduction to Graphing Equations

Definition of a Cartesian Plane

The rectangular coordinate system consists of a horizontal line called the x-axis, a vertical line called the y-axis, and their point of intersection called the origin.

• Ordered Pair: A solution to an equation if, when substituted, it makes the equation true.

• Graph of an Equation: The set of all ordered pairs (x, y) that satisfy the equation.

Example: For the equation $y = x^2 + 5x + 4$, check if (0, 4) and (-1, 1) are solutions.

• For (0, 4): $y = (0)^2 + 5(0) + 4 = 4$ (True)

• For (-1, 1): $y = (-1)^2 + 5(-1) + 4 = 1 - 5 + 4 = 0$ (False, since $y \neq 1$)

Graphing Equations by Plotting Points

• Choose values for x, calculate corresponding y values, and plot the points.

• Connect the points to reveal the graph's shape.

Example: For $y = x^2 + 5x + 4$, plot points for $x = -2, -1, 0, 1, 2$.

x	-2	-1	0	1	2
y	0	0	4	10	18

Standard Viewing Window

The standard viewing window for graphing calculators is typically:

x minimum	x maximum	y minimum	y maximum	x scale	y scale
-10	10	-10	10	1	1

Adjusting the window settings changes the visible portion of the graph.

X and Y Intercepts

• X-intercept: Where the graph crosses the x-axis ($y = 0$).

• Y-intercept: Where the graph crosses the y-axis ($x = 0$).

Example: For a given graph, record intercepts as (x, 0) and (0, y).

Section 1.2 – Linear and Rational Equations

Solving Equations Using a Graphing Utility (Zero or Root Method)

1. Write the equation in the form $expression = 0$.

2. Graph the equation.

3. Use the ZERO (or ROOT) feature to find the x-intercepts (solutions).

Example: Solve $x^2 + 2x - 8 = 0$ using the zero method. Answers: $x = -4, 2$.

Solving Equations Using Intersection Utility

1. Write each side of the equation as a separate function.

2. Graph both functions and use the INTERSECT feature to find the solution.

Example: Solve $x^2 = 5x - 1$ by graphing $y_1 = x^2$ and $y_2 = 5x - 1$ and finding their intersection points.

Linear Equations

• A linear equation can be written as $ax + b = 0$ where $a, b$ are real numbers.

• To solve algebraically: simplify, isolate the variable, and check your answer.

Example: Solve $8x - 5x - 12 = 3x - 10$.

• Combine like terms: $3x - 12 = 3x - 10$

• Subtract $3x$ from both sides: $-12 = -10$ (No solution)

Rational Equations

• A rational equation contains a variable in the denominator.

• To solve algebraically:

  1. Identify the domain restrictions (values that make the denominator zero).

  2. Multiply both sides by the least common denominator (LCD).

  3. Solve the resulting equation.

  4. Check for extraneous solutions (solutions that do not satisfy the original equation).

Example: Solve $\frac{x}{x-2} = 5$.

• LCD: $x-2$

• Multiply both sides: $x = 5(x-2) \Rightarrow x = 5x - 10 \Rightarrow -4x = -10 \Rightarrow x = \frac{5}{2}$

• Check that $x \neq 2$ (domain restriction).

Extraneous Solutions

• An extraneous solution is a solution that does not satisfy the original equation after solving.

Example: Solve $\frac{x}{x+1} = \frac{7}{x+1}$.

• LCD: $x+1$

• Multiply both sides: $x = 7$

• Check that $x \neq -1$ (domain restriction).

Solving for the Indicated Variable

• To solve for a variable, isolate it on one side of the equation.

Example: Solve $A = P + Prt$ for $t$.

• Subtract $P$ from both sides: $A - P = Prt$

• Divide both sides by $Pr$: $t = \frac{A - P}{Pr}$

Summary Table: Rational Equation Solution Steps

Step	Description
1	Identify domain restrictions (values that make denominator zero)
2	Multiply both sides by the LCD
3	Solve the resulting equation
4	Check for extraneous solutions

Additional info: These notes cover foundational Precalculus topics including graphing, intercepts, solving equations algebraically and graphically, and handling rational equations with attention to domain and extraneous solutions. Examples and calculator instructions are included for practical application.