1.1 Graphs and Graphing Utilities Pre-calculus notes Date:

Draw the rectangular coordinate system (Cartesian coordinate system).

Label the four quadrants, the axes, origin, etc.

Ex. 1: Plot the points A(-2,4), B(4,-2), C(-3,0), and D(0,-3).

Graphs of equations in two variables—the set of all points (x,y) whose coordinates satisfy the equation.

Ex. 2: Graph 𝑦 = 4 − 𝑥. Select integers for x, starting with -3 and ending with 3.

Ex. 3: Graph 𝑦 = |𝑥 + 1|. Select integers for x, starting with -4 and ending with 2.

Graphing calculators: interpreting the viewing window [xmin, xmax, xstep].

Ex. 4: What is the meaning of a [−2,3,0.5] 𝑏𝑦 [−10,20,5]. Create a figure that illustrates the viewing

window.

x-intercept: x-coordinate of a point where the graph intersects the x-axis (y-coordinate is 0)

y-intercept y-coordinate of a point where the graph intersects the y-axis (x-coordinate is 0)

Additional Examples: Pgs. 98-100: #34,38,40,44,46,50,53,57-60,81,83

Homework: Pgs. 97: #18-54(mult of 3), 55-56, 65,71-80(all), 84,85

Quiz #2 over sections 1.1-1.4 will be Monday 9/19

1.2 Linear Equations and Rational Equations Pre-calculus notes Date:

Def: A linear equation in one variable x is of the form: ____________________

where _________________________

To SOLVE means to find the SOLUTION SET. Answers go in braces { }.

Examples: Solve and check.

Ex. 1: 4𝑥 + 5 = 29.

Ex. 2: 2(𝑥 − 3) − 17 = 13 − 3(𝑥 + 2).

Linear Equations with fractions: Multiply all terms on both sides by the LCD.

Ex. 3: 𝑥−3

4=

5

14−

𝑥+5

7.

Solving a Rational Equation (equation with variables in the denominator): Pay attention to restrictions

on the domain. If a value is NOT in the domain, then it CANNOT be in the solution set!

Ex. 4: 5

2𝑥=

17

18−

1

3𝑥.

Ex. 5: 𝑥

𝑥−2=

2

𝑥−2−

2

3.

Ex. 6: If 𝑦1 =1

𝑥+4+

1

𝑥−4 𝑎𝑛𝑑 𝑦2 =

22

𝑥2−16, 𝑤ℎ𝑒𝑛 𝑖𝑠 𝑦1 = 𝑦2?

Types of Equations:

1) Identity:

2) Conditional equation:

3) Inconsistent equation:

Ex. 7: Identify each of the following equations as an identity, conditional equation, or inconsistent

equation.

a) 4𝑥 − 7 = 4(𝑥 − 1) + 3

b) 2𝑥 + 8 = 2(𝑥 + 3) + 2

c) 4𝑥 + 5 = 29

Homework due Wed.: Pgs. 112-116: #3,9,13,17,21,25,29,33,37,41,

45,49,53,57,61,65,69,73,77,85,89,93,97-98, 110-114, 131,132

1.3 Models and Applications Pre-calculus notes Date:

Complete the following problems on pages 126-129 in your textbook. You may use this page or your

own notebook paper. Label each problem with the number from your textbook.

In class practice (notes): Pgs. 126-129: #3,9,13,17,19,27,29,37,41,60,69,71,73,80

Homework: Pgs. 126-129: #2 – 18(e), 20 – 26, 28, 30, 38, 40, 44, 62, 66, 70, 80 – 86(e) (Define all

variables before using them.)

1.4 Complex Numbers Pre-calculus notes Date:

Def: The imaginary unit, i, is defined as:

Ex: √−36 =

Complex numbers: the set of all numbers in the form _______________, where a and b are real

numbers

The real number a is called the _________________ and the real number b is called the

_____________________ of the complex number ____________.

If 𝑏 ≠ 0, the complex number is called an _______________________.

An imaginary number in the form ________ (a = 0) is called a ______________________________.

Examples of complex numbers:

“𝒊" 𝒄𝒉𝒂𝒓𝒕

𝑖 =

𝑖2 =

𝑖3 =

𝑖4 =

Based on this pattern, find each of the following.

𝑖50 = , 𝑖101 = . 𝑖400 = , 𝑖31 =

A complex number is said to be simplified if it is expressed in the standard form ________________.

The number is not considered to be simplified if it has any powers of i in it. (Use the “i” chart to

simplify.)

The statement 𝑎 + 𝑏𝑖 = 𝑐 + 𝑑𝑖 if and only if ______________ and ______________.

Adding and Subtracting Complex Numbers

*

*

Example 1: Perform the indicated operations, writing the result in standard form:

a) (5 − 2𝑖) + (3 + 3𝑖)

b) (2 + 6𝑖) − (12 − 𝑖)

Multiplication of complex numbers is performed the same way as multiplication of polynomials, using

the distributive property and the FOIL method.

Example 2: Find the products:

a) 7𝑖(2 − 9𝑖)

b) (5 + 4𝑖)(6 − 7𝑖)

Multiply 𝑎 + 𝑏𝑖 𝑏𝑦 𝑎 − 𝑏𝑖:

This illustrates that it is possible to multiply two imaginary numbers and obtain a real number.

The numbers 𝑎 + 𝑏𝑖 𝑎𝑛𝑑 𝑎 − 𝑏𝑖 are called ______________________________.

Since the definition of i indicates that it is a radical, an expression is not simplified if there is an i in

the denominator. The goal of the division process is to obtain a real number in the denominator.

Example 3: Divide and express the result in standard form: 5+4𝑖

4−𝑖

For any positive real number b, the principal square root of the negative number -b is defined by:

√−𝑏 =

When performing operations with square roots of negative numbers, begin by expressing all square

roots in terms of i.

Consider the problem: √−25 ∙ √−9

Correct: Incorrect:

Example 4: Perform the indicated operations and write the result in standard form:

a) √−27 + √−48 b) (−2 + √−3)2

c) −14+√−12

2 d) (3√−5)(−4√−12)

e) (2 − 3𝑖)(1 − 𝑖) − (3 − 𝑖)(3 + 𝑖) f) Evaluate 𝑥2 − 2𝑥 + 2 for 𝑥 = 1 + 𝑖

1.4 Homework: MathXL: 1.4

Quiz Monday over sections 1.1-1.4.

1.5 Quadratic Equations Pre-calculus notes Date:

A quadratic equation in x is an equation that can be written in the general form:

_________________________________,

where a,b, and c are real numbers with ________.

A quadratic equation in x is also called a _________________________________ in x.

Solving Quadratic Equations by Factoring

Zero-Product Principle: If AB = 0, then ________ or ________.

To solve a quadratic equation by factoring:

1. Rewrite the equation with a zero on one side.

2. Factor completely.

3. Apply the zero-product principle (set each factor equal to zero).

4. Solve the equations in step 3.

Ex. 1: Solve by factoring:

a) 3𝑥2 − 9𝑥 = 0. b) 2𝑥2 + 𝑥 = 1.

The real solutions of a quadratic equation are the x-intercepts of the graph.

Solving Quadratic Equations by the Square Root Property

If 𝑢2 = 𝑑, 𝑡ℎ𝑒𝑛 ______________ or ________________.

Ex. 2: Solve by the square root property:

a) 3𝑥2 − 21 = 0. b) 5𝑥2 + 45 = 0. c) (𝑥 + 5)2 = 11.

Completing the Square

If 𝑥2 + 𝑏𝑥 is a binomial, then by adding (𝑏

2)

2, which is the square of half the coefficient of x, a

perfect-square trinomial will result. That is,

𝑥2 + 𝑏𝑥 + (𝑏

2)

2

=

Ex. 3: What term should be added to each binomial so that is becomes a perfect-square trinomial?

Write and factor the trinomial.

a) 𝑥2 + 6𝑥 b) 𝑥2 − 5𝑥 c) 𝑥2 +2

3𝑥

Ex. 4: Solve by completing the square: 𝑥2 + 4𝑥 − 1 = 0.

In order to complete the square, the leading coefficient must be one.

Ex. 5: Solve by completing the square: 2𝑥2 + 3𝑥 − 4 = 0.

Solving Quadratic Equations Using the Quadratic Formula

The Quadratic Formula: The solutions of a quadratic equation in general form 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0,

with 𝑎 ≠ 0, are given by: 𝑥 = ________________________________________

Ex. 6: Solve using the quadratic formula: 2𝑥2 + 2𝑥 − 1 = 0.

Ex. 7: Solve using the quadratic formula: 𝑥2 − 2𝑥 + 2 = 0.

The Discriminant

The quantity ______________ is called the discriminant. The discriminant determines the number and

type of solutions.

1) Positive

2) Zero

3) Negative

Ex. 8: For each equation, compute the discriminant. Then determine the number and type of solutions:

a) 𝑥2 + 6𝑥 + 9 = 0 b) 2𝑥2 − 7𝑥 = 4 c) 3𝑥2 + 4 = 2𝑥

Homework: Pgs. 152-155: #7,9,13,21,25,33,37,41,45,53,57,61,63,67,71,73,75,81,

89,95,99,105,109,117,121,139,141,147,153-158,160,164-172even

1.6 Other Types of Equations Pre-calculus notes Date:

Polynomial Equations:

Ex. 1: Solve by factoring: 4𝑥4 = 12𝑥2.

Ex. 2: Solve by factoring: 2𝑥3 + 3𝑥2 = 8𝑥 + 12.

Radical Equations:

Isolate the radical

Raise both sides to the nth power for nth roots

Solve the equation. May need to repeat the process if there is another radical.

Check all proposed solutions in the original equation—There may be extraneous solutions

Ex. 3: Solve: √𝑥 + 3 + 3 = 𝑥.

Ex. 4: Solve: √𝑥 + 5 − √𝑥 − 3 = 2.

Equations with Rational Exponents:

Solving Radical Equations of the form 𝑥𝑚

𝑛 = 𝑘.

Isolate the expression with the rational exponent.

Raise both sides of the equation to the _________ power.

o If m is even: 𝑥 = ±𝑘𝑛

𝑚.

o If m is odd: 𝑥 = 𝑘𝑛

𝑚.

Ex. 5: Solve:

a) 5𝑥3

2 − 25 = 0. b) 𝑥2

3 − 8 = −4.

Equations that are Quadratic in Form:

Some equations that are not quadratic can be written as quadratic equations using an appropriate

substitution.

Ex. 6: Solve: 𝑥4 − 5𝑥2 + 6 = 0. Let u = ______________

Ex. 7: Solve: 3𝑥2

3 − 11𝑥1

3 − 4 = 0. Let u = ________________

Equations involving Absolute Value

Rewriting an Absolute Value Equation without Absolute Value Bars: If c is a positive real number and u

represents any algebraic expression, then |𝑢| = 𝑐 is equivalent to 𝑢 = 𝑐 𝑜𝑟 𝑢 = −𝑐.

Ex. 8: Solve: |2𝑥 − 1| = 5.

Note: Rewriting the equation should only be done AFTER isolating the absolute value expression on one

side of the equation.

Ex. 9: Solve: 4|1 − 2𝑥| − 20 = 0.

Ex. 10: Solve: |3𝑥 − 2| = −2.

Look at #79-84 on pg. 169.

Homework 1.6: MathXL 1.6

Homework 1.7: pg. 185 #2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86,

90, 94, 98, 102, 119, 124, 130, 132

Ch. 1 Test Friday

1.7 Linear Inequalities (Interval Notation) Pre-calculus notes Date:

Interval Notation: Let a and b be real numbers such that a < b.

Interval Notation Set-Builder Notation Graph

(𝑎, 𝑏)

[𝑎, 𝑏]

[𝑎, 𝑏)

(𝑎, 𝑏]

(𝑎, ∞)

[𝑎, ∞)

(−∞, 𝑏)

(−∞, 𝑏]

(−∞, ∞)

Ex. 1: Express each interval in set-builder notation and graph:

a) [−2,5) b) [1,3.5] c) (−∞, −1)

Finding Intersections and Unions of Two Intervals

1. Graph each interval on a number line.

2. To find the intersection, take the portion of the number line that the two graphs have in

common. Symbol: ∩

3. To find the union, take the portion of the number line representing the total collection of

numbers in the two graphs. Symbol: ∪

Ex. 2: Use graphs to find each set:

a) [1,3] ∩ (2,6) b) [1,3] ∪ (2,6)

Properties of Inequalities:

Addition/Subtraction Property of Inequality:

Positive Multiplication/Division Properties of Inequality:

Negative Multiplication/Division Properties of Inequality:

Ex. 3: Solve and graph the solution set on a number line: 2 − 3𝑥 ≤ 5.

Ex. 4: Solve and graph the solution set on a number line: 3𝑥 + 1 > 7𝑥 − 15.

Ex. 5: Solve and graph the solution set on a number line:

𝑥−4

2≥

𝑥−2

3+

5

6

Ex. 6: Solve each inequality:

a) 3(𝑥 + 1) > 3𝑥 + 2 b) 𝑥 + 1 ≤ 𝑥 − 1

Solving Compound Inequalities – Goal: Isolate 𝑥 in the middle of the inequality.

Ex. 7: Solve and graph the solution set on a number line:

1 ≤ 2𝑥 + 3 < 11

Solve Inequalities with Absolute Value

Solving an Absolute Value Inequality

If 𝑢 is an algebraic expression and 𝑐 is a positive number.

1. The solutions of |𝑢| < 𝑐 are the numbers that satisfy ___________________________________

2. The solutions of |𝑢| > 𝑐 are the numbers that satisfy ___________________________________

These rules are valid if < is replaced with ≤ and > is replaced with ≥

Ex. 8: Solve and graph the solution set on a number line:

|𝑥 − 2| < 5

Ex. 9: Solve and graph the solution set on a number line:

−3|5𝑥 − 2| + 20 ≤ −19

Ex. 10: Solve and graph the solution set on a number line:

18 < |6 − 3𝑥|

Ex. 11: A car can be rented from Company A for $260 per week with no extra charges for mileage.

Company B charges $80 per week plus 25 cents for each mile driven to rent the same car. How many

miles must be drive in a week to make the rental cost for Company A a better deal than Company B’s?

Homework 1.7: pg. 185 #2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86,

90, 94, 98, 102, 119, 124, 130, 132

Homework (due Thursday 9/22): pg. 195 #1 – 44