P.1 Real Numbers

PreCalculus

P.1 REAL NUMBERS Learning Targets for P1 1. Describe an interval on the number line using inequalities 2. Describe an interval on the number line using interval notation (closed vs. open) 3. Switch between interval notation and inequality notation 4. Simplify exponential expressions 5. Identify Algebraic Properties (Commutative, Associative, Distributive, Identity, Inverse) Inequalities and Intervals When you first graphed inequalities you used an open circle or a solid dot. Interval notation is another way to describe intervals instead of using an inequality. The ends of an interval are either OPEN or CLOSED. When using interval notation use ___________________ instead of open circles, and use _____________________ instead of solid dots. Example 1: Graph the following on a number line and write your answer using interval notation. a) 5 < x

b) x < 2

c) 7 < x < 12

Example 2: Graph the following on a number line. a) (3, 5]

b)

[ −12, ∞ )

c)

( −∞, 7 )

Example 3: Use interval notation to describe each interval on the x – axes below.

a

b

a

b

a

b

a

b

a

b

c

a

Unit 1 - 1

b

c

P.1 Real Numbers

PreCalculus

Properties of Algebra Commutative Property

Associative Property

Inverse Property

Identity Property

Distributive Property

Unit 1 - 2

P.3 Linear Equations and Inequalities

PreCalculus

P.3 LINEAR EQUATIONS AND INEQUALITIES Learning Targets for P.3 1. Solve multiple step linear equations 2. Solve multiple step linear inequalities 3. Solve equations with fractions Solving Linear Equations Example 1: Solve each of the following equations.. a)

2 (3 – 4b) – 5 (2b + 3) = b – 17

c)

x 5 2x  1 4 15 25

b)

x  3 x 1  1 4 6

Solving Linear Inequalities Example 2: Solve each inequality, graph the solution on a number line, and write the answer in interval notation. a)

2 y  3 3 y 1   y 1 2 5

b)

1 5  x  4  x  3  x  4 2

Example 3: Solve the following inequality and graph the solution on a number line:

2  3x  4  5

Unit 1 - 3

Factoring

PreCalculus

FACTORING Learning Targets for Factoring 1. Factor out a GCF 2. Factor difference of squares 3. Factor a quadratic expression of the form ax2 + bx + c 4. Completely Factor an expression (including grouping) Greatest Common Factor (GCF) The first step in factoring is to factor out a GCF. We did this in P.1. Example 1: Factor out the GCF from the expression. 5 x 4 − 7 x 3 + 3 x 2

Factoring Quadratic Expressions of the Form ax2 + bx + c Example 2: Factor each expression completely.

−18 x 2 + 57 x − 45

Difference of Two Perfect Squares: a 2 − b 2 = (a + b)(a − b) Example 3: Factor each expression. a)

b) 36 x 2 + 49

9 x 2 − 25 y 2

Example 4: Completely factor each expression. a) 3 ( 2a − 3) + 17 ( 2a − 3) + 10 .

b) 4 x 4 + 24 x 3 + 5 x + 30

2

Unit 1 - 4

P.5 Solving Equations Graphically, Numerically, and Algebraically

PreCalculus

P.5 SOLVING EQUATIONS GRAPHICALLY, NUMERICALLY, AND ALGEBRAICALLY Learning Targets for P.5 1. Solve equations graphically with a calculator (2 ways) 2. Solve quadratic equations algebraically: x2 but no x, x2 and x but no c, x2 and x and c, undo (…)2 3. Complete the square 4. Solve Absolute value equations Being able to solve equations in multiple ways allows you to answer a question using the easiest method possible. Another advantage of having multiple ways to solve an equation is that it gives you an opportunity to check your work. The three methods are obvious from the title of the section … Graphically, Numerically, and Algebraically. Solve an Equation by Graphing There are two keys to solving an equation by graphing. 1. Understanding WHERE the solution(s) is(are). 2. Finding those solutions using your calculator (or a hand drawn graph). Solutions are located in _________ possible places, either the _____________________or _________________________. Example 1: Solve the equation x 3 − x = 1 by graphing the left side of the equation and the right side of the equation. a) Where is(are) the solution(s) to this equation located?

b) What is the solution to this equation?

Example 2: Solve the equation x 3 − x = 1 by setting one side equal to 0, and graphing the other side. (This is equivalent to finding the ZEROS of a function) a) Where is(are) the solution(s) to this equation located?

b) What is the solution to this equation?

Solve an Equation Numerically Typically, you will be given a table of values when asked to solve an equation numerically. You could also create the table yourself. Example 3: The table below shows ordered pairs for f  x   x 3  x  1 . Use the table to approximate the solution to the equation 0 =− x 3 + x + 1 . x 1.1 1.2 1.3 1.4 1.5 1.6 1.7

f (x) 0.769 0.472 0.103 – 0.344 – 0.875 – 1.496 – 2.213

Unit 1 - 5

P.5 Solving Equations Graphically, Numerically, and Algebraically

PreCalculus

Solve an Equation Algebraically This is the method you have spent the most time using in math classes so far. We focused on linear equations earlier in chapter P. In this section we are going to focus on various methods of solving quadratic equations. Polynomials, rational equations, exponential, logarithmic, and trigonometric equations will be done in later sections throughout the year. Method 1: Square Root Both Sides Example 4: Solve by extracting square roots:

2 ( 2 y + 3) − 9 = 23 2

Method 2: Complete the Square Example 5: Solve by completing the square. a) x 2 + 6 x = 7

b) 2 x 2 + 5 x − 12 = 0

Method 3: Quadratic Formula Example 6: Solve using the quadratic formula: 3 x 2 − 6 x − 7 = 2 x + 3

Unit 1 - 6

P.5 Solving Equations Graphically, Numerically, and Algebraically

PreCalculus

Method 4: Factoring Example 7: Solve by factoring. a) 7 x 2 = 3 x

b) 4 x 2 + 3 = 8x

Definition of Absolute Value Absolute Value is best viewed as a distance. When finding the absolute value of a single number, you are finding the _______________________ that number is from _______________. We will talk about the algebraic definition more when we graph functions in the next chapter.

Example 8: Solve for all values of x:

x 7

Example 9: Consider the equation 2 3 x + 5 − 8 = 19 . a) Solve the equation algebraically.

b) Solve the equation graphically.

Unit 1 - 7

P.6 Solving Inequalities Algebraically and Graphically

PreCalculus

P.6 SOLVING INEQUALITIES ALGEBRAICALLY AND GRAPHICALLY Learning Targets for P.6 1. Solve Absolute Value Inequalities using correct notation and vocabulary 2. Solve quadratic or cubic inequalities by finding zeros on a graph 3. Apply solving inequalities to context including but not limited to projectile motion

Rules to Remember When Solving Absolute Value Inequalities Let u be an algebraic expression in x and let a be a real number greater than 0. 1. u < a if and only if −a < u < a 2. u > a if and only if u < −a or u > a

Informally, what these means is if you are represented by , then  represents your distance from zero. Example 1: If   5 , use a number line to represent where YOU are allowed to be.

Example 2: If   5 , use a number line to represent where YOU are allowed to be.

Example 3: Solve each inequality algebraically. Graph your solution & write your solution in interval notation. a) 2 x − 1 > 35

b) 13 − 4 3 − 4 x ≥ 9

Unit 1 - 8

P.6 Solving Inequalities Algebraically and Graphically

PreCalculus

Example 4: Graphically verify your solutions to example 3 using your calculator. a)

b)

Solving Inequalities Without Absolute Values We will spend much more time solving inequalities algebraically later on in the year, but for now, our focus is on solving graphically. Example 6: Graphically solve x 3 + x 2 − 6 x < 0

Unit 1 - 9

2.5 Complex Numbers

PreCalculus

2.5 COMPLEX NUMBERS Learning Targets: 1. Understand that 1 is an imaginary number denoted by the letter i. 2. Evaluate the square root of negative numbers. 3. Understand that complex numbers look like a + bi, where a is real and bi is imaginary. 4. Know how to add and subtract complex Numbers. 5. Know how to multiply complex numbers and simplify powers of i. 6. Know how to find a complex conjugate. 7. Know how to use the complex conjugate to divide complex numbers. 8. Know how to solve a quadratic equation using a domain of complex numbers. 9. Graph complex numbers in the complex plane. 10. Find the absolute value of a complex number. 11. Find the distance and midpoint of a segment connected by two complex numbers.

Review from Alg 2

Complex Numbers are made up of real and imaginary parts. They look like ____________________. Whenever you add, subtract, multiply, or divide complex numbers, your answer will be in the form of a complex number as well.

Complex Conjugates: The complex conjugate of a + bi is _______________. Complex conjugates are used for two reasons and they are both related to the fact that when you multiply complex conjugates the result is a real number. 1. Divide complex numbers using the _____________________ of the denominator. Using the complex conjugate eliminates the imaginary part of the denominator, thus rationalizing the denominator. 2. Complex solutions (zeros) to polynomial functions always occur as ________________________ pairs.

Example 1: Divide

5  8i … be sure to write your answer as a complex number. 7  2i

Example 2: Write a quadratic function in standard form that has x = 3 – 2i as a zero.

Unit 1 - 10

2.5 Complex Numbers

PreCalculus

We use the complex plane to visualize the location of complex numbers. The complex plane has real numbers along the horizontal axis and imaginary numbers along the vertical axis. For example, the point represents the complex number w = –3 + 2i Imaginary axis

Recall that absolute value is defined as the distance from zero…this concept is true on a coordinate plane as well. w Example 3: Find −3 + 2i

Real axis

Some of the geometric measurements, such as distance and midpoint, from the Cartesian plane actually become simpler on the complex plane. Given two complex numbers u and v… The distance between the points u and v is d=

The midpoint between the points u and v is

u−v .

u+v 2

.

Example 3: Given the complex numbers u = 4 – 3i and v = 1+ 2i, answer the questions below… a) Plots points in the complex plane.

Imaginary axis

Real axis

b) Find the distance between u and v.

c) Find the midpoint of the segment connecting u and v.

Unit 1 - 11

2.6 Complex Numbers, Complex Zeros and the Fundamental Theorem of Algebra

Pre Calculus

2.6 COMPLEX ZEROS AND THE FUNDAMENTAL THEOREM OF ALGEBRA Learning Targets: 1. Know how many and what type (real or non-real) of zeros a polynomial can have. 2. Understand that complex zeros come in pairs. 3. Given the zeros (real and complex) of a polynomial, find the standard form of the polynomial. 4. Given a polynomial, find all the real and complex zeros. 5. Use the zeros of a polynomial to write a polynomial as a product of linear and irreducible quadratic factors.

First … you should become familiar with the following theorems and concepts … •

Fundamental Theorem of Algebra (NEW): A polynomial function of degree n > 0 has n complex zeros. Some of these zeros may be repeated.

•

Rational Zeros Theorem (Alg 2): The only possible rational zeros of a function are p/q, where p is a divisor of the constant term and q is a divisor of the leading coefficient.

•

Multiplicity (Alg 2): The number of times a zero or a factor is repeated.

•

Complex Conjugate Zeros (NEW): If x = a + bi is a zero of a polynomial, then x = a – bi (the conjugate) is also a zero.

•

Factor Theorem (Alg 2): If c is a zero, then (x – c) is a factor and vice-versa.

•

Factors of a Polynomial with Real Coefficients: Every polynomial function with real coefficients can be written as a product of linear factors and irreducible quadratic factors*, each with real coefficients. *An irreducible quadratic factor is a quadratic with NO real solutions (i.e. 2 complex solutions).

Example 1: Suppose you have a polynomial function with zeros x = –2 and x = 3 + i. a) What is the minimum degree of this polynomial?

b) Write a polynomial function in factored form with these given zeros. Use linear factors.

c) Write your polynomial function as the product of linear and irreducible quadratic factors with real coefficients.

d) Write your polynomial function in standard form (with with zeros x = –2 and x = 3 + i.)

Unit 1 - 12

2.6 Complex Numbers, Complex Zeros and the Fundamental Theorem of Algebra

Pre Calculus

Now that we have gone from the zeros to the equation of the polynomial in standard form, let’s revisit using the standard form of a polynomial to find the zeros as they relate to multiplicity and complex zeros.

Example 2: Given the function f ( x)  3 x 4  x 3  33 x 2  71x  20 . a) List the possible rational zeros using the rational root theorem.

b) Use your graphing calculator to determine which of these possible zeros could be zeros. Verify them. Then, find ALL zeros of the function.

c) Write the function from part (b) as a product of linear and irreducible quadratic factors.

Unit 1 - 13

2.8 Solving Rational Equations

PreCalculus

2.8 SOLVING RATIONAL EQUATIONS Learning Targets for 2.8 1. Be able to find the LCD of 3 polynomials 2. Solve a Rational Equation by clearing the fractions 3. Understand when a solution to a rational equation is extraneous and identify them when found Example 1: When we solved equations involving fractions, we eliminated the fractions by

Example 2: Find the least common denominator of the following pairs of fractions: a)

1 1 and 2 3t 5t

b)

4 2 and 3 3h 2 h

c)

5z 4 and z +1 2z + 2z

d)

4 3+ y and y+2 y −1

2

Example 3: Solve the following equations. a)

a a + = 4 3 5

b)

Unit 1 - 14

5 2 1 + = 4x 3 x

2.8 Solving Rational Equations

PreCalculus

Whenever you solve rational equations (or square root equations or logarithmic equations) you create the possibility of extraneous solutions. You are NOT allowed to have a value of 0 in the denominator of a fraction, so IF your algebra leads you to a solution that would give you a value of 0 in the denominator of the original equation, that answer is extraneous. Example 4: Solve the following equations. Check for extraneous solutions. a)

x 3 + = −1 x +1 x − 3

b)

5 15 = 2x − 2 x2 − 1

c)

x 1 1 + = 2 x − 3 x − 4 x − 7 x + 12

d)

2 3 2x − 2 − = 2 x + 3 4 − x x − x − 12

Unit 1 - 15

2.9 Solving Inequalities in One Variable

PreCalculus

2.9 Solving Inequalities in One Variable Learning Targets: 1. Make a sign chart of a function using the values of x where the function is zero or undefined 2. Use a sign chart (or a graph) to determine the intervals where a function is positive or negative. 3. Understand how … ( … and … [ … relate to … < … and … < when solving inequalities.

In the next unit, we will graph polynomials like f  x    x  2  x  3 ( x  5) using the zeros, multiplicity, and end 2

behavior. Polynomial functions can be positive, negative or zero. 

To solve f ( x ) > 0 …



To solve f ( x ) < 0 …

Example 1: Use the graph above to answer the following inequalities. Write your answer in interval notation. a) Find the values of x so that  x  2  x  3 ( x  5)  0 . 2

b) Find the values of x so that  x  2  x  3 ( x  5)  0 . 2

c) Find the values of x so that  x  2  x  3 ( x  5)  0 . 2

d) Find the values of x so that  x  2  x  3 ( x  5)  0 . 2

Sign charts are simply number lines with (pos) or (neg) signs on them to represent whether or not the function’s y-values are positive or negative. For polynomial functions, the critical numbers for the sign chart are the zeros of the function. Example 2: Make a “sign chart” for the function f  x    x 1  x  7 (3 x  5) 2

Example 3: Solve the following inequality:

 x 1  x  7 (3x  5)  0 2

(Use your sign chart from example 2)

Unit 1 - 16

2.9 Solving Inequalities in One Variable

PreCalculus

For the remainder of this lesson, we will deal with rational functions (a.k.a. fraction of polynomials). Rational functions can be positive, negative, zero or undefined. : A fraction equals 0 when the numerator equals 0. A fraction is undefined when the denominator equals 0. **Always use ( ) where the fraction is undefined. For this reason, we need to include BOTH of these values in our sign charts. Example 4: Make a sign chart for g  x  

3x  1 . Indicate which values are zeros and which values are undefined.  x  1 x  3

Example 5: Answer the following inequalities using the sign chart from example 5: a) Find the values of x so that

3x  1 0.  x 1 x  3

b) Find the values of x so that

3x  1 0.  x 1 x  3

c) Find the values of x so that

3x  1 0.  x 1 x  3

d) Find the values of x so that

3x  1 0  x 1 x  3

In order to make the sign chart, one side of the inequality MUST be written as a single fraction. Example 6: Solve the following inequality:

1 2  0 x 1 x  3

Unit 1 - 17

3.5 Equation Solving and Modeling

PreCalculus

3.5 EQUATION SOLVING (Exponential and Logarithmic) Learning Targets: 1. Solve exponential and logarithmic equations.

From Algebra 2 you should already know… •

A logarithmic function is simply an inverse of an exponential function. The following definition relates the two functions: log b y  x if and only if b x  y NOTE: You can only switch between exponential and logarithmic forms from the above formats.

•

Two Special Logarithms  

•

A logarithm with base 10 is called a common logarithm and is written log x A logarithm with base e is called a natural logarithm and is written ln x

3 Rules of to simplify Logarithms 

log b (= MN ) log b M + log b N



M log= log b M − log b N b ( N )



log b M k = k ⋅ log b M

(

)

When you solve an equation, you “undo” what has been done … addition to undo subtraction, multiplication to undo division. Since exponents and logarithms are inverses of each other, it follows that in order to solve a logarithmic equation, you can write it as an exponent to “undo” the logarithm, and if you are solving for an exponent, you write the equation as a logarithm.

 

blogb M  M … and … log b b M  M

Example 1: Solve the following equations: a) e x2  19

b) 3  5  2  14 3x

Unit 1 - 18

3.5 Equation Solving and Modeling

PreCalculus

c) log 2  x   4

d) 7  3 log( x)  5

e) log x  log  x  21  2

f) log 3  x  4  log 3  x  5  2

Unit 1 - 19