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## I. Expressions, Equations, and Inequalities

### Patterns and Expressions

Example: What is the next number in the pattern: 3, 6, 9, 12, _ ?

### Lesson: Properties of Real Numbers

Example: Which property justifies the equation $3(2x-5y)+1=6x-15y+1$?

### Lesson: Algebraic Expressions

Example: Which algebraic expression models the phrase “10 less than a number m”?

### Solving Equations

Example: What is the value of $x$ in $3x-7=11$?

### Solving Inequalities

Example: Write the inequality that represents “3 less than a number is greater than 5”

### Absolute Value Equations and Inequalities

Example: Evaluate $|-x|=12$

## II. Functions, Equations, and Graphs

### Relations and Functions

Example: What is the output of the function $f(x)=x^2 -3x -2$ if $x=5$?

### Direct Variation

Example: If $y$ varies directly with $x$, and $y=5$ when $x=2$, then what is the value of $y$ when $x=6$?

### Linear Functions and Slope-Intercept Form

Example: Find the slope of the line passing through the points (2,4) and (4,-2)

Example: Write the equation of the line with a slope of 3 that goes through the point (5,0).

### Using Linear Models

Example: Can the values 2, 4, 6, 8, 10 be modeled by a linear function?

### Families of Functions

Example: What is the equation of the line $y=x$ after it undergoes a transformation 2 units up and 3 units left?

### Absolute Value Functions and Graphs

Example: Solve for $x$: $|4x-1|=15$

### Two Variable Inequalities

Example: Draw the graph of $y \textless 3x-1$

## III. Linear Systems

### Lesson: Solving Systems of Equations Using Tables and Graphs

Example: Solve the following system of equations graphically: $y=3x-1$ and $y=-x+5$

### Solving Systems Algebraically

Example: Solve the following system of equations algebraically: $3x-4y=-3$ and $y=2x+1$

### Lesson: System of Inequalities

Example: Solve the following system of inequalities: $y \textless 2x-1$ and $y \geq -x$

### Linear Programming

Example: What values for $x$ and $y$ maximize the objective function $P=2x+4y$?

### Systems with Three Variables

Example: Solve the following system of equations: $3x-2y+z=0$, $-2x+5y+4z=12$, and $6x-y-z=-8$

### Solving Systems Using Matrices

Example: Solve the following system of equations by using matrices: $3x-2y=9$ and $-x+6y=-1$

## IV. Quadratic Functions and Equations

Example: What is the vertex of the function $y=3(x-2)^2 +3$?

### Standard Form of a Quadratic Equation

Example: What is the vertex of the function $y=x^2-5x-24$?

Example: Do the points (3,-1), (4,0), and (5,-1) lie on the function $y=-(x-4)^2$?

Example: Factor the expression $x^2 - 5x -36$

Example: Solve for $x$: $x^2 +15 +50$

### Lesson: Completing the Square

Example: What are the solutions to $y=x^2 -12x -14 = 9$

Example: Solve for $x$ using the quadratic formula: $x^2+2x+14=0$

### Complex Numbers

Example: What is equivalent to $\sqrt{-25}$?

Example: Solve the following system of equations: $y=2x-1$ and $y=x^2+3x-4$

## V. Polynomials and Polynomial Functions

### Polynomial Functions

Example: Classify the following polynomial by its degree and number of terms: $3x^5 -2x+1$

### Polynomials, Linear Factors, and Zeros

Example: What is the factored form of $x^3 -x^2 -12x$

### Solving Polynomial Equations

Example: Solve for $x$: $x^3 -8 =0$

### Lesson: Dividing Polynomials

Example: What is the solution to $\dfrac{x^2+20+44}{x-2}$?

### Rational Root Theorem

Example: What are the possible rational zeroes for $f(x)=x^3 -2x^2 +12x -1$?

### The Fundamental Theorem of Algebra

Example: Solve for $x$: $x^4 +x^3 +4x^2 -8x +25=0$

### The Binomial Theorem

Example: How can you expand $(3x-2y)^3$?

### Polynomial Models in the Real World

Example: What are three points that lie on $f(x)=x^3 -7x^2 +4$?

### Transforming Polynomial Functions

Example: What is the equation of $y=x^3$ after it is reflected in the y-axis and then translated 4 units down and 3 units left?

## VI. Radical Functions and Rational Exponents

Example: What values are square roots of 25?

### Lesson: Multiplying and Dividing Radical Expressions

Example: Evaluate $\sqrt{288} \times \sqrt{4}$

Example: Evaluate $\sqrt{50} + \sqrt{8}$

### Rational Expressions

Example: Evaluate $(27x^6)^{\frac{1}{3}}$

### Lesson: Solving Square Root and Other Rational Expressions

Example: Solve for $x$: $\sqrt{x-2} +12 = 14$

### Function Operations

Example: Find $(f-g)(x)$ if $f(x)=12x-12$ and $g(x)=-x^2 +2x +8$

### Inverse Relations and Functions

Example: What is the inverse function of $f(x)=\dfrac{3x-5}{2} -3$?

Example: Draw the graph of $y=\sqrt{3x}$

## VII. Exponential and Logarithmic Functions

### Exploring Exponential Models

Example: How can you describe the function $y=3(0.6)^x$?

### Properties of Exponential Models

Example: What transformation changed $y=4^x$ to $y=4^{x-1}$?

### Logarithmic Functions as Inverses

Example: What is the inverse function of $y=2+4^x$?

### Properties of Logarithms

Example: Expand $log_3 (x\sqrt{y} - z)$

### Exponenital and Logarithmic Equations

Example: Solve for $x$: $3^x -1.8 = 5.2$

### Natural Logarithms

Example: Evaluate $ln 4$

## VIII. Rational Funtions

### Lesson: Inverse Variation

Example: Suppose $x$ and $y$ vary inversely. Write a function for when $x=4$ and $y=20$

### The Reciprocal Family Functions

Example: What is the y-intercept of $y=\dfrac{2}{x}$

### Lesson: Rational Functions and their Graphs

Example: Draw the graph of $y=\sqrt{x-1}$

Example:

### Lesson: Adding and Subtracting Rational Expressions

Example: Simplify $\dfrac{3x}{2x^2} + \dfrac{3+x}{x}$

### Lesson: Solving Rational Equations

Example: Solve for $x$: $\dfrac{1}{2x^2} = \dfrac{3}{x} + \dfrac{5}{2}$

## IX. Sequences and Series

### Mathematical Patterns

Example: Continue the pattern with the next three values: 4, 12, 36, 108, …

### Arithmetic Sequences

Example: What is the explicit formula for the sequence 12, 15, 18, 21, …?

### Geometric Sequences

Example: What is the explicit formula for the sequence 2, -6, 18, -54, …?

### Arithmetic Series

Example: What is the sum of the series 32, 43, 54, 65, 76?

### Geometric Series

Example: What is the sum of the series -3 -6 -12 -24 …, n=6?

## X. Quadratic Relations and Conic Sections

### Exploring Conic Sections

Example: Which conic section is associated with the equation $4x^2 +36y^2 = 16$?

### Parabolas

Example: What is the vertex of the function $y=3x^2 -6x +9$?

### Circles

Example: What is the radius of the circle $(x-4)^2 + (y+1)^2 = 12$?

### Ellipses

Example: What are the vertices of the ellipse $\dfrac{(x-9)^2}{9} + (y-9)^2 = 1$?

### Hyperbolas

Example: What are the vertices of the hyperbola $\dfrac{x^2}{9} - \dfrac{(y-1)^2}{9} = 1$?

### Translating Conic Sections

Example: Write the standard form equation of an ellipse with vertices at (-2,3) and (5,3), and a focus at (2,3)

## XI. Probability and Statistics

### Lesson: Permutation and Combinations

Example: Evaluate $_7 P _2$

### Lesson: Probability

Example: Bob flipped heads 6 out of 8 times. What is the experimental probability that he flips tails on his next coin toss?

### Lesson: Probability of Multiple Events

Example: What is the probability of A and B if they are independent events and the probability of A is $\dfrac{3}{10}$ and the probability of B is $\dfrac{2}{3}$?

### Conditional Probability

Example: If the freshman students are 45% male and 55% female, and 86% of the males passed this semester while 89% of the females did, what is the probability that a student is male who failed this semester?

### Probability Models

Example: The weather channel tells you that there is a 30% chance of rain on each of the next three days. What is the chance that it does not rain on all three days?

### Analyzing Data

Example: What is the median of {28 26, 40, 37, 39, 29, 31, 33, 33, 33, 31, 36}?

### Standard Deviations

Example: What is the standard deviation of {28 26, 40, 37, 39, 29, 31, 33, 33, 33, 31, 36}?

### Lesson: Samples and Surveys

Example: If a principal wants to conduct a survey to determine which subject is the favorite in his school, so he samples the physics class to get results, is this a biased sample?

### Binomial Distributions

Example: What is the fourth term in $(2x-y)^5$?

### Normal Distributions

Example: What percent of data is in the interval from 45 to 50 in a data set with a mean of 55 and a standard deviation of 5?

## XII. Matrices

### Lesson: Adding and Subtracting Matrices

Example: Simplify $\begin{bmatrix} 4 & 1 & -8 & -2 \\ \end{bmatrix} + \begin{bmatrix} 3 & 6 & -5 & -1 \\ \end{bmatrix}$

### Lesson: Matrix Multiplication

Example: Simplify $6 \begin{bmatrix} 1 & 3 & 0 & -9 \\ \end{bmatrix}$

### Determinants and Inverses

Example: $\begin{vmatrix} 4 & -2\\ 8 & -6\\ \end{vmatrix}$

### Inverse Matrices and Systems

Example: Solve the following system of equations using matrices: $4x-2y=12$ and $3x+y=1$

### Geometric Transformations

Example: Dilate the coordinates forming a traingle given in the following matrix by a scale factor of 3. $\begin{bmatrix} -3 & -2 & -1 \\ 5 & -3 & -3 \\ \end{bmatrix}$

### Vectors

Example: If a vector starts at the point (2, 3) and extends to (4, 6), what is its component form?

## XIII. Periodic Functions and Trigonometry

### Exploring Periodic Data

Example: What is the amplitude and the period of $y=2 \sin{x}$?

### Angles and the Unit Circle

Example: If an angle extends outwards from the x-axis and touches the x-axis on the other side, what is the degree measure of the angle formed?

Example: What is 50 degrees in terms of radians?

### The Sine Function

Example: What is the amplitude and the period of $y=3 \sin{4x}$?

### The Cosine Function

Example: What is the amplitude and the period of $y= 2 \cos{2x}$?

### The Tangent Function

Example: What is the amplitude and the period of $y= 4 \tan{x}$?

### Translating Sine and Cosine Functions

Example: What is the amplitude and the period of $y= 5 \sin{3x} -3$?

### Reciprocal Trigonometric Functions

Example: Evaluate $\csc{\dfrac{2\pi}{3}}$

## XIV. Trigonometric Identities and Equations

### Trigonometric Identities

Example: What is equivalent to $\sin ^2 \theta + \cos ^2 \theta$

### Solving Trigonometric Equations Using Inverses

Example: Solve for $\theta$: $\dfrac{\sqrt{2}}{2} - 2\sin \theta = -\sin \theta$

### Right Triangles and Trigonometric Ratios

Example: In a 30-60-90 triangle, what is $\tan 30 \textdegree$?

### Area and Law of Sines

Example: In $\triangle ABC$, $\angle A = 34 \textdegree$, $b=12$, and $c=30$. Find $m\angle C$

### The Law of Cosines

Example: In $\triangle ABC$, $a=5$, $b=13$, and $m\angle C = 57 \textdegree$. Find $c$

### Angle Identities

Example: Evaluate $\sin 75\textdegree$

### Double-Angle and Half-Angle Identities

Example: Find $\sin 2\theta$ is $0 \textless \theta \textless 90$ and $\cos \theta = \dfrac{\sqrt{5}}{4}$

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