Functions and Their Representations

This section delves into how functions can be represented using graphs and tables.

The concept of function machines is introduced, showing how inputs are transformed into outputs. An example is provided where f$x$ = 3x^2, and when x = 3, the output f$3$ = 27.

Highlight: The page notes two important points for graphing functions:

1. To find the x-intercept, substitute y = 0
2. To find the y-intercept, substitute x = 0

The section also covers the domain and range of functions, explaining that the domain represents the x inputs and the range represents the y outputs of a function.

Example: A table is shown with input and output values to illustrate a function.

The vertical line test is mentioned as a method to determine if a relation is a function. If a vertical line intersects the graph more than once, it is not a function.

Definition: Function - a relation where each input value $x$ corresponds to exactly one output value $y$

Practice problems are included, asking students to calculate outputs for given functions and determine if certain relations are functions.

Domain and Range Questions

This section focuses on domain and range concepts and introduces rules for working with exponents.

The page begins with questions about identifying domain and range in various function representations, including graphs and tables. It emphasizes the importance of understanding these concepts in relation to functions.

Example: A graph is shown, asking students to determine if it represents a function and identify its domain and range.

The section then transitions to rules of exponents, covering several key concepts:

Vocabulary: Product Rule - When multiplying terms with the same base, add the exponents $x^a * x^b = x^(a+b$) Vocabulary: Quotient Rule - When dividing terms with the same base, subtract the exponents $x^a / x^b = x^(a-b$) Vocabulary: Power Rule - When raising a power to another power, multiply the exponents $(x^a$^b = x^$ab$) Vocabulary: Zero Rule - Any number $except 0$ raised to the power of 0 equals 1 $a^0 = 1, where a ≠ 0$ Vocabulary: Negative Rule - For negative exponents, the base can be moved to the denominator with a positive exponent $x^-a = 1/x^a$

The page also introduces the concepts of x-intercept and y-intercept:

Definition: X-intercept - the point where a line or curve crosses the x-axis Definition: Y-intercept - the point where a line or curve crosses the y-axis

Highlight: The page reminds students of the order of operations: Parentheses, Exponents, Multiplication/Division $left to right$, Addition/Subtraction $left to right$

Exponent Questions and Practice

This final section provides practice problems focusing on exponents and algebraic expressions.

The page presents several questions that require students to apply the rules of exponents learned in the previous section. These problems involve simplifying expressions with exponents and solving equations.

Example: One problem asks students to simplify the expression $x^3y^2$^3, which requires applying the power rule for exponents.

Another problem involves factoring a quadratic expression:

Example: Simplify $x-2$$x+3$

The page also includes a fill-in-the-blank question about the parts of an exponential expression, reinforcing vocabulary related to exponents.

Vocabulary: Base - the number being raised to a power Vocabulary: Exponent - the power to which a base is raised Vocabulary: Power - the result of raising a base to an exponent

The section concludes with encouragement for students who have completed the practice problems, emphasizing the importance of understanding and applying these concepts in algebra.

Highlight: "GOOD JOB!! You're ALL Done!" - This positive reinforcement aims to boost student confidence after completing the challenging problems.

Page 5: Advanced Function Concepts

This page delves deeper into function representation and analysis techniques.

Example: Function machine example: f$x$=3x² with input x=3 yields output f$3$=27.

Definition: X-intercept is found by substituting y=0, while y-intercept is found by substituting x=0.

Page 6: Domain & Range Analysis and Exponent Rules

This page focuses on domain and range questions while introducing fundamental rules of exponents.

Definition: X-intercept is the point where a line crosses the x-axis, and y-intercept is where it crosses the y-axis.

Highlight: The page introduces PEMDAS $Parentheses, Exponents, Multiplication, Division, Addition, Subtraction$.

Page 7: Exponent Rules and Mathematical Operations

This page provides detailed explanations of exponent rules and their applications.

Definition: Product Rule: x^a * x^b = x^$a+b$ Definition: Quotient Rule: x^a/x^b = x^$a-b$ Definition: Power Rule: $x^a$^b = x^$ab$

Page 8: Practice Problems

This page presents practice problems focusing on exponent simplification and algebraic expressions.

Example: Problems include simplifying expressions like $x-2$$x+3$ and $x³y²$³.

Page 9: Final Solutions and Conclusion

This concluding page provides solutions to practice problems and ends with positive reinforcement.

Highlight: The page includes final answers and concludes with "GOOD JOB!!" to encourage students.

Growth of Patterns

This section explores how patterns grow and introduces different types of functions.

The page introduces inverse variation and direct variation functions. It explains that inverse variation is represented by the equation y = k/x, where k is a constant, and results in a decreasing pattern. Direct variation, on the other hand, shows a proportional relationship between y and x.

Example: For inverse variation, y = -15/x is given as an example.

The concept of exponential functions is also introduced, represented by the equation y = ab^x + k, where a is the initial value and b is the multiplier. This type of function results in an increasing pattern.

Highlight: The page emphasizes that a function can have repeating y values, but each input value must have only one output value to be considered a function.

Vocabulary: Domain - the set of input values $x$ for a function Vocabulary: Range - the set of output values $y$ for a function

The page includes several practice problems covering various function types and asks students to determine domain and range.