Mathematics Topic Assessment Notes

Set Z and Negative Numbers

• Set Z: Z = {-34, -28, -16, -2, 4, 8, 12, 26}

• Question: Identify the elements of Set Z that are both negative numbers and multiples of 4.

• Options:

  • A: {-28, -16} ✔

  • B: {-28, -16, 4, 8, 12}

  • C: {-28, 16, 2, 4, 8, 12}

  • D: {-34, -28, -16, -2, 4, 8, 12, 26}

• Correct Answer: A

Ordering Numbers

• Numbers to Order: 45, 11, 4.5, √20, √45

• Expression for Order: Convert to decimal forms or approximate values:

  • $45 = 45$

  • $11 = 11$

  • $4.5 = 4.5$

  • $√20 ≈ 4.47$

  • $√45 ≈ 6.71$

• Ordered List (Least to Greatest): 4.5, √20, 11, 45

Rational Numbers from Expressions

• Question: Find which expressions produce rational numbers:

  • A: $5 + rac{√12}{6}$

  • B: $ rac{5√12}{3}$

  • C: $5 + rac{5√12}{3}$

  • D: $56$

• Correct Answers: All options (A, B, C, D) yield rational expressions since they include whole numbers or rational operations.

Rational Exponents

• Task: Write $√10$ using rational exponents.

  • Answer: $10^{1/2}$

Solving Equations

Example 1: Solving for x

• Equation: $ rac{1}{x+2} = 49$

  • Two Cases:

  • Case A: $x = -7$

  • Case B: $x = (7)x - 3$

  • No Solution if it leads to non-real solutions.

Example 2: Exponential Function Asymptotes

• Function: $f(x) = 10^x$

  • Domain: All real numbers.

  • Range: $y > 0$

  • Asymptote: Horizontal line $y = 0$.

  • Y-intercept: At (0,1).

Graphing Exponential Functions

• Function: $f(x) = 6^x$

  • Points to Graph:

  • For $x = 0, f(0) = 6(2^0) => 6$

  • For $x = 1, f(1) = 6(2^1) => 12$

  • For $x = 2, f(2) = 6(2^2) => 24$

  • Continue for $x = 3, 4$…

Characteristics of Polynomials

Third-Degree Binomials

• Options: Select expressions that are third-degree:

  • A: $2y - xy^3 + 7$ (Has $y^3$) ✔

  • B: $3x^2y + 5xy$ (No third-degree term)

  • C: $3y^3 + 3x^3y^4$ (Has $y^3$, but also $y^4>3$)

  • D: $3xy - 3xy^2$ (No third-degree term)

Write in Standard Form

• Polynomial: $3x^3 + 5x + 7x^4 - 9 - x^2$

  • Standard Form: $7x^4 + 3x^3 - x^2 + 5x - 9$

Simplifying Expressions

Example 1: First Expression Simplification

• Expression: $(3x^2 + 7x - 1) + (4x^3 - 9x^2 + 1)$

  • Result: Combine terms to reach $7x^3 - 9x^2 + 4x + 0$ in standard form.

Example 2: Solve the polynomial (-5x + 7 - (x^2 - 3x + 2))

• Solution: $-x^2 + 2x + 5$ in simplified form.

Framed Portrait Calculation

• Height: 1.5 times width $w$; width of frame: 4 in.

• Area:

  • Area of framed portrait = $(1.5w + 8)(w + 8) - 1.5w^2$.

Quadratic Functions and Their Properties

Opening Direction of Graphs

• Function: $f(x) = ax^2$, where $a < 0 ightarrow$ Graph opens downwards. Check for wider or narrower graphs than $f(x) = x^2$.

  • Possible values of a = {-10, -0.1} (wider).

Average Rate of Change

• Function: $f(x) = 2x^2 - x - 4$ over interval (-4 ≤ x ≤ 2).

• Calculation: Average Rate of Change formula = $\frac{f(b) - f(a)}{b - a}$.

Systems of Equations

• Function: $y = x^2 - 3x$ intersecting lines must be analyzed to check which $ ext{y}$-equation intersects at two points.

Polynomial Factoring Techniques

1. Equation: $2x^2 + 7x - 4 = 0$: Factors into $(x-4)(x + 0.5)$ yielding roots: -4, 0.5.

2. Vertex: Functions in vertex forms lead to calculation of max/min values.

3. Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ necessary to solve when factoring fails.