Algebra 2B Comprehensive Study Guide

Exam Overview and Course Materials

• The University of Texas at Austin UT High School Algebra 2B Credit by Exam (CBE) for Credit Recovery (CR) or Acceleration (A) is designed to test proficiency in the subject matter.
• Students must be thoroughly familiar with the subject matter before attempting the exam.
• The exam is strictly derived from the Texas Essential Knowledge and Skills (TEKS) mandated standards. The relevant TEKS are found in section (c), Knowledge and skills, 1A–8d of the Texas Administrative Code.
• Materials Required:
  • A graphing calculator is mandatory. Students may bring their own from home, subject to the UTHS Calculator Testing Policy.
  • For print exams, a #2 pencil is required to complete the computer-graded answer sheet.
  • Scratch paper is provided by the exam proctor.
• Exam Structure:
  • Duration: 3 hours.
  • Format: 72 equally weighted multiple-choice questions.
  • Coverage: The exam spans 6 key topics identified by the UT High School.

Scholastic Honesty and Testing Rules

• Students must sign the "Examinee’s Certification" agreeing to all rules or credit will not be awarded.
• Scholastic dishonesty is defined as a serious academic violation including, but not limited to:
  • Copying from another student’s work.
  • Using an unauthorized proctor or testing location.
  • Using or possessing unauthorized materials such as notes, books, or lessons.
  • Knowing the contents of an unadministered test or soliciting those contents.
  • Collaborating with or seeking aid from another student during the test.
  • Substituting for another person or allowing a substitute to take the exam.
  • Buying, stealing, or transporting the contents of an unadministered test, rubric, or homework answer.
• Consequences: Evidence of dishonesty results in an F on the exam and an F in the course (if applicable).

Topic 1: Plan and Solve Real-world Problems

• This topic focuses on TEKS 1A–1G, relating to the ability to create problem-solving plans and explain reasoning for everyday life scenarios.
• Core Competencies:
  • Solving word problems related to the workplace and daily societal interactions.
  • Justifying solutions and explaining why a solution is or is not reasonable.
  • Creating and utilizing representations such as tables, graphs, number lines, and Venn Diagrams to organize information.
• Sample Scenario: An absolute value equation representing climate variation. If an island averages $87.8^{\circ}$ Fahrenheit and varies by no more than $8.2^{\circ}$, the governing equation for minimum and maximum temperatures is $|x - 87.8| = 8.2$.

Topic 2: Graph Parent Functions and Inverse Functions

• This topic focuses on TEKS 2A–2D, covering the basic shapes and properties of specific function families and their inverses.
• Function Families: Square root, inverse, cubic, cube root, exponential, absolute value, and logarithmic functions.
• Core Competencies:
  • Graphing parent functions for all specified families.
  • Identifying and analyzing key properties: domain, range, intercepts, asymptotic behavior, and maximum/minimum values.
  • Determining the inverse of a function and graphing it.
  • Describing the relationship between a function and its inverse both numerically and graphically.
  • Using the composition of two functions to verify if they are inverses.
• Graph Identification: Students must recognize graphs by sight, such as identifying a cube root function graph based on its characteristic shape.
• Inverse Example: For an exponential function $f(x) = 2 \times 5^x$, the inverse is calculated and graphed as $f^{-1}(x) = \log_5(\frac{x}{2})$.

Topic 3: Exponential and Logarithmic Functions

• This topic focuses on TEKS 5A–5E, addressing the notation, bases, and real-world applications of these functions.
• Core Competencies:
  • Graphing logarithmic parent functions: $y = \log_2(x)$, $y = \log(x)$, and $y = \ln(x)$.
  • Graphing exponential parent functions: $y = 2^x$, $y = 10^x$, and $y = e^x$.
  • Describing and graphing transformations of these functions.
  • Converting expressions between logarithmic and exponential forms (e.g., $\ln(x) = 2$ is equivalent to $e^2 = x$, and $\log_5(x) = 2$ is equivalent to $5^2 = x$).
  • Writing equations for real-world situations, such as population growth. For example, a deer population of $275$ with a $3.1\%$ annual increase is modeled by $y = 275 \times 1.031^x$.
  • Solving logarithmic and exponential equations with real solutions and checking for reasonableness.

Topic 4: Rational, Cubic, and Cube Root Functions

• This topic focuses on TEKS 6A, 6B, and 6G–6L, emphasizing transformations, modeling, and solving equations.
• Core Competencies:
  • Graphing parent functions for cubic ($y = x^3$), cube root ($y = \sqrt[3]{x}$), and rational ($y = \frac{1}{x}$) functions.
  • Describing and graphing transformations for each.
  • Solving cube root equations. Example: $\sqrt[3]{x + 7} - 5 = 3$ leads to $x = 505$.
  • Writing rational equations for real-world problems. Example: A win percentage problem where Zack has won 10 out of 12 matches. To reach a $90\%$ win rate by winning $m$ matches in a row, the equation is $\frac{10 + m}{12 + m} = 0.9$, requiring $8$ more wins.
  • Solving rational equations and checking solutions.
  • Identifying and determining the effect of asymptotes on the domain and range of rational functions.
  • Solving equations involving inverse variation.

Topic 5: Operations on Rational Expressions, Polynomials, and Complex Numbers

• This topic focuses on TEKS 7A–7I, dealing with arithmetic and simplification of complex expressions.
• Core Competencies:
  • Performing addition, subtraction, and multiplication of complex numbers. This requires familiarity with powers of $i$; for example, $\sqrt{-1} = i$ and $i^2 = -1$.
  • Complex Multiplication Example: $(-2i + 4)(17i)$ simplifies to $-34i^2 + 68i = 34 + 68i$.
  • Adding, subtracting, and multiplying polynomials.
  • Determining quotients and remainders for polynomials of degree 3 and 4.
  • Determining linear and quadratic factors of polynomials. Example: For $x^4 + 2x^3 - 3x^2$, the factors are $x^2$, $(x - 1)$, and $(x + 3)$.

Topic 6: Using Data

• This topic focuses on TEKS 8A–8D, centered on data analysis and regression modeling.
• Core Competencies:
  • Analyzing data sets to determine if a linear, quadratic, or exponential model is the best fit.
  • Utilizing the regression feature on a graphing calculator to generate a best-fit equation.
  • Making predictions and solving problems using the regression model.
• Data Example: A laboratory study of bacteria growth over time (minutes: 0, 2, 4, 6, 8; counts: 219, 251, 287, 329, 376). Using regression technology, the best fit is an exponential model $y = 219 \times 1.07^x$. To reach $5,000$ bacteria, it would take approximately $47$ minutes.

Algebra 2 Formula Sheet Reference

Factoring:

• Perfect square trinomials (Subtraction): $a^2 - 2ab + b^2 = (a - b)^2$
• Perfect square trinomials (Addition): $a^2 + 2ab + b^2 = (a + b)^2$
• Difference of squares: $a^2 - b^2 = (a + b)(a - b)$
• Sum of cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
• Difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Properties of Exponents:

• Product of powers: $a^m a^n = a^{m+n}$
• Quotient of powers: $\frac{a^m}{a^n} = a^{m-n}$
• Power of a power: $(a^m)^n = a^{mn}$
• Rational exponent: $a^{m/n} = \sqrt[n]{a^m}$
• Negative exponent: $a^{-n} = \frac{1}{a^n}$

Quadratic Equations and Parabolas:

• Parabola (Vertical axis): $(x - h)^2 = 4p(y - k)$
• Parabola (Horizontal axis): $(y - k)^2 = 4p(x - h)$
• Standard form: $f(x) = ax^2 + bx + c$
• Vertex form: $f(x) = a(x - h)^2 + k$
• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• Axis of symmetry: $x = \frac{-b}{2a}$

Sample Questions Answer Key