Algebra I – August 2024 Practice Flashcards

Factoring Quadratic Trinomials

Finding the correct binomial factors for $ax^2 + bx + c$.
  - Example: Factor $x^2 + 4x - 12$.
  - Logic: Two numbers multiply to $-12$ (constant) and add to $4$ (linear coefficient).
    - Numbers: $6$ and $-2$ (since $6 \times -2 = -12$ and $6 + (-2) = 4$).
  - Result: $(x - 2)(x + 6)$.
  - Options:
    1. $(x + 3)(x - 4)$
    2. $(x - 3)(x + 4)$
    3. $(x + 2)(x - 6)$
    4. $(x - 2)(x + 6)$.

Equivalent Algebraic Expressions

Simplifying expressions by distributing and combining like terms.  - Example: Identify the equivalent expression for $3(x^2 - 2x + 3) - (4x^2 + 3x - 1)$.  - Steps:   - Distributing: $3x^2 - 6x + 9 - 4x^2 - 3x + 1$.   - Combine like terms: $(3x^2 - 4x^2) + (-6x - 3x) + (9 + 1) = -x^2 - 9x + 10$.  - Note: Transcript states option (4) is $2x^2 - 9x + 10$, indicating a discrepancy.

Polynomial Multiplication

 - Example: For $x = 4a^2 - a + 3$ and $y = a - 5$, find the product.  - Calculation: $(4a^2 - a + 3)(a - 5)$.  - Multiply: $4a^3 - a^2 + 3a - 20a^2 + 5a - 15$.  - Combine: $4a^3 - 21a^2 + 8a - 15$.  - Correct Option: Option (4).

Linear vs. Exponential Models

• Linear Functions: Represented by a constant rate of change (addition or subtraction of a fixed value over equal intervals).

• Exponential Functions: Represented by a constant percentage rate of change (doubling, halving, or growth/decay by a percentage).

Transcript Question 2 Analysis:

 1. Printer output: One page every three seconds. This is a constant rate ($\frac{1}{3} \text{ pages/second}$), identifying it as a linear function.  2. Bank Interest: 0.5\$ interest compounded annually. This is exponential growth.  3. Cellular Growth: Organism cells doubling every four days. This is exponential growth (multiplier of 2).  4. Sports Attendance: Decreasing by 1.5\%$annually. This is exponential decay.</p><h4 id="f32fb85d-26d8-45f9-97eb-22708c0d7a5d" data-toc-id="f32fb85d-26d8-45f9-97eb-22708c0d7a5d" collapsed="true" seolevelmigrated="true">Algebraic Modeling and Word Problems</h4><h5 id="ca05b286-7288-46c7-aa89-f491b37c5271" data-toc-id="ca05b286-7288-46c7-aa89-f491b37c5271" collapsed="false" seolevelmigrated="true">Transcript Question 4 Scenario:</h5><ul><li><p>Constructing an equation based on defined variables and known totals: Adelynn's party guests brought nickels, dimes, and quarters totaling$28.00.  - Definitions and Quantities:   - Number of nickels: x.   - Number of dimes: Twice as many as nickels = 2x.   - Number of quarters: 12$more than nickels =$x + 12.  - Values:   - Nickels: 0.05x   - Dimes: 0.10(2x)   - Quarters: 0.25(x + 12)  - The Model: 0.05x + 0.10(2x) + 0.25(x + 12) = 28$.</p></li></ul><h4 id="68e2b016-7afa-4ff4-80c2-12cfca835f22" data-toc-id="68e2b016-7afa-4ff4-80c2-12cfca835f22" collapsed="false" seolevelmigrated="true">Properties and Polynomial Classification</h4><h5 id="0b23ac9a-26f5-4d54-bfea-5785e76c5cc6" data-toc-id="0b23ac9a-26f5-4d54-bfea-5785e76c5cc6" collapsed="true" seolevelmigrated="true">Transcript Question 5:</h5><ul><li><p>Identify a fourth-degree trinomial with a leading coefficient of$2$and a constant of$5.  - Definitions:   - Fourth-degree: The highest exponent of the variable is 4.   - Trinomial: There are exactly three terms.   - Leading Coefficient: The number in front of the highest degree term.   - Constant: The term with no variable.  - Analysis of Options:   - Option (1): 2x^4 + 3x^2 + 5$. Fits all criteria.</p></li></ul><h5 id="6454a5fe-4348-4661-81d4-976b833456cb" data-toc-id="6454a5fe-4348-4661-81d4-976b833456cb" collapsed="true" seolevelmigrated="true">Justifying Steps in Equation Solving:</h5><ul><li><p><strong>Transcript Question 6</strong>: Moving from$4x^2 - 16 = 0$to$4x^2 = 16.
 - Action: Adding 16 to both sides of the equation.
 - Property: Addition Property of Equality.

Solving Quadratic Equations

• Completing the Square:
    - Q8: Same solutions as x^2 + 6x - 18 = 0.    1. Move constant: x^2 + 6x = 18.    2. Half coefficient to square: 3^2 = 9, then add to both sides.    3. Result: (x + 3)^2 = 27.

• Intersection of Functions:
   - Q16: Where f(x) = g(x)$with given functions leads to solutions$x = 8$and$x = -2.

Statistics and Data Representation

• Box-and-Whisker Plot:
    - Q9: Heights in inches: 69, 70, 71, 72, 74, 76, 78.    - Summary: Min: 69$, Q1:$70$, Median:$71.5$, Q3:$75$, Max:$78.

• Two-Way Frequency Tables:   - Q14: Survey results for prom themes, beach party (86 girls, 123 boys). Percentage of girls: Approx. 46rac{1}{4} ext{ ext{ 5}}.

Rate of Change and Function Interpretation

• Average Rate of Change:   - Q10: Profit change in bookstore: Rate = rac{375}{250} = 1.50.

• Linear Intercept Interpretation:   - Q12: Plumber's charge p(h) = 45 + 90h$; flat fee of$45.

Function Transformations

• Shifting Graphs:   - Q11: Shift f(x) = x^2$yields the new function:$h(x) = (x - 4)^2 - 3.

Inequalities and Literals

• Linear Inequalities:
    - Q13: Solve 2m - 4 ext{ to } 3(2m + 4)$, leading to$m ext{ values} ext{ at least } -4..

• Variable Rearrangement:
    - Q20: Solve 6 - ax = ax - 2$for$x = rac{4}{a}.

Arithmetic and Geometric Sequences

• Geometric Sequence:   - Q19: Identify sequence with ratio r = 3.   - Example: -2, -6, -18, -54.

• Arithmetic Sequence:   - Q23: Given a_1 = 4$and$a_3 = -2$finds$d = -3.

Radicals and Exponents

• Adding Radicals:   - Q15: 2 ext{-radicals sum} o 8 ext{(Sqrt)}.

• Function Zeros:
    - Q21: Zeros -1, 3, -4$yield factors:$(x + 1)(x - 3)(x + 4).

• Exponential Equivalencies:   - Q22: Evaluate 5^{a + 2b}$, simplified to$5^a imes 25^b$$.