Algebra Functions and Graph Identification

Functions and Parabolic Graphs

Vertex Form of a Quadratic Function:
- The vertex form of a parabola is defined as: $f(x) = a(x - h)^2 + k$ where $(h, k)$ represents the coordinates of the vertex, and $a$ determines the direction and vertical stretch or compression of the parabola.
- If $a > 0$ , the parabola opens upwards and has a minimum value at the vertex.
- If $a < 0$ , the parabola opens downwards and has a maximum value at the vertex.
Graph Identification and Analysis:
- Vertex Identification: Examining the graph reveals a turning point (vertex) located at $(-2, 3)$ .
- Direction of Opening: The parabola opens downwards, indicating that the leading coefficient must be negative ( $a = -1$ ).
- Function Formulation: $f(x) = -(x - (-2))^2 + 3 = -(x + 2)^2 + 3$
- Key Intercepts and Points:
  - When $x = 0$ : $f(0) = -(0 + 2)^2 + 3 = -(4) + 3 = -1$ , matching the $y$ -intercept at $(0, -1)$ .
  - When $x = -4$ : $f(-4) = -(-4 + 2)^2 + 3 = -(-2)^2 + 3 = -4 + 3 = -1$ , matching the symmetric point at $(-4, -1)$ .
  - When $x = -1$ : $f(-1) = -(-1 + 2)^2 + 3 = -(1)^2 + 3 = 2$ .
  - When $x = -3$ : $f(-3) = -(-3 + 2)^2 + 3 = -(-1)^2 + 3 = 2$ .
- Comparison with Other Function Families:
  - $f(x) = -|x + 2| + 3$ : An absolute value function that forms a V-shaped graph with straight linear branches, not a smooth parabolic curve.
  - $f(x) = -\sqrt{x + 2} + 3$ : A square root function that only exists for $x \ge -2$ and represents a single curved branch rather than a two-sided curve.
  - $f(x) = (x + 2)^2 + 3$ : A quadratic function with vertex at $(-2, 3)$ that opens upwards, with a range of $[3, \infty)$ .
Polynomial Function Evaluation:
- Evaluating a function at a specific domain value requires substituting the input value for every occurrence of the independent variable.
- Given the quadratic function: $f(x) = 3x^2 - 3x + 3$
- Evaluation at $x = -2$ : $f(-2) = 3(-2)^2 - 3(-2) + 3$ $f(-2) = 3(4) - (-6) + 3$ $f(-2) = 12 + 6 + 3 = 21$

Polynomial Factoring Techniques

Greatest Common Factor (GCF) Extraction:
- Factoring out the greatest common factor is the reverse of applying the distributive property: $ab + ac = a(b + c)$
- Single-Variable Binomial Factorization:
  - Expression: $7x^2 + 28x$
  - Find the greatest common numerical factor: $\gcd(7, 28) = 7$ .
  - Find the greatest common variable factor: $\gcd(x^2, x) = x$ .
  - Combine to determine the overall GCF: $7x$ .
  - Divide each term by the GCF: $\frac{7x^2}{7x} = x$ $\frac{28x}{7x} = 4$
  - Completely factored form: $7x(x + 4)$
- Multivariate Binomial Factorization:
  - Expression: $2x^3y + 12x^2y^2$
  - Find the greatest common numerical factor: $\gcd(2, 12) = 2$ .
  - Find the lowest power for each shared variable:
    - For $x$ : $\min(3, 2) = 2 \implies x^2$
    - For $y$ : $\min(1, 2) = 1 \implies y$
  - Combine to determine the GCF: $2x^2y$ .
  - Divide each term by the GCF: $\frac{2x^3y}{2x^2y} = x$ $\frac{12x^2y^2}{2x^2y} = 6y$
  - Completely factored form: $2x^2y(x + 6y)$
Difference of Two Squares Identity:
- A binomial consisting of two perfect square terms separated by a subtraction operator factors according to the algebraic identity: $A^2 - B^2 = (A - B)(A + B)$
- Application to $9x^4 - 4y^2$ :
  - Identify the square root of the first term: $A = \sqrt{9x^4} = 3x^2$ .
  - Identify the square root of the second term: $B = \sqrt{4y^2} = 2y$ .
  - Substitute into the difference of squares identity: $9x^4 - 4y^2 = (3x^2 - 2y)(3x^2 + 2y)$
  - The two prime factors of the polynomial are $(3x^2 - 2y)$ and $(3x^2 + 2y)$ .
Factoring Quadratic Trinomials with $a \neq 1$ (The $ac$ -Method):
- Expression: $15x^2 + 29x - 14$
- Standard form identification: $a = 15$ , $b = 29$ , and $c = -14$ .
- Step 1: Compute the product $ac$ : $ac = 15 \times (-14) = -210$
- Step 2: Find two integer factors of $-210$ that sum to $b = 29$ :
  - Testing factor pairs of $-210$ :
    - $-6 \times 35 = -210$
    - $-6 + 35 = 29$
- Step 3: Rewrite the middle linear term using the two identified factors: $15x^2 + 35x - 6x - 14$
- Step 4: Factor by grouping:
  - Group the first two terms: $15x^2 + 35x = 5x(3x + 7)$
  - Group the last two terms: $-6x - 14 = -2(3x + 7)$
  - Combine the grouped binomial factors: $(3x + 7)(5x - 2)$
- Step 5: Verification via FOIL expansion: $(3x + 7)(5x - 2) = 15x^2 - 6x + 35x - 14 = 15x^2 + 29x - 14$
Factoring Monic Quadratic Trinomials ( $a = 1$ ):
- For trinomials of the form $x^2 + bx + c$ , factoring requires finding two numbers $p$ and $q$ such that: $p \cdot q = c$ $p + q = b$
- The factored form is $(x + p)(x + q)$ .
- Case analysis for products yielding $c = \pm 28$ :
  - For $x^2 - 3x - 28$ : Factors of $-28$ that add to $-3$ are $4$ and $-7$ , yielding $(x + 4)(x - 7)$ .
  - For $x^2 + 3x - 28$ : Factors of $-28$ that add to $3$ are $-4$ and $7$ , yielding $(x - 4)(x + 7)$ .
  - For $x^2 + 11x + 28$ : Factors of $28$ that add to $11$ are $4$ and $7$ , yielding $(x + 4)(x + 7)$ .
  - For $x^2 - 11x + 28$ : Factors of $28$ that add to $-11$ are $-4$ and $-7$ , yielding $(x - 4)(x - 7)$ .
Expansion Matching and Missing Term Determination:
- Given an identity where a trinomial equals the product of two binomials: $7x^2 + 19xy + M = (x + 2y)(7x + 5y)$
- Expand the right-hand side using the FOIL method: $(x + 2y)(7x + 5y) = x(7x) + x(5y) + 2y(7x) + 2y(5y)$ $= 7x^2 + 5xy + 14xy + 10y^2$ $= 7x^2 + 19xy + 10y^2$
- Equate coefficients of corresponding terms: $7x^2 + 19xy + M = 7x^2 + 19xy + 10y^2$ $M = 10y^2$

Linear Equations and Systems

Single-Variable Linear Equations:
- Equation to solve: $5(x + 8) = 4x + 42$
- Step 1: Apply the distributive property to eliminate parentheses: $5x + 40 = 4x + 42$
- Step 2: Subtract $4x$ from both sides to collect variable terms on one side: $5x - 4x + 40 = 42$ $x + 40 = 42$
- Step 3: Subtract $40$ from both sides to isolate the variable: $x = 42 - 40$ $x = 2$
- Step 4: Check by substitution:
  - Left-hand side: $5(2 + 8) = 5(10) = 50$
  - Right-hand side: $4(2) + 42 = 8 + 42 = 50$
  - Both sides equal $50$ , confirming $x = 2$ .
Systems of Linear Equations in Two Variables:
- Given the simultaneous system: $3x + 5y = 30$ $4x + 3y = 29$
- Step 1: Choose a variable to eliminate. To eliminate $y$ , find the least common multiple of the coefficients of $y$ ( $5$ and $3$ ), which is $15$ .
- Step 2: Multiply the first equation by $3$ and the second equation by $5$ : $3(3x + 5y) = 3(30) \implies 9x + 15y = 90$ $5(4x + 3y) = 5(29) \implies 20x + 15y = 145$
- Step 3: Subtract the first transformed equation from the second: $(20x + 15y) - (9x + 15y) = 145 - 90$ $11x = 55$ $x = \frac{55}{11} = 5$
- Step 4: Substitute $x = 5$ into the first original equation to solve for $y$ : $3(5) + 5y = 30$ $15 + 5y = 30$ $5y = 30 - 15$ $5y = 15$ $y = 3$
- Step 5: Write the unique solution as an ordered pair: $(x, y) = (5, 3)$
- Step 6: Verification using the second original equation: $4(5) + 3(3) = 20 + 9 = 29$
Modeling Real-World Word Problems with Linear Equations:
- Problem Context: A shop sells notebooks for $\$3$ and binders for $\$7$ . A student purchases a total of $10$ items.
- Variable Definitions:
  - Let $n$ be the number of notebooks purchased.
  - Let $b$ be the number of binders purchased.
- Quantity Constraint Equation: $n + b = 10$
- Cost Function / Constraint Equation: $\text{Total Cost} = 3n + 7b$

Exponent Rules and Monomial Simplification

Product Rule of Exponents:
- When multiplying like bases, keep the base and add the exponents: $a^m \cdot a^n = a^{m+n}$
- For constants, multiply the numerical coefficients directly: $(c_1 \cdot x^a)(c_2 \cdot x^b) = (c_1 \cdot c_2)x^{a+b}$
Simplifying a Product of Several Monomials:
- Expression: $(5z^2x)(3z^2)(3zx^3)$
- Step 1: Group and multiply the numerical coefficients: $5 \times 3 \times 3 = 45$
- Step 2: Group and combine terms containing the base $z$ : $z^2 \cdot z^2 \cdot z^1 = z^{2 + 2 + 1} = z^5$
- Step 3: Group and combine terms containing the base $x$ : $x^1 \cdot x^3 = x^{1 + 3} = x^4$
- Step 4: Combine the simplified components: $45z^5x^4$

Descriptive Statistics and Circle Graphs

Arithmetic Mean and Missing Value Calculations:
- The arithmetic mean (average) $\bar{x}$ of a set of $n$ observations is given by: $\bar{x} = \frac{\sum_{i=1}^n x_i}{n} = \frac{x_1 + x_2 + \dots + x_n}{n}$
- This formula can be rearranged to find the total sum required for a given average: $\sum_{i=1}^n x_i = n \cdot \bar{x}$
- Problem Breakdown:
  - Known quiz scores: $60, 84, 79, 77, 70$
  - Number of completed quizzes: $5$
  - Sum of the first 5 quizzes: $\sum_{i=1}^5 x_i = 60 + 84 + 79 + 77 + 70 = 370$
  - Total number of quizzes after adding the next quiz: $n = 6$
  - Target average score across all 6 quizzes: $\bar{x} = 74$
  - Required total sum of all 6 scores: $\sum_{i=1}^6 x_i = 6 \times 74 = 444$
  - Calculation of the required sixth quiz score ( $x_6$ ): $x_6 = \left(\sum_{i=1}^6 x_i\right) - \left(\sum_{i=1}^5 x_i\right)$ $x_6 = 444 - 370 = 74$
Circle Graphs (Pie Charts) and Proportional Analysis:
- Data Breakdown of After-School Sports Participation:
  - Total population surveyed: $N = 300$ students
  - Soccer: $25\%$
  - Track: $20\%$
  - Baseball: $25\%$
  - Tennis: $30\%$
  - Verification of total percentage: $25\% + 20\% + 25\% + 30\% = 100\%$
- Absolute Student Count per Activity:
  - Tennis: $0.30 \times 300 = 90$ students
  - Baseball: $0.25 \times 300 = 75$ students
  - Soccer: $0.25 \times 300 = 75$ students
  - Track: $0.20 \times 300 = 60$ students
- Comparative Difference Between Groups:
  - To find how many more students participate in tennis than in baseball: $\text{Difference} = \text{Students in Tennis} - \text{Students in Baseball}$ $\text{Difference} = 90 - 75 = 15\text{ students}$
  - Alternative calculation via percentage difference: $\Delta \% = 30\% - 25\% = 5\%$ $\text{Difference} = 5\% \times 300 = 0.05 \times 300 = 15\text{ students}$