Algebraic Functions and Factoring Study Notes

General Student and Assessment Information

• Student Name: Janessa Johnson.
• Assessment Guidelines:     * Students must show all work used to solve each problem to receive credit.     * The use of aids is restricted to a 4-function calculator only.

Problem 1: Graphical Analysis of Functions

• Problem Statement: Use the provided graph of $y = f(x)$ to answer specific questions regarding coordinates, solutions, domain, and range.
• Evaluating Specific Points:     * Task: Evaluate $f(-2)$.     * Procedure: Locate the value on the x-axis where $x = -2$ and identify the corresponding y-coordinate on the function's curve.     * Result: $f(-2) = -3$.
• Solving for x (Horizontal Intercepts):     * Task: Write all solutions to $f(x) = -1$.     * Procedure: Determine the x-values where the graph intersects the horizontal line $y = -1$.     * Solutions: $x = -3$ and $x = 1$.
• Determining Domain:     * Task: Write the domain of $f$ using interval notation.     * Definition: The domain is the set of all possible input values (x-values) for which the function is defined.     * Observation: Based on the graph's extension to the left and its termination on the right at $x = 4$.     * Interval Notation: $(-\text{∞}, 4)$.
• Determining Range:     * Task: Write the range of $f$ using interval notation.     * Definition: The range is the set of all possible output values (y-values) the function can produce.     * Observation: The lowest point (minimum) on the graph occurs at $y = -3$, and the graph extends upwards infinitely.     * Interval Notation: $[-3, \text{∞})$. (Note: While the student's handwritten note indicates $[3, \text{∞})$, the graph and part (a) support $-3$ as the minimum value).

Problem 2: Factoring Algebraic Expressions

• Problem Statement: Factor every given expression completely using appropriate algebraic techniques.
• Factoring by Greatest Common Factor (GCF):     * Expression: $6x^2 - 12x$     * Work Shown: The student initially factored out $2x$, resulting in $2x(3x - 6)$.     * Complete Factorization: To factor completely, the GCF of $6$ must be pulled out. The final result is $6x(x - 2)$.
• Difference of Squares:     * Expression: $x^2 - 16$     * Technique: Use the formula $a^2 - b^2 = (a - b)(a + b)$, where $a = x$ and $b = 4$.     * Step-by-Step Verification: $(x - 4)(x + 4) = x^2 - 4x + 4x - 16 = x^2 - 16$.     * Result: $(x - 4)(x + 4)$.
• Trinomial Factoring (Splitting the Middle Term):     * Expression: $x^2 - 10x - 24$     * Requirement: Find two numbers that multiply to $-24$ and add to $-10$. Those numbers are $-12$ and $2$.     * Steps:         1. Rewrite the middle term: $x^2 + 2x - 12x - 24$         2. Group the terms: $(x^2 + 2x) - (12x + 24)$         3. Factor out a GCF from each group: $x(x + 2) - 12(x + 2)$         4. Factor out the common binomial: $(x - 12)(x + 2)$     * Result: $(x - 12)(x + 2)$.
• Perfect Square Trinomials:     * Expression: $9x^2 - 12x + 4$     * Analysis: This expression follows the pattern $a^2 - 2ab + b^2 = (a - b)^2$.         * The first term is a perfect square: $(3x)^2$         * The last term is a perfect square: $2^2$         * The middle term is twice the product of the square roots: $2 \times (3x) \times 2 = 12x$     * Result: $(3x - 2)^2$. (Note: The student was unable to complete this problem on the page).

Questions & Discussion

• Student Inquiry (2d): In response to the problem $9x^2 - 12x + 4$, the student noted "idk…", indicating difficulty with recognizing perfect square trinomials or factoring quadratic expressions with a lead coefficient greater than 1.