MCR3U - Mastery Sampler #4

Expression Summation: To simplify an expression consisting of identical quadratic terms, add the coefficients: * $4x^2 + 4x^2 + 4x^2 = 12x^2$
Monomial Multiplication: To multiply monomials, multiply the coefficients and add the exponents of the variables: * $(4x^2)(4x^2)(4x^2) = 64x^6$
Polynomial Expansion and Distribution: * Simplified form of $(x + 1)(x + 3)x$ : Expand the binomials first to get $(x^2 + 4x + 3)x$ , then distribute $x$ to obtain $x^3 + 4x^2 + 3x$ . * Simplified form of $2a(a^2 + 3a^3)$ : Distribution results in $2a^3 + 6a^4$ . * Expansion of squared binomials: $(4c + 2d)^2 = (4c)^2 + 2(4c)(2d) + (2d)^2 = 16c^2 + 16cd + 4d^2$ .
Further Expansion Exercises: * $2x(x + 3) = 2x^2 + 6x$ * $3(x + 1)(x - 2) = 3(x^2 - x - 2) = 3x^2 - 3x - 6$ * $2x(x^2 + 2x + 1) = 2x^3 + 4x^2 + 2x$ * $2(x + 3)^2 = 2(x^2 + 6x + 9) = 2x^2 + 12x + 18$

Function Evaluation and Inverses: Given the function set $f(x) = {(2, 3), (-1, 2), (-4, 0)}$ : * Value of f(2): $3$ * Value of f^{-1}(2): The input that yields an output of $2$ is $-1$ . * Solving for x: When $f(x) = 0$ , then $x = -4$ .
Equation Writing for Transformations: * For a function subject to a shift right by $2$ units and a horizontal compression by a factor of $\frac{1}{4}$ , the resulting equation is $y = f(4x - 2)$ . * A shift of left $3$ units and down $4$ units is represented by the general transformation $f(x + 3) - 4$ .
Multi-Step Transformation Analysis: For the function $y = 9 - 4f(7 - \frac{1}{5}x)$ , the specific transformations applied in order are: 1. Vertical stretch by a factor of $4$ . 2. Vertical reflection in the $x$ -axis. 3. Vertical shift up $9$ units. 4. Horizontal shift left $7$ units. 5. Horizontal reflection in the $y$ -axis. 6. Horizontal stretch by a factor of $5$ .
Point and Mapping Rules for Transformations: * Original Point: $(10, -2)$ * Transformed Point: $(-15, 13)$ * Mapping Rule: $(x, y) \rightarrow (5(x - 7), -4y + 9)$
Specific Parent Function Transformations: * $y = x^2$ translated up $3$ units: $y = x^2 + 3$ * $y = x^3$ vertical stretch of factor $2$ : $y = 2x^3$ * $y = \sqrt{x}$ translation right $2$ units: $y = \sqrt{x - 2}$ * $y = |x|$ reflected on $y$ -axis: Based on the sampler's results, the model provided is $y = -|x|$

Comprehensive Factoring Techniques: * Common Factoring: $2x^2 - 16x = 2x(x - 8)$ * Perfect Square Trinomials: $x^2 + 6x + 9 = (x + 3)^2$ * Difference of Squares: $x^2 - 49 = (x - 7)(x + 7)$ * Trinomial Factoring ( $ax^2 + bx + c$ ): $6x^2 - 13x + 6 = (3x - 2)(2x - 3)$
Roots and Quadratic Forms: * Factored form of an equation with roots $-1$ and $2$ : $(x + 1)(x - 2) = 0$ * Standard form of an equation with roots $-1$ and $2$ : $x^2 - x - 2 = 0$
Completing the Square: To make $x^2 - 8x$ a perfect square, the term needed is $16$ , resulting in $(x - 4)^2$ .
The Discriminant ( $D = b^2 - 4ac$ ): * For the equation $4x^2 - 3x + 2 = 0$ , the value of the discriminant is $(-3)^2 - 4(4)(2) = 9 - 32 = -23$ . * If the discriminant of a quadratic equation is negative (e.g., $-16$ ), the graph of its related function would have zero ( $0$ ) or no $x$ -intercepts.
Tile Diagrams: The factored form of $x^2 + 2x - 8$ is $(x + 4)(x - 2)$ , which can be visualized using a tile diagram.

Numerical Evaluation: * Subtraction: $\frac{3}{5} - \frac{1}{6} = \frac{18}{30} - \frac{5}{30} = \frac{13}{30}$ * Multiplication: $\frac{2}{3} \times 4 = \frac{8}{3}$ * Division: $\frac{5}{6} \div \frac{2}{3} = \frac{5}{6} \times \frac{3}{2} = \frac{15}{12} = \frac{5}{4}$
Equivalent Rational Fractions: To solve $\frac{4}{3y} - \frac{1}{4x}$ , find the common denominator $12xy$ . The missing numerator values are $16x$ and $3y$ , resulting in $\frac{16x - 3y}{12xy}$ .
Simplifying with Restrictions: * Expression: $\frac{(x - 3)(x + 2)}{(x - 3)}$ simplifies to $x + 2$ ; Restriction: $x \neq 3$ . * Expression: $\frac{(x + 2)(7x - 1)}{(x - 4)(x + 2)^2} \times \frac{(x - 4)}{(7x - 1)}$ simplifies to $\frac{1}{(x + 2)}$ ; Restrictions: $x \neq 4, x \neq \frac{1}{7}, x \neq -2$ .

Solving Basic and Quadratic Equations: * $x^2 = 64$ results in $x = \pm 8$ * $x = \frac{4}{x}$ results in $x^2 = 4$ , therefore $x = \pm 2$ * $x^2 + x = 12 \rightarrow x^2 + x - 12 = 0 \rightarrow (x + 4)(x - 3) = 0$ , thus $x = -4, 3$ * $x(x + 1) - 2(x + 1) = 0 \rightarrow (x - 2)(x + 1) = 0$ , thus $x = -1, 2$
Solving Inequalities: * Absolute value: $|x| < 2$ implies $-2 < x < 2$ . (Note: Transcript answer key states $-2 < x > 2$ ). * Quadratic inequalities: $x^2 > 16$ implies $x > 4$ or $x < -4$ . * Linear inequalities: $2 - 3x > 5 \rightarrow -3x > 3 \rightarrow x < -1$ .

Primary Trigonometric Ratios: For a given angle $\theta$ with side components $x = 6, y = 8, r = 10$ : * $\sin(\theta) = \frac{8}{10} = \frac{4}{5}$ * $\cos(\theta) = \frac{6}{10} = \frac{3}{5}$ * $\tan(\theta) = \frac{8}{6} = \frac{4}{3}$
Standard Position and Quadrants: If the sine ratio for $\angle \theta$ is positive, the terminal arm must be located in Quadrant I or Quadrant II.
Finding Possible Angles ( $0^\circ \leq \theta \leq 360^\circ$ ): * If $\tan(\theta) = 1$ , then $\theta = 45^\circ, 225^\circ$ * If $\sin(\theta) = 0.5$ , then $\theta = 30^\circ, 150^\circ$
Law of Cosines Expressions (assuming sides are $8$ and $12$ with angle $A$ opposite side $a$ ): * Side length: $a = \sqrt{8^2 + 12^2 - 2(8)(12)\cos(A)}$ * Angle: $A = \cos^{-1}(\frac{a^2 - 8^2 - 12^2}{-2(8)(12)})$
Law of Sines Expressions: * Side length: $a = \frac{12\sin(A)}{\sin(B)}$ or $a = \frac{8\sin(A)}{\sin(C)}$ * Angle: $\sin(B) = \frac{12\sin(A)}{a}$ or $\sin(B) = \frac{12\sin(C)}{8}$
Sinusoidal Properties for $f(x) = 2\sin(3x)$ : * Amplitude: $2$ * Maximum Value: $2$ * Period: $\frac{360^\circ}{3} = 120^\circ$
Analyzing Graphs of Sinusoidal Functions: * For a standard cycle shown in the graph: * Period: $360^\circ$ * Amplitude: $2$ * Modeling Equations: * Sine model: $f(x) = 2\sin(x)$ * Cosine model: $f(x) = 3\cos(x)$ (Note: Amplitude change in answer key relative to sine model).