enVision Florida B.E.S.T. Algebra 1 EOC Practice Test Form C: Complete Study Notes

Function Transformations and Algebraic Representations

• Vertical Transformations and Reflections:     - The function $f(x) = -2x^3 + x + 1$ is transformed to create $g(x) = kf(x)$.     - When $k = 4$, the graph of $g(x)$ is the graph of $f(x)$ stretched vertically by a factor of $4$.     - When $k = -1$, the graph of $g(x)$ is the graph of $f(x)$ reflected over the x-axis.

• Radical Expressions and Equivalencies:     - The expression $\frac{\sqrt{225}}{\sqrt{5}}$ is examined for equivalence.     - Calculation: $\frac{\sqrt{225}}{\sqrt{5}} = \sqrt{\frac{225}{5}} = \sqrt{45}$.     - Simplifying $\sqrt{45}$: $\sqrt{9 \times 5} = 3\sqrt{5}$.     - The equivalent expression is $3\sqrt{5}$.

• Polynomial Division:     - Dividing the expression $(30x^4y^3 + 42x^3y^9)$ by $6x^3y^3$:     - Step 1: $\frac{30x^4y^3}{6x^3y^3} = 5x$.     - Step 2: $\frac{42x^3y^9}{6x^3y^3} = 7y^6$.     - Result: $5x + 7y^6$.

Systems of Equations and Exponential Forms

• Solving Systems of Linear Equations:     - System:         - $y = \frac{1}{2}x - 3$         - $y = -3x + 4$     - Solution Method (Substitution): $\frac{1}{2}x - 3 = -3x + 4$     - Add $3x$ to both sides: $3.5x - 3 = 4$     - Add $3$ to both sides: $3.5x = 7$     - Solve for $x$: $x = 2$     - Substitute $x = 2$ back: $y = -3(2) + 4 = -2$     - Solution set: $(2, -2)$.

• Exponential Equivalence:     - The expression $1.9^{2x+1}$ is analyzed.     - Using the property $a^{b+c} = a^b \times a^c$: $1.9^{2x} \times 1.9^1$.     - Using the property $a^{bc} = (a^b)^c$: $(1.9^2)^x \times 1.9$.     - Calculation: $1.9^2 = 3.61$.     - Equivalent form: $1.9(3.61)^x$.

Interest, Savings, and Financial Models

• Yearly Compounded Interest (Ainsley's Account):     - Model: $A = 680(1.031)^t$.     - Initial Investment: The principal amount is represented by the coefficient outside the parentheses, which is $\$680$.     - Annual Interest Rate: The growth factor is $1.031$. The rate $r$ is found by $1 + r = 1.031$, thus $r = 0.031$, or $3.1\%$.

• Monthly Compounded Growth (Savings Account):     - Initial deposit: $\$1000$.     - Monthly increase: $0.25\%$, which is expressed as the decimal $0.0025$.     - Function for Account Balance: $B(t) = 1000(1.0025)^t$, where $t$ is the number of months.     - Growth Factors (Nearest Thousandth):         - Every month: $1.003$ (rounded from $1.0025$).         - Every year (12 months): $(1.0025)^{12} \approx 1.030$.         - Every 3 years (36 months): $(1.0025)^{36} \approx 1.094$.

• Standard Investment Modeling:     - Curtis invests $\$475$ at $2.9\%$ compounded annually.     - Type of function: Exponential.     - Reasoning: The investment is modeled by an exponential function because the balance grows by the same percent (rate) each year.

• Bank Fees (Fred's Checking Account):     - Monthly fee: $0.25\%$.     - Initial balance: $\$875$.     - Growth model (decay): Since it is a fee, the balance decreases. $B(t) = 875(1 - 0.0025)^t$.

Literal Equations and Linear Algebra

• Solving for x in Literal Equations:     - Equation: $6x - 2ax = 9 - 3(ax + 2)$.     - Step 1: Distribute: $6x - 2ax = 9 - 3ax - 6$.     - Step 2: Combine constants: $6x - 2ax = 3 - 3ax$.     - Step 3: Move all $x$ terms to one side: $6x - 2ax + 3ax = 3$, which simplifies to $6x + ax = 3$.     - Step 4: Factor out $x$: $x(6 + a) = 3$.     - Step 5: Isolate $x$: $x = \frac{3}{a + 6}$.

• Identifying Algebraic Errors (Isaac's Solution):     - Original equation: $5(x + 4) + 3y = 4 - 2y$.     - Isaac's Step 1: $5x + 4 + 3y = 4 - 2y$.     - Mistake: Isaac failed to use the Distributive Property correctly on the left side. He should have multiplied $5$ by both $x$ and $4$ to get $5x + 20$.

• Lines and Geometry:     - Graphing: The equation $y = \frac{2}{3}x - 1$ represents a line with a y-intercept at $(0, -1)$ and a slope of $\frac{2}{3}$.     - Perpendicular Lines: To find a line through $(20, 38)$ perpendicular to $y = -\frac{2}{5}x - 2$:         - Perpendicular slope ($m_{\perp}$): The negative reciprocal of $-\frac{2}{5}$ is $\frac{5}{2}$.         - Use point-slope form: $y - 38 = \frac{5}{2}(x - 20)$.         - Simplify: $y - 38 = \frac{5}{2}x - 50$.         - Add $38$: $y = \frac{5}{2}x - 12$.

Rational Exponents and Quadratic Modeling

• Rational Exponents and Radicals:     - Property: $5^{\frac{1}{2}} = \sqrt{5}$.     - Logic: $(5^{\frac{1}{2}})^2 = 5^{\frac{1}{2} \times 2} = 5^1$. Because squaring $5^{\frac{1}{2}}$ results in $5$, it is equivalent to the square root of $5$.\n

• Projectile Motion (Ball off a Cliff):     - Function: $h(t) = 13 + 9.8t - 4.9t^2$.     - Vertex: The vertex is at $(1, 17.9)$.     - Interpretation of X-coordinate ($x = 1$): Represents the time it takes for the ball to reach its maximum height.     - Interpretation of Y-coordinate ($y = 17.9$): Represents the ball's maximum height.

• Polynomial Operations (Standard Form):     - Expression: $(3x - 2)(x + 1) + 2(x - 5)^2 - 25$.     - Expand $(3x - 2)(x + 1)$: $3x^2 + 3x - 2x - 2 = 3x^2 + x - 2$.     - Expand $2(x - 5)^2$: $2(x^2 - 10x + 25) = 2x^2 - 20x + 50$.     - Combine all: $(3x^2 + x - 2) + (2x^2 - 20x + 50) - 25 = 5x^2 - 19x + 23$.

• Quadratic Equations from Tables:     - Given table: $(3, 13), (4, 16), (5, 17), (6, 16), (7, 13)$.     - The vertex is clearly $(5, 17)$ due to symmetry.     - Vertex form: $f(x) = a(x - 5)^2 + 17$.     - Using point $(4, 16)$: $16 = a(4 - 5)^2 + 17 \rightarrow 16 = a(1) + 17 \rightarrow a = -1$.     - Standard form: $f(x) = -1(x^2 - 10x + 25) + 17 = -x^2 + 10x - 8$.

• Quadratic Real-World Application (Stock Market):     - Model: $P(d) = \frac{13}{100,000}d^2 - \frac{37}{1,000}d + \frac{156}{10}$.     - The price is declining during the interval prior to the vertex (since $a \gt 0$).     - Vertex $d = -\frac{b}{2a} = -\frac{-0.037}{2(0.00013)} \approx 142.3$.     - Declining period (rounded to nearest day): 0 < d < 142.

Properties of Functions and Relations

• End Behavior of Absolute Value Functions:     - Function: $f(x) = -|x + 3| + 4$.     - As it is a negative absolute value function ($a = -1$), the graph opens downward.     - Behavior: As $x \to \infty$, $f(x) \to -\infty$; and as $x \to -\infty$, $f(x) \to -\infty$.

• Population Growth/Decay:     - Function: $f(x) = 698(1 - 0.004)^x$.     - The base of the exponent is $0.996$. Since 0.996 < 1, the function represents exponential decay.

• Domain and Range:     - Function: $f(x) = (x + 1)^2 - 6$.     - This is a parabola opening upward with vertex at $(-1, -6)$.     - Domain: All real numbers.     - Range: $y \ge -6$.

• Determining Function Types from Tables:     - Table 23: $x = 0, 1, 2, 3, 4$; $f(x) = \frac{1}{4}, 2, 16, 128, 1024$.     - The values of $f(x)$ increase by a factor of $8$ ($\frac{1}{4} \times 8 = 2$, $2 \times 8 = 16$).     - Function: $f(x) = \frac{1}{4}(8)^x$.     - Table 36: $x = 0, 1, 2, 3, 4$; $f(x) = 2, 5, 11, 29, 83$.     - Type: Exponential (rate of change is not constant, nor is the second difference, but growth is rapid).

• Real Solutions of Quadratics (Discriminant $b^2 - 4ac$):     - A. $x^2 + 3x + 2 = 0 \to 9 - 4(2) = 1$ (Two real solutions).     - B. $-x^2 + 4x - 4 = 0 \to 16 - 4(-1)(-4) = 0$ (Exactly one real solution).     - C. $2x^2 + x - 1 = 0 \to 1 - 4(2)(-1) = 9$ (Two real solutions).     - D. $4x^2 + 4x + 1 = 0 \to 16 - 4(4)(1) = 0$ (Exactly one real solution).     - E. $x^2 - 5x + 7 = 0 \to 25 - 4(7) = -3$ (Zero real solutions).

Data Analysis, Statistics, and Probability

• Test Score Data:     - Scores: $76, 83, 83, 78, 89, 90, 95, 77, 83, 82, 90, 80$.     - Data Type: Numerical.     - Display Method: Line plot is best for showing frequency and distribution of numerical test scores.

• Categorical Data Analysis:     - Hot drink orders sample: "tea with sugar and milk", "coffee with milk", etc.     - Type: Categorical univariate (one variable: the type of drink order described by categories).

• Frequency Tables (Cafeteria Sales):     - Total items: $172$.     - Burgers: $108$ ($90$ regular bun, $18$ gluten-free).     - Hot dogs: $64$ ($43$ regular bun, $21$ gluten-free).     - Total regular: $90 + 43 = 133$.     - Total gluten-free: $18 + 21 = 39$.

• Statistical Estimation and Margin of Error:     - Season ticket sample ($n=125$): Average of $3.2$ tickets, margin of error $\pm 0.6$.     - Range: $[3.2 - 0.6, 3.2 + 0.6] = [2.6, 3.8]$.     - Population projection ($N=450$):         - Lower bound: $2.6 \times 450 = 1170$.         - Upper bound: $3.8 \times 450 = 1710$.     - Estimated tickets: Between 1170 and 1710 tickets.

• Residuals and Correlation:     - Jonas's pay vs. hours: Residual plot shows data distributed evenly around the line of fit (no visible pattern).     - Fit Quality: Yes; the data are distributed evenly around the line of fit.     - Temperature vs. Months: Scatter plot shows a downward trend over 5 months.     - Conclusion: An increase in the number of months is associated with a decrease in average daily temperature.

Graphing and Functional Behavior

• Increasing Intervals:     - Square root function $f(x)$ is increasing over the interval $0 \le x \lt \infty$.     - Linear function $g(x)$ is increasing over the interval $-\infty \lt x \lt \infty$.

• Absolute Value Graphs:     - The graph of $f(x) = 2|x + 4|$ has a vertex at $(-4, 0)$ and a vertical stretch of $2$ (V-shape opening up).

• Line of Fit (County Fair Ribbons):     - Best model for relationship between categories ($x$) and ribbons ($y$): $y = x + 1$.     - Slope Interpretation: For every additional category entered, students earn an additional 1 ribbon on average.

• Function Evaluation and Rate of Change:     - Evaluate $f(x) = \frac{x}{2} + 8$ when $x = -14$: $f(-14) = \frac{-14}{2} + 8 = -7 + 8 = 1$.     - Evaluate $f(x) = 3x^5 - 100$ when $x = 4$: $f(4) = 3(4^5) - 100 = 3(1024) - 100 = 3072 - 100 = 2972$.     - Average rate of change of $Q(t)$ from $t=0$ to $t=b$ is $-2.4$. Given $(b, 15)$ and initial value $21$ at $t=0$:         - $\frac{15 - 21}{b - 0} = -2.4 \to \frac{-6}{b} = -2.4 \to b = \frac{-6}{-2.4} = 2.5$.

Inequalities and Word Problems

• Desk Payment Plan:     - $(3, \$580)$ and $(6, \$406)$.     - Slope (monthly payment): $m = \frac{406 - 580}{6 - 3} = \frac{-174}{3} = -58$.     - Y-intercept: $580 = -58(3) + b \to 580 = -174 + b \to b = 754$.     - Equation: $y = -58t + 754$.

• Cleaning Rooms for Money:     - Model: $y = 6x + 5$.     - Target: $y = \$41$.     - Calculation: $41 = 6x + 5 \to 36 = 6x \to x = 6$.     - Ira needs to clean 6 rooms.

• Distance and Biking (Inequalities):     - Your distance to grocery store: $d = 20\,min \times 625\,ft/min = 12,500\,ft$.     - Friend lives closer. Inequality: $d \lt 12,500$.

• Solving Absolute Value Equations:     - Equation: $0 = |8x - 4| - 20 \to |8x - 4| = 20$.     - Case 1: $8x - 4 = 20 \to 8x = 24 \to x = 3$.     - Case 2: $8x - 4 = -20 \to 8x = -16 \to x = -2$.     - Values: -2 and 3.

• Baseball Tickets System:     - Afternoon ($x$) cost $\$17.50$, Evening ($y$) cost $\$32$.     - Constraint 1 (Tickets): $x + y \ge 5$.     - Constraint 2 (Cost): $17.5x + 32y \le 250$.

• Graphing Linear Inequalities:     - Graph has a solid line crossing $(-1, 0)$ and $(0, 0.5)$, shading below.     - Slope: $m = \frac{0.5 - 0}{0 - (-1)} = 0.5$.     - Inequality: $y \le 0.5x + 0.5$.

• Systems of Inequalities Graphing:     - Graph shows intersection of a solid horizontal line at $y = -6$ and a dashed angled line.     - Shaded area is below the dashed line and above the solid line.     - Example solution: $y \ge -6$ and y < \frac{1}{2}x + 3.