Algebra 1 EOC 2024 Released Items Study Notes

Item 37: Analysis of the Function $f(x) = x^2 + 12x + 20$

Function Defined: The given function is a quadratic function in standard form, where $a = 1$ , $b = 12$ , and $c = 20$ .
Leading Coefficient Analysis: Since the leading coefficient ( $a = 1$ ) is positive, the parabola opens upward.
End Behavior Definition: End behavior describes what happens to the function values ( $f(x)$ ) as the input values ( $x$ ) move toward positive infinity ( $+\infty$ ) and negative infinity ( $-\infty$ ).
Behavior as $x$ approaches $\infty$ : For a quadratic with a positive leading coefficient, as the input $x$ increases indefinitely, the output $f(x)$ also increases indefinitely. Therefore, as $x \rightarrow \infty$ , $f(x) \rightarrow \infty$ .
Behavior as $x$ approaches $-\infty$ : Because the degree of the function is even (degree 2), both ends of the graph point in the same direction. As the input $x$ decreases toward negative infinity, the output $f(x)$ increases toward positive infinity. Therefore, as $x \rightarrow -\infty$ , $f(x) \rightarrow \infty$ .

Item 38: Dipping Sauce Preference Survey

Context: Margo surveyed $150$ high school students, specifically juniors and seniors, to determine their favorite dipping sauces.
Data Table (Verbatim Values): * Juniors: * Barbeque: $23$ * Honey Mustard: $19$ * Ranch: $26$ * Sweet and Sour: $11$ * Seniors: * Barbeque: $5$ * Honey Mustard: $7$ * Ranch: $22$ * Sweet and Sour: $30$ * Additional Data Point: $12$
Categorical Totals (calculated based on question requirements): * Total Honey Mustard: $19\,(\text{Juniors}) + 7\,(\text{Seniors}) = 26$ * Total Sweet and Sour: Depending on the interpretation of the extra data points ( $30$ and $12$ ), the baseline total appears to be $11\,(\text{Juniors}) + 30\,(\text{Seniors}) = 41$ .
Comparison Task: To determine how many more students chose Honey Mustard than Sweet and Sour (or vice-versa), compute the difference. If Sweet and Sour was chosen by $41$ students and Honey Mustard by $26$ , then $41 - 26 = 15$ more students chose Sweet and Sour.

Item 40: Graphs and Tables Comparison

Function $g(x)$ (Exponential): Defined by a table of values. * $(-2, 0.25)$ * $(-1, 0.5)$ * $(0, 1)$ * $(1, 2)$ * $(2, 4)$ * Functional Form: This follows the form $g(x) = 2^x$ .
Function $f(x)$ (Graphed): The graph shows a function $y = f(x)$ . Based on the visible data, the $y-intercept$ of $f(x)$ is at $(0, 2)$ .
Comparison Statements: * Y-Intercepts: The $y-intercept$ for $g(x)$ is $1$ (where $x=0$ ). The $y-intercept$ for $f(x)$ is $2$ . Therefore, the value of the $y-intercept$ of $y = f(x)$ is greater than the value of the $y-intercept$ of $y = g(x)$ . * Long-term Growth: Because $g(x)$ is an exponential growth function with a base greater than $1$ ( $b=2$ ), it will eventually grow at a faster rate than any linear or lower-degree function. As the values of $x$ increase, $g(x)$ will eventually be greater than $f(x)$ .

Item 41: Budget Constraints for Hot Wings and Soda

Variables: * $x$ represents the number of boxes of hot wings. * $y$ represents the number of bottles of soda.
Constraint Inequality: $40 \geq 6x + 2y$ * This implies a maximum budget of $\$40$ . * Hot wings cost $\$6$ per box. * Soda costs $\$2$ per bottle.
Graphing the Solution: * Find the boundary line intercepts by setting the equation to $6x + 2y = 40$ . * x-intercept: Set $y=0 \Rightarrow 6x = 40 \Rightarrow x = \frac{20}{3} \approx 6.67$ . * y-intercept: Set $x=0 \Rightarrow 2y = 40 \Rightarrow y = 20$ . * Shading: Since the inequality is "less than or equal to" ( $\leq$ ), and the point $(0,0)$ satisfies the inequality ( $0 \leq 40$ ), the shaded region must be below the line toward the origin.

Item 42: Janelle's Fruit Purchase

Parameters: * Blackberries ( $b$ ) cost $\$8$ per pound. * Raspberries ( $r$ ) cost $\$9$ per pound. * The maximum budget is $\$40$ .
Inequality Construction: The sum of the cost of blackberries and the cost of raspberries cannot exceed $\$40$ . * Expression: $8b + 9r \leq 40$

Item 43: Value of Tools over Time

Given Data Points: * Value after year 1 ( $t=1$ ): $V(1) = \$380$ * Value after year 2 ( $t=2$ ): $V(2) = \$361$
Determining the Common Ratio ( $r$ ): * $r = \frac{V(2)}{V(1)} = \frac{361}{380} = 0.95$ * This indicates the tools retain $95\%$ of their value each year, or decay at a rate of $5\%$ ( $0.05$ ).
Determining the Initial Value ( $a$ ): * $V(t) = a \times (r)^t$ * $380 = a \times (0.95)^1$ * $a = \frac{380}{0.95} = 400$
Final Function Equation: * $V(t) = 400(1 - 0.05)^t$

Item 44: Identifying the Equation of a Parabola

Observations from Graph: * Vertex: The peak of the graph is located at the point $(-3, 4)$ . * Direction: The parabola opens downward, indicating a negative leading coefficient ( $a < 0$ ). * Vertical Stretch/Compression: The graph passes through specific points, such as $(-1, -4)$ . By substituting the vertex into vertex form: $y = a(x + 3)^2 + 4$ . * Using $(-1, -4)$ : $-4 = a(-1 + 3)^2 + 4 \Rightarrow -4 = 4a + 4 \Rightarrow -8 = 4a \Rightarrow a = -2$ .
Resulting Equation: * $y = -2(x + 3)^2 + 4$

Item 45: Numerical Equivalencies

Task: Identify values equivalent to the provided expression (likely $\sqrt{16}$ or associated constant).
Available Values to Evaluate: * $\sqrt{16} = 4$ * $3$ * $0$ * $-20$ * $2$

Here are some questions based on the provided notes:

What is the standard form of the quadratic function discussed, and what are the values of the coefficients $a$ , $b$ , and $c$ ?
How does the sign of the leading coefficient affect the direction in which a parabola opens?
Describe the end behavior of the quadratic function $f(x) = x^2 + 12x + 20$ as $x$ approaches $+ ext{infinity}$ .
As $x$ approaches $- ext{infinity}$ , what happens to the value of $f(x)$ for this quadratic function?
In the Dipping Sauce Preference Survey, how many students preferred Honey Mustard over Sweet and Sour?
If Margo surveyed 150 students, what percentage of those students chose Ranch as their favorite dipping sauce?
What is the functional form of the exponential function given in the notes?
How does the growth rate of an exponential function compare to that of a linear function as $x$ increases?
Describe the budget constraint for Janelle's fruit purchases in terms of the costs of blackberries and raspberries.
Based on the exponential decay example, what is the common ratio ( $r$ ) for the value of tools after 2 years?