Unit 1 June 2024 — Unit Plan

A ball was launched into the air, and its height above the ground was recorded each second, as shown in the table below.

Based on these data, which statement is a valid conclusion?

(1) The ball lands on the ground at 4 seconds.
(2) The ball reaches a maximum height of 11 feet.
(3) The ball was launched from a height of 0 feet.
(4) The ball reaches its maximum height at 2 seconds.

A tour bus can seat, at most, 48 passengers. An adult ticket costs $18 and a child ticket costs $12. The bus company must collect at least $650 to make a profit. If $a$ represents the number of adult tickets sold and $c$ represents the number of child tickets sold, which system of inequalities models this situation if they make a profit?

(1) $a + c < 48$
$\phantom{(1) }18a + 12c > 650$

(2) $a + c \leq 48$
$\phantom{(2) }18a + 12c \geq 650$

(3) $a + c < 48$
$\phantom{(3) }18a + 12c < 650$

(4) $a + c \leq 48$
$\phantom{(4) }18a + 12c \leq 650$

Which equation is always true?

(1) $x^2 \cdot x^3 = x^5$
(2) $3^x \cdot 3^2 = 9^{2x}$
(3) $-z^2 = z^2$
(4) $7^a \cdot 7^b = 7^{ab}$

The expression $-2(x^2 - 2x + 1) + (3x^2 + 3x - 5)$ is equivalent to

(1) $x^2 + x - 4$
(2) $x^2 - x - 7$
(3) $x^2 + 7x - 4$
(4) $x^2 + 7x - 7$

Which sum is irrational?

(1) $-2\sqrt{12} + \sqrt{100}$
(2) $-\sqrt{4} + \frac{1}{3}\sqrt{900}$
(3) $\frac{1}{2}\sqrt{25} + \sqrt{64}$
(4) $\sqrt{49} + 3\sqrt{121}$

The solution to $\frac{4(x - 5)}{3} + 2 = 14$ is

(1) 15
(2) 14
(3) 6
(4) 4

On an island, a rare breed of rabbit doubled its population each month for two years. Which type of function best models the increase in population at the end of two years?

(1) linear growth
(2) linear decay
(3) exponential growth
(4) exponential decay

What is the degree of the polynomial $2x - x^2 + 4x^3$?

(1) 1
(2) 2
(3) 3
(4) 4

The zeros of the function $f(x) = x(x - 5)(3x + 6)$ are

(1) $0, -5, \text{ and } 2$
(2) $0, 5, \text{ and } -2$
(3) $-5 \text{ and } 2$, only
(4) $5 \text{ and } -2$, only

What is the $y$-intercept of the line that passes through the points $(-1, 5)$ and $(2, -1)$?

(1) $-1$
(2) $-2$
(3) 3
(4) 5

Nancy has just been hired for her first job. Her company gives her four choices for how she can collect her annual salary over the first eight years of employment.

Each function below represents the four choices she has for her annual salary in thousands of dollars, where $t$ represents the number of years after she is hired.

$$a(t) = 2^t + 25$$

$$b(t) = 10t + 75$$

$$c(t) = \sqrt{400t} + 80$$

$$d(t) = 2(t + 1)^2 - 10t + 50$$

Which pay plan should Nancy choose in order to have the highest salary in her eighth year?

(1) $a(t)$
(2) $b(t)$
(3) $c(t)$
(4) $d(t)$

The third term in a sequence is 25 and the fifth term is 625. Which number could be the common ratio of the sequence?

(1) $\frac{1}{5}$
(2) 5
(3) $\frac{1}{25}$
(4) 25

The box plot below summarizes the data for the amount of snowfall, in inches, during the winter of 2021 for 12 locations in western New York.

Image Description: A box-and-whisker plot on a number line labeled "Winter of 2021 Snowfall (inches)" from 0 to 140 in increments of 20. The minimum is at 50. The box extends from 60 (Q1) to 110 (Q3), with a median line at 80. The maximum is at 120.

What is the interquartile range?

(1) 30
(2) 50
(3) 80
(4) 110

Four quadratic functions are represented below.

• I: $$a(x) = (x - 3)^2 - 7$$
• II: Image Description: A graph of $b(x)$ on a coordinate grid. The parabola opens upward with a vertex at $(0, -5)$. The curve passes through $(-3, 4)$ and $(3, 4)$. The y-axis is labeled $b(x)$.
• III: $$c(x) = x^2 + 6x + 3$$
• IV:

  $x$	$d(x)$
  $-4$	$-1$
  $-3$	$-4$
  $-2$	$-5$
  $-1$	$-4$
  $0$	$-1$

Which function has the smallest minimum value?

(1) I
(2) II
(3) III
(4) IV

The equation that represents the sequence $-2, -5, -8, -11, -14, \ldots$ is

(1) $a_n = -3 + (-2)(n - 1)$
(2) $a_n = -2 + (-3)(n - 1)$
(3) $a_n = 3 + (-2)(n - 1)$
(4) $a_n = -2 + (3)(n - 1)$

The dot plot below shows the number of goals Jessica scored in each lacrosse game last season.

Image Description: A dot plot on a number line labeled "Goals Scored per Game" from 0 to 6. The dots are stacked vertically at each value: 0 has 3 dots, 1 has 3 dots, 2 has 4 dots, 3 has 5 dots, 4 has 2 dots, 5 has 2 dots, and 6 has 1 dot.

Which statement about the dot plot is correct?

(1) mean $>$ mode
(2) mean $=$ median
(3) mode $=$ median
(4) median $>$ mean

The students in Mrs. Smith's algebra class were asked to describe the graph of $g(x) = 2(x - 3)^2$ compared to the graph of $f(x) = x^2$.

Which student response is correct?

(1) Ashley said that the graph of $g(x)$ is wider and shifted left 3 units.
(2) Beth said that the graph of $g(x)$ is narrower and shifted left 3 units.
(3) Carl said that the graph of $g(x)$ is wider and shifted right 3 units.
(4) Don said that the graph of $g(x)$ is narrower and shifted right 3 units.

One Saturday, Dave took a long bike ride. The graph below models his trip.

Image Description: A line graph on a coordinate grid with the x-axis labeled "Hours" (from 0 to 6) and the y-axis labeled "Miles Traveled" (from 0 to 60, in increments of 10). Plotted points are connected by line segments, forming a piecewise linear function. The plotted points are (0, 0), (0.5, 5), (1, 20), (2.5, 35), (3, 35), and (5.5, 55). The graph rises steeply from the origin, has a flat segment from (2.5, 35) to (3, 35), then rises steadily to the endpoint at (5.5, 55).

What was Dave's average rate of change, in miles per hour, on this trip?

(1) 10
(2) 11
(3) 11.6
(4) 14.5

Which expression is equivalent to $(x - 5)(2x + 7) - (x + 5)$?

(1) $2x^2 - 2x - 30$
(2) $2x^2 - 2x - 40$
(3) $2x^2 - 4x - 30$
(4) $2x^2 - 4x - 40$

The functions $f(x)$ and $g(x)$ are graphed on the set of axes below.

Image Description: A coordinate grid showing two functions. $f(x)$ is a parabola opening upward with its vertex at $(2, -4)$. $g(x)$ is a linear function with a slope of 2 and a y-intercept at $(0, -5)$. The two functions intersect at $(1, -3)$ and $(5, 5)$.

What is the solution to the equation $f(x) = g(x)$?

(1) 1 and 5
(2) $-5$ and 0
(3) $-3$ and 5
(4) 0 and 4

When babysitting, Nicole charges an hourly rate and an additional charge for gas. She uses the function $C(h) = 6h + 5$ to determine how much to charge for babysitting. The constant term of this function represents

(1) the additional charge for gas
(2) the hourly rate Nicole charges
(3) the number of hours Nicole babysits
(4) the total Nicole earns from babysitting

When solved for $x$ in terms of $a$, the solution to the equation $3x - 7 = ax + 5$ is

(1) $\frac{12}{3a}$
(2) $\frac{12}{3 - a}$
(3) $\frac{3a}{12}$
(4) $\frac{3 - a}{12}$

Wayde van Niekerk, a runner from South Africa, ran 400 meters in 43.03 seconds to set a world record. Which calculation would determine his average speed, in miles per hour?

(1) $\frac{400 \text{ m}}{43.03 \text{ sec}} \cdot \frac{1000 \text{ m}}{0.62 \text{ mi}} \cdot \frac{1 \text{ hr}}{3600 \text{ sec}}$

(2) $\frac{400 \text{ m}}{43.03 \text{ sec}} \cdot \frac{0.62 \text{ mi}}{1000 \text{ m}} \cdot \frac{1 \text{ hr}}{3600 \text{ sec}}$

(3) $\frac{400 \text{ m}}{43.03 \text{ sec}} \cdot \frac{0.62 \text{ mi}}{1000 \text{ m}} \cdot \frac{3600 \text{ sec}}{1 \text{ hr}}$

(4) $\frac{400 \text{ m}}{43.03 \text{ sec}} \cdot \frac{1000 \text{ m}}{0.62 \text{ mi}} \cdot \frac{3600 \text{ sec}}{1 \text{ hr}}$

Which function has a domain of all real numbers and a range greater than or equal to three?

(1) $f(x) = -x + 3$
(2) $g(x) = x^2 + 3$
(3) $h(x) = 3^x$
(4) $m(x) = |x + 3|$

Solve $5(x - 2) \leq 3x + 20$ algebraically.

Given $g(x) = x^3 + 2x^2 - x$, evaluate $g(-3)$.

Given the relation $R = \{(-1,1), (0,3), (-2,-4), (x,5)\}$.

State a value for $x$ that will make this relation a function.

Explain why your answer makes this a function.

A survey of 150 students was taken. It was determined that $\frac{2}{3}$ of the students play video games.

Of the students that play video games, 85 also use social media.
Of the students that do not play video games, 20% do not use social media.

Complete the two-way frequency table.

Use the method of completing the square to determine the exact values of $x$ for the equation $x^2 + 10x - 30 = 0$.

Factor $20x^3 - 45x$ completely.

Graph the following system of equations on the set of axes below.

$$y = x^2 - 3x - 6$$

$$y = x - 1$$

Image Description: A blank coordinate grid with x-axis and y-axis. The grid extends approximately from $-9$ to $9$ in both directions.

State the coordinates of all solutions.

The table below shows the amount of money a popular movie earned, in millions of dollars, during its first six weeks in theaters.

Write the linear regression equation for this data set, rounding all values to the nearest hundredth.

State the correlation coefficient to the nearest hundredth.

State what this correlation coefficient indicates about the linear fit of the data.

Use the quadratic formula to solve the equation $3x^2 - 10x + 5 = 0$. Express the answer in simplest radical form.

Graph the system of inequalities on the set of axes below.

$$3y + 2x \leq 15$$

$$y - x > 1$$

Image Description: A blank coordinate grid with x-axis and y-axis. The grid extends approximately from $-9$ to $9$ in both directions.

State the coordinates of a point in the solution to this system. Justify your answer.

Courtney went to a coffee shop to purchase lattes and donuts for her friends. One day she spent a total of $15.50 on four lattes and two donuts. The next day she spent a total of $18.10 on three lattes and five donuts. All prices included tax.

If $x$ represents the cost of one latte and $y$ represents the cost of one donut, write a system of equations that can be used to model this situation.

Courtney thinks that one latte costs $2.75 and one donut costs $2.25. Is Courtney correct? Justify your answer.

Use your equations to determine algebraically the exact cost of one latte and the exact cost of one donut.