Unit 3 January 2025 — Unit Plan

When factored, the expression $x^3 - 36x$ is equivalent to

(1) $(x + 6)(x - 6)$
(2) $(x + 18)(x - 18)$
(3) $x(x + 6)(x - 6)$
(4) $x(x + 18)(x - 18)$

Which equation represents the line that passes through the points $(-1,8)$ and $(4,-2)$?

(1) $y = -2x + 6$
(2) $y = -2x + 10$
(3) $y = -0.5x + 7.5$
(4) $y = -0.5x + 8.5$

A geometric sequence is shown below.

$$\frac{1}{2}, 2, 8, 32, \ldots$$

What is the common ratio?

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

What is the constant term of the polynomial $2x^3 - x + 5 + 4x^2$?

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

A landscaping company charges a set fee for a spring cleanup, plus an hourly labor rate. The total cost is modeled by the function $C(x) = 55x + 80$. In this function, what does the 55 represent?

(1) the set fee for the cleanup
(2) the hourly labor rate for a cleanup
(3) the profit earned by the company for one cleanup
(4) the number of hours of labor required for one cleanup

Which expression is equivalent to $(5x^2 - 2x + 4) - (3x^2 + 3x - 1)$?

(1) $2x^2 + x + 3$
(2) $2x^2 - 5x + 5$
(3) $2x^4 + x^2 + 3$
(4) $2x^4 - 5x^2 + 5$

A system of inequalities is graphed on the set of axes below.

Image Description: A coordinate plane shows two dashed lines with shaded regions. One line has a negative slope with a y-intercept at $(0, 4)$ and a slope of $-3$, with shading below. The other line has a positive slope with a y-intercept at $(0, -1)$ and a slope of $2$, with shading above. The two lines intersect at $(1, 1)$. The solution region (double-shaded area) is between the two lines.

Which point is a solution to this system?

(1) $(1,1)$
(2) $(2,-2)$
(3) $(1,8)$
(4) $(4,2)$

In an arithmetic sequence, the first term is 25 and the third term is 15. What is the tenth term in this sequence?

(1) $-20$
(2) $-25$
(3) $70$
(4) $75$

When the formula $p = 2l + 2w$ is solved for $w$, the result is

(1) $w = \frac{2l + p}{2}$
(2) $w = \frac{p - 2l}{2}$
(3) $w = \frac{p}{2} + l$
(4) $w = l - \frac{p}{2}$

Market Street Pizza kept a record of pizza sales for the month of February. The results are shown in the table below.

Of all the pizzas sold in February, what percent were plain, deep-dish pizzas?

(1) 20%
(2) 30%
(3) 40%
(4) 50%

When solving $-2(3x - 5) = \frac{9}{2}x - 2$ for $x$, the solution is

(1) $\frac{8}{7}$
(2) $\frac{10}{11}$
(3) $-\frac{16}{21}$
(4) $-\frac{16}{3}$

The expression $x^{2a + b}$ is equivalent to

(1) $x^{2a} + x^{b}$
(2) $x^{a} + x^{a + b}$
(3) $x^{a} \cdot x^{a + b}$
(4) $x^{a + b} \cdot x^{a + b}$

The inputs and outputs of a function are shown in the table below.

This function can best be described as

(1) linear
(2) quadratic
(3) exponential
(4) absolute value

Stephanie is solving the equation $x^2 - 12 = 7x - 8$. Her first step is shown below.

Given: $x^2 - 12 = 7x - 8$
Step 1: $x^2 - 4 = 7x$

Which property justifies her first step?

(1) associative property
(2) commutative property
(3) distributive property
(4) addition property of equality

What is the sum of $8\sqrt{3}$ and $\sqrt{3}$?

(1) $8\sqrt{6}$
(2) $9\sqrt{6}$
(3) $7\sqrt{3}$
(4) $9\sqrt{3}$

The dot plots below represent test scores for 20 students on a math test.

Image Description: Four dot plots labeled I, II, III, and IV are shown. Each dot plot has a horizontal axis ranging from 60 to 100 in increments of 5, representing test scores. Each dot represents one student's score.

Dot Plot I: 65(1), 70(1), 75(1), 80(6), 85(2), 90(5), 95(3), 100(1)

Dot Plot II: 60(1), 65(1), 70(4), 75(2), 80(3), 85(6), 95(2), 100(1)

Dot Plot III: 60(1), 65(2), 70(2), 75(2), 80(6), 85(2), 90(2), 95(2), 100(1)

Dot Plot IV: 70(3), 75(3), 80(4), 85(6), 90(4)

The mode for this math test is 80 and the median is 85. Which dot plot correctly represents this data?

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

A function is graphed on the set of axes below.

Image Description: A coordinate plane is shown with the x-axis and f(x)-axis. A curve begins at an open circle at approximately $(-2, -4)$ and increases, passing through the origin area and continuing to the right. The curve resembles a square root function that has been shifted. The open circle indicates the point is not included in the function.

The domain of this function is

(1) $\{x | x > -2\}$
(2) $\{x | x \geq -2\}$
(3) $\{x | x > -4\}$
(4) $\{x | x \geq -4\}$

Which ordered pair is a solution to the equation $y - 1 = 2\left(x + \frac{1}{4}\right)$?

(1) $(0.75, 0)$
(2) $(1.25, 4)$
(3) $(2.5, -6.5)$
(4) $(4, -9.5)$

Elena's fastest time for the 50-meter dash is 7 seconds. She wants to know how fast this is in inches per minute. Which expression can Elena use for a correct conversion?

(1) $\frac{7 \text{ sec}}{50 \text{ meters}} \cdot \frac{60 \text{ sec}}{1 \text{ min}} \cdot \frac{1 \text{ meter}}{39.37 \text{ in}}$
(2) $\frac{7 \text{ sec}}{50 \text{ meters}} \cdot \frac{1 \text{ min}}{60 \text{ sec}} \cdot \frac{39.37 \text{ in}}{1 \text{ meter}}$
(3) $\frac{50 \text{ meters}}{7 \text{ sec}} \cdot \frac{60 \text{ sec}}{1 \text{ min}} \cdot \frac{1 \text{ meter}}{39.37 \text{ in}}$
(4) $\frac{50 \text{ meters}}{7 \text{ sec}} \cdot \frac{60 \text{ sec}}{1 \text{ min}} \cdot \frac{39.37 \text{ in}}{1 \text{ meter}}$

The table below shows the highest temperatures recorded in August for several years in one town.

The interquartile range of these data is

(1) 7
(2) 10
(3) 11
(4) 18

The function $f(x) = x^2$ is multiplied by $k$, where $k < -1$. Which graph could represent $g(x) = kf(x)$?

Image Description: Four graphs of parabolas on coordinate grids are shown, each with vertex at the origin.

• (1) An upward-opening parabola that is wider than $x^2$.
• (2) A downward-opening parabola that is wider than $x^2$.
• (3) An upward-opening parabola that is narrower than $x^2$.
• (4) A downward-opening parabola that is narrower than $x^2$.

Which graph is the solution to the inequality $6.4 - 4x \geq -2.8$?

Image Description: Four number lines are shown, each marked from 2.1 to 2.5.

• (1) Open circle at 2.3 with an arrow pointing to the right.
• (2) Closed circle at 2.3 with an arrow pointing to the right.
• (3) Open circle at 2.3 with an arrow pointing to the left.
• (4) Closed circle at 2.3 with an arrow pointing to the left.

The number of fish in a pond is eight more than the number of frogs. The total number of fish and frogs in the pond is at least 20. If $x$ represents the number of frogs, which inequality can be used to represent this situation?

(1) $x + 8x \geq 20$
(2) $2x + 8 \geq 20$
(3) $x + 8x \leq 20$
(4) $2x + 8 \leq 20$

Which graph below represents a function that is always decreasing over the entire interval $-3 < x < 3$?

Image Description: Four graphs are shown on coordinate grids.

• (1) A piecewise linear function that increases to a maximum at $x = 5$, then decreases with a slope of $-1$ after $x = 5$.
• (2) A downward-opening parabola with vertex at $(0, 2)$, crossing the x-axis near $x = -2$ and $x = 2$.
• (3) A square root function starting at $(-3, 0)$ and increasing to the upper right.
• (4) A piecewise linear function. The first piece is a line with slope $-\frac{1}{2}$ and y-intercept $3$, with an open circle at $x = 4$. The second piece starts at $x = 4$ with an x-intercept at $(4, 0)$ and has a slope of $-2$.

The graph below models Sally's drive to the store.

Image Description: A coordinate plane graph with the x-axis labeled "Time (in minutes)" ranging from 0 to 10, and the y-axis labeled "Speed (miles per hour)" ranging from 0 to 50. The graph shows a piecewise linear function with line segments connecting the following points: $(0, 0)$ to $(2, 10)$, $(2, 10)$ to $(5, 35)$, $(5, 35)$ to $(9, 35)$, and $(9, 35)$ to $(10, 0)$.

State an interval when Sally is traveling at a constant speed.

Explain your reasoning.

Graph the function $f(x) = x^2 + 4x + 3$.

Image Description: A coordinate plane with x-axis and f(x)-axis, with gridlines. The axes are labeled x and f(x).

State the equation of the axis of symmetry of $f(x)$.

The function $f(x)$ is shown in the table below.

State an appropriate value for $m$ in the table, so that $f(x)$ remains a function.

Explain your reasoning.

Solve $x^2 + 8x = 33$ for $x$ by completing the square.

If $f(x) = \frac{-3x - 5}{2}$, algebraically determine the value of $x$ when $f(x) = -22$.

Rationalize the denominator of the fraction below. Express the solution in simplest form.

$$\frac{4}{\sqrt{2}}$$

Alex had $1.70 in nickels and dimes on his desk. There were 25 coins in all.

Write a system of equations that could be used to determine both the number of nickels, $n$, and the number of dimes, $d$, that Alex had.

Use your system of equations to algebraically determine both the number of nickels and the number of dimes that he had.

The table below shows the average heart rate, $x$, and Calories burned, $y$, for seven men on an Olympic rowing team during a one-hour workout class.

Write the linear regression equation that models these data, rounding all values to the nearest tenth.

State the correlation coefficient, rounded to the nearest tenth.

State what the correlation coefficient suggests about the linear fit of these data.

Using the quadratic formula, solve $x^2 + 4x - 3 = 0$.

Express your solution in simplest radical form.

Solve the following system of equations algebraically for all values of $x$ and $y$.

$$y = x^2 - 7x + 12$$

$$y = 2x - 6$$

Anna plans to spend $30 on balloons and party hats for her daughter's birthday party. Including tax, balloons cost $2 each and party hats cost $1.50 each. The number of party hats Anna needs is twice as many as the number of balloons.

If $x$ represents the number of balloons and $y$ represents the number of party hats, write a system of equations that can be used to represent this situation.

Use your system of equations to algebraically determine the number of balloons and the number of party hats Anna can buy.