This page shows questions in the Algebra I 2024 public release module at MSDE. Algebra 1
"Algebra I 2024"

Function $f$ is defined as $f\left(x\right)=x^2-6x+14\text{.}$

What is the minimum value of $f\left(x\right)\text{?}$

What value of $x$ is the solution of the equation $\frac{3x-7}{5}=x+1$ ?

Enter your answer in the space provided.

$x=$

The function f \( f \) is defined by f of x equals 7 fifths x minus 7. \( f(x) = \frac{7}{5}x - 7. \)

Graph y equals f of x \( y = f(x) \) on the coordinate plane shown.

To graph the line, select two points on the coordinate plane. A line will be drawn through the points.

Which of the following expressions can be written as $(ax+b)(ax-b)$ , where $a$ and $b$ are integers?

Select all that apply.

Let $r$ , $s$ , and $w$ represent three distinct numbers.

• $r$ is a nonzero rational number.

• $s$ is a nonzero rational number.

• $w$ is an irrational number.

For each expression in the following table, indicate whether the value of the expression must be rational or irrational.

Select one box per row.

The equation $x^2-8x-5=0$ can be transformed into the equation ${\left(x-p\right)}^2=q,$ where $p$ and $q$ are real numbers.

What are the values of $p$ and $q?$

Enter your answers in the spaces provided.

$p=$

$q=$

Use the given information to answer the question.

Mr. Carnegie will buy pencils and pens for his students for the school year. Use the following information to determine the number of pencils and the number of pens he can buy.

• He can spend no more than $30 on pencils and pens.

• Pencils cost $0.15 each, and pens cost $0.75 each.

• He needs to buy at least 3 times as many pencils as pens.

Complete the system of inequalities that can be used to determine the number of pencils, $x$ , and the number of pens, $y$ , that Mr. Carnegie can buy.

Move a number into each box.

x plus \( x \,\, + \, \) y is less than or equal to 30 \( y \, \le 30 \)

y less than or equal to \( y \,\, \le \) x \( x \)

Function f \( f \) is defined by f uv x equals, open parenthesis, x minus 5, close parenthesis, squared, plus 1. \( f(x) = (x - 5)^2 + 1. \) Function g \( g \) is defined by g uv x equals, f uv x, minus 2. \( g(x) = f(x) - 2. \)

Which quadrants in the xy-plane contain points on the graph of y equals, g uv x? \( y = g(x)? \)

Select all that apply.

Samantha has $\$35$ in her savings account. At the end of each week, she will add $\$20$ to the account.

Which equation describes the total amount $S$ , in dollars, that Samantha will have in her account at the end of week $w$ ?

Select one answer.

A system of equations is given.

$\left\{\begin{array}{l}y=9x-1\\ y=2x+3\end{array}\right.$

What is the value of $x$ in the solution $(x,y)$ to the system?

Enter your answer as a fraction in the spaces provided.

fraction bar

$$ \rule[0.4cm]{3.5cm}{0.5mm} $$

A student is saving money in a bank account to buy a computer. The function $f\left(w\right)=35w+10$ represents the total amount of money, $f\text{,}$ in dollars, in the student’s account $w$ weeks after the student started saving for the computer.

What does the number $10$ in the function represent in this context?

Two quadratic relationships, J and K, are represented in the following table and graph, respectively.

In the following table, indicate which of the relationships have each feature.

Move a phrase into each box.

The expression ${2x+(x-7)}^2$ is equivalent to $x^2+bx+49$ for all values of $x$ .

What is the value of $b$ ?

Enter your answer in the space provided.

$b=$

A collection of data is shown in the scatterplot.

Which value is the best estimate of the correlation coefficient of a line of best fit for the data?

A sequence is defined by $a_n=-5+(n-1)14\text{,}$ where $n$ is a positive integer.

What is the value of $a_{10}\text{?}$

Enter your answer in the space provided.

$a_{10}=$

Consider the equation ${(x-a)}^2-b=0\text{,}$ where $a$ and $b$ are both positive real numbers.

Which statement about the solution or solutions to the equation must be true?

An office manager is ordering this year’s supply of pencils and pens for the office. Each pencil costs $\$0.10\text{,}$ each pen costs $\$0.75\text{,}$ and the manager will spend a total of $\$235$ on pencils and pens.

Which additional piece of information is sufficient to determine the number of pencils and the number of pens that the manager will order?

The system of equations shown is graphed in the xy-plane.

$$\{\begin{array}{l}x+y=1\\ 2x-3y=17\end{array}$$

Determine the coordinates of the solutions of the system algebraically. Then show that your solution is valid. Show your work or explain how you found your answer.

Enter your answer and your work or explanation in the space provided. You may also use the drawing tool to help explain or support your answer.

Drawing Box

The table shows the distance, in miles, that an athlete will run each week for $6$ weeks to prepare for a half marathon.

The equation $y=3.25x+8.5$ models the distance, $y\text{,}$ in miles, that the athlete will run in week $x\text{.}$

For which weeks is the number of miles given by the equation greater than the actual number of miles the athlete will run?

Select all that apply.

The function $f$ is defined by $f\left(x\right)={(x-5)}^2-20$ for all real numbers $x\text{.}$

What is the range of $f\text{?}$

Select from the drop-down menus to correctly complete the sentence.

Two functions, $f$ and $g\text{,}$ are graphed in the xy-plane shown.

In which intervals do the solutions of the equation $f\left(x\right)=g\left(x\right)$ exist?

Select all that apply.

A student read a 303-page book in 7 days. The table shows the total number of pages the student had read at the end of each day for the 7 days.

What was the average number of pages the student read per day from the end of day $2$ to the end of day $6\text{?}$

John has to type a 2,000-word business research report, and he types at a rate of $40$ words per minute. The function $R\left(m\right)=\left(\frac{2\text{,}000}{40}\right)-m$ can be used to calculate the number of minutes required to finish after $m$ minutes of typing.

What is the domain for function $R$ in this context?

Select one answer.

The owner of a company bought 150 tickets for a circus performance. The owner paid $10 for each adult ticket and $6 for each child ticket.

Write an equation to represent y, the total amount, in dollars, the owner paid for the tickets in terms of x, the number of adult tickets the owner bought.

Enter your equation in the space provided. Enter only your equation.

The quadratic function $f$ is defined as $f\left(x\right)=-{(x+2)}^2+3\text{.}$

The quadratic functions g \( g \) and h \( h \) are transformations of the function f. \( f. \)

Function g \( g \) is defined as g of x equals f of x plus k, \( g(x)=f(x)+k, \) where k \( k \) is a constant.

Function h \( h \) is defined as h of x equals f of open parenthesis x plus n close parenthesis, \( h(x)=f(x+n), \) where n \( n \) is a constant.

• Determine whether there are values of k \( k \) for which the graph of y equals g of x \( y=g(x) \) has no x-intercepts. Justify your answer.

• Determine whether there are values of n \( n \) for which the graph of y equals h of x \( y=h(x) \) has no x-intercepts. Justify your answer.

Enter your answers and your justifications in the space provided. You may also use the drawing tool to help explain or support your answer.

Drawing Box

A system of inequalities is given, where $a$ and $b$ are positive integers.

$$\{\begin{array}{l}y<ax-b\\ y>-ax+b\end{array}$$

Which region in the xy-plane shown contains solutions of the system?

Select one region.

For each one-year period after a car was purchased, its value at the end of the year was $15\%$ less than its value at the beginning of the year.

Part A

State whether the value of the car as a function of time after it was purchased is best modeled with a linear function, a quadratic function, or an exponential function, and explain why.

Enter your answer and your work or explanation in the space provided. You may also use the drawing tool to help explain or support your answer.

Part B

If the value of the car $2$ years after it was purchased is $\text{\$}17\text{,918}$ , what was the value of the car when it was purchased? Show your work or explain your answer.

Enter your answer and your work or explanation in the space provided. You may also use the drawing tool to help explain or support your answer.

Drawing Box

A ball is hit from the ground. When the ball has traveled a horizontal distance of $d$ meters, its height, $h\text{,}$ in meters, can be modeled by the function $h\left(d\right)=-\frac{1}{125}{d}^2+d\text{.}$

What is the horizontal distance from the point where the ball is hit to the point where the ball lands on the ground?

Enter your answer in the space provided.

meters

The functions $f$ and $g$ are defined as follows.

• $f\left(x\right)=-x+2$

• $g\left(x\right)=x^2-1$

The point $(b,c)$ is on both the graph of $f$ and the graph of $g$ in the xy-plane.

Which of the following inequalities must be true?

Select one answer.

The librarian at a local school collected data on the number of students using the library after school each week during the first $22$ weeks of school. The librarian counted the number of students in the library each day and then calculated the total for each week. The weekly totals were plotted in a graph and then a trend line was fit to the data.

Students in the Library

For which week does the trend line overpredict the number of students using the library by the greatest amount, and for which week does the trend line underpredict the number of students using the library by the greatest amount?

Select from the drop-down menus to correctly complete each sentence.

An object is dropped off the roof of a building. The function $h\left(x\right)=-16x^2+64$ models the height, $h\left(x\right)\text{,}$ in feet, of the object above the ground $x$ seconds after it is dropped.

What is the best description of the meaning of $h\left(2\right)=0$ in terms of the context ?

A student claims that if an equation has no exponents greater than $1\text{,}$ then the equation is linear.

Which equation can be used to show that the student’s claim is false ?

A nursing student works at a doctor’s office for $\$15$ per hour and tutors other students for $\$25$ per hour.

The student cannot work more than $20$ hours each week.

The student wants to earn at least $\$375$ each week.

• Define two variables and write a system of inequalities that represents the given constraints.

• Suppose the student wants to work at the doctor’s office for as many hours as possible for the experience, and suppose the student can only work a whole number of hours at the doctor’s office. How many hours should the student work at the doctor’s office each week? Show your work or explain how you found your answer.

Enter your answers and your work or explanation in the space provided. You may also use the drawing tool to help explain or support your answer.

Drawing Box

The quadratic function $f$ is graphed in the xy-plane shown.

The function will be transformed to form a second function, $g\text{.}$

For each transformation of $f\text{,}$ indicate whether the equation $f\left(x\right)=g\left(x\right)$ has no solutions, one solution, or two solutions.

Select the boxes to show the number of solutions to each equation. Select only one box per row.

What are the two solutions of the equation $x^2+14x-32=0$ ?

Enter your two answers in the spaces provided.

$x=$

$x=$

A person deposited $10,000 into a bank account and made no other withdrawals or deposits. Each year, interest was added to the account. The table shows the account balance, $f\left(x\right)\text{,}$ in dollars, $x$ years after the person deposited the money for selected values of $x\text{.}$

Which function best represents this situation?