By Michael Avidon, math editor
A good part of the work we do here is to write and edit items (questions) for high-stakes statewide exams, and one of which is for Algebra. Those items cannot be released to the public, so the items here are not from actual exams, but are written in a similar style and align with Common Core standards. Solutions, and additional useful information, are found in the last section.
Types of Items
In the past, all items on standardized tests were multiple choice, and many of the items in these documents will be in that format.
With the advent of online exams, other formats are also used. Some of these are known as technology enhanced items (TEIs). Such items cannot be properly presented in a document, but some are presented in a mock format. These are described below.
Multiple select/multiple response: appears to have the same format as multiple choice (with 4 or more options), but has more than one correct answer. The test-taker is asked to select all of these. Sometimes these items state how many options to select and sometimes they do not.
Multiple choice drop-down/Inline choice: will have any number of drop-down menus in the middle of sentences, equations, labeling a diagram, etc. The test-taker must choose one option from each menu to correctly complete the sentences, equations, etc.
In this document, the existence of a drop-down menu is indicated with braces:
Start of sentence {option 1/option 2/option 3} end of sentence.
Drag and drop: have blank spaces or boxes in sentences, equations, etc. in which are placed correct options (e.g., numbers, words) that are found elsewhere (the bottom or side). The test-taker points to options, clicks on them, and drags them across the screen, dropping them into the appropriate spaces. In this document, underlined blank spaces will generally indicate drop zones and a labeled list of options will be presented at the bottom.
Fill in the blank: The test-taker usually has to provide a whole number or decimal.
Constructed response: requires a complete, hand-written solution.
One type of TEI that you will not find here is a hot spot item. In such an item, a student must point and click to indicate a correct answer (e.g., select a point on a number line).
MORE IN THIS SERIES: How to prepare for Grade 8 state assessment tests | Grade 7 tests | Grade 6 tests | Print out this lesson
Sample Test Items
Standard: A-APR.3
Item Type: Multiple choice drop-down
- Select from the drop-down menus to correctly complete the sentence.
The zeros of the function f(x) = x^{2} + 5x – 24
are {–24, –8, –6, –5, –4, –3} and {3, 4, 5, 6, 6, 24}.
Standard: A-CED.1
Item Type: Multiple choice
- A ball is shot into the air from the ground with an initial velocity of 32 feet per second. The height, in feet, of the ball can be modeled by the equation h(t) = 32t – 16t^{2}, where t is the time in seconds after the ball is thrown.
Which inequality could be used to find the range of time when the ball is at least 10 feet above the ground?
- a. 16t^{2} + 32t + 10 ≥ 0
- b. 16t^{2} – 32t + 10 ≥ 0
- c. –16t^{2} – 32t – 10 ≥ 0
- d. –16t^{2} + 32t – 10 ≥ 0
Standard: A-CED.3, A.REI.6
Item Type: Constructed response
- For a school trip, the teacher bought 138 snacks for a total of $540. Some of the snacks are chocolate bars, which cost $3 each, and the rest are trail mix packages, which cost $5 each. What is the difference between the number of chocolate bars and the number of trail mix packages that the teacher bought? Show your work.
Standard: A-REI.4
Item Type: Constructed response
- An equation is shown: 16x^{2} – 8x – 3 = 0
- a. Factor the left side of the equation into a product of two binomials.
- b. Find the roots of the equation. Show or explain how you got your answer.
Robin wants to investigate how the roots of similar equations depend on the constant term. So he writes this equation: 16x^{2} – 8x – 3 = 0
- c. Find the value of c for which the equation will have exactly one root. Show or explain how you got your answer.
Standard: A-REI.7
Item Type: Multiple select
- The line y = mx + 10 is tangent to the graph of y = 1 – x^{2}.
What are the possible values of m? Select all that apply.
- -√44
- –6
- –3
- 3
- 6
- √44
Standard: A-REI.D
Item Type: Constructed response
- a. Marla has $10 to spend at the bead store. She needs to buy a total of at least 22 blue or green beads. Blue beads cost $0.50 each, and green beads cost $0.40 each. Write a system of inequalities to represent these conditions.
- b. Graph the solution to your system on the coordinate grid below.
- c. Explain what g = 10 and g = 20.5 and their corresponding b-values within the solution mean in the context of the problem.
Standard: A-SSE.1b
Item Type: Multiple choice
- George wants to know the time t, in seconds, when a ball thrown will reach a certain height. He uses the quadratic formula to solve the equation he set up and gets the following:
t = ^{-10 + √100 + 64(2)} ⁄ _{-32}
Based on this information, what can George conclude?
- The ball will be at that height at two different times.
- The ball will never be at that height.
- The ball will reach that height when t is between 0 and 1.
- The ball will reach that height when t is between 1 and 2.
Standard: A-SSE.3b
Item Type: Multiple select
- Which statements concerning the function f(x) = 4x^{2 }+ 16x + 5 are true?
Select all that apply.
- a. It can be rewritten as f(x) = (2x +4)^{2} – 11.
- b. It can be rewritten as f(x) = 4(x+2)^{2} + 1.
- c. Its minimum value occurs at x = 2.
- d. Its minimum value occurs at x = -2.
- e. Its minimum value is 1.
- f. Its minimum value is 11.
Standard: F-BF.1a
Item Type: Multiple choice
- A numeric sequence is shown below.
1, 3, 9, 19, 33, 51, …
Which explicit formula represents the sequence, where n represents the position of a number in the sequence?
- a. f(n) = 2n – 1
- b. f(n) = 2n + 1
- c. f(n) = 2n^{2} + 1
- d. f(n) = 2(n – 1)^{2} + 1
Standard: F-BF.2
Item Type: Constructed response
- The formula shown represents a sequence: f(n) = 5n – 2
- a. Write the first six terms in the sequence.
- b. Show that the sequence is arithmetic.
- c. Write a recursive formula to model the sequence.
Show or explain how you got your answer.
- d. Describe a scenario that can be modeled by the formula.
Standard: F-IF.5
Item Type: Multiple select
- Patrick is graphing a function that has a domain that includes both positive and negative numbers. Which of the following could be the label of the x-axis? Select all that apply.
- a. Temperature (°F)
- b. Length (inches)
- c. Amount Earned ($)
- d. Altitude (meters)
- e. Time from start (seconds)
Standard: F-IF.6
Item Type: Drag and drop
- Drag and drop values from below to complete the table showing the function’s rate of change for each given interval.
A function is defined below.
Interval | Average Rate of Change |
15 < x < 18 | |
–1 ≤ x ≤ 1 | |
–20 < x < –10 |
OPTIONS
2 –2 0 1/3 3 undefined
Standard: F-IF.7a
Item Type: Multiple choice drop-down
- Select from the drop-down menus to correctly complete the statements.
The parabolic graph of a function with x-intercepts 0 and 4 is shown.
The equation representing the function is {y = 16 – x^{2}/y = 4x + x^{2}/y = 4x – x^{2}}.
This function has a {maximum/minimum} value. This value is {0/2/4}.
Standard: F-LE.2, F-IF.3
Item Type: Multiple choice drop-down
- Select from the drop-down menus to correctly complete the sentence.
Anya creates the following sequence of numbers: 7, 13, 31, 85, 247, …
This sequence is a translated {linear/quadratic/exponential} function and the formula for the nth term is {3n + 4/3n + 7/2n^{2} + 5/3^{n} + 4/3^{n} + 7}.
Standard: F-LE.2
Item Type: Multiple select
- Two input-output pairs are given for a function below.
x | f(x) |
5 | 16 |
6 | 32 |
Which formulas could describe this function? Select all that apply.
- a. f(x) = x + 11
- b. f(x) = 16x
- c. f(x) = 16(x – 4)
- d. f(x) = 2^{x}^{–1}
- e. f(x) = 2^{x}
Standard: F-LE.5
Item Type: Multiple choice
- The number of grams of a chemical compound is modeled by the function
b(t) = 1850(0.997)^{t}, where t equals the time in years after the measurement starts. According to the model, which statement is correct?
- a. The original amount of the compound was 1850(0.997) grams.
- b. The amount of the compound will eventually be 0 grams.
- c. The amount of the compound decreases by 0.003% each year.
- d. Each year, the amount of the compound is 99.7% of the previous year’s amount.
Standard: N-Q.3
Item Type: Multiple choice
- The actual length of a field is 300 feet long. Someone measured the field and reported that it is 309 feet long.
The formula for percent error is
Percent Error = ^{|actual – reported measurement|} ⁄ _{actual measurement} x 100%
Which is the best approximation for percent of error for this measurement?
- a. 0.03%
- b. 0.3%
- c. 3%
- d. 30%
Standard: N-RN.2
Item Type: Fill in the blank
- The expression √ x^{4}(x + y)*(x – y)*(x^{9})^{⅓ }is evaluated at y = 0 and then multiplied by another expression. The final result is x^{10}.
The expression that it was multiplied by is _____.
Standard: N-RN.3
Item Type: Multiple choice
- Which statement about the values of the function f(x) = Πx is true?
- a. The value of f(x) is irrational for all values of x.
- b. The value of f(x) is irrational only when x is an integer.
- c. The value of f(x) is rational when x is rational.
- d. The value of f(x) is irrational for all rational x other than 0.
Standard: S-ID.2
Item Type: Multiple choice
- A class took a 10-point quiz. The results were as follows:
- 1 student scored 5 points,
- 5 students scored 6 points,
- 3 students scored 7 points,
- 5 students scored 8 points, and
- 1 student scored 9 points.
What was the standard deviation of the scores?
- a. √4
- b. √1.2
- c. √1.5
- d. √2.0
Standard: S-ID.6, G-GPE.5
Item Type: Multiple choice
- An engineer measured the strength of metal bars and put the results on a scatterplot. The x-axis represented the thickness of each bar. The y-axis represented the force required to break each bar. A computer determined that the line of best fit is y = 0.2563x + 42.701. But the engineer discovered an error, and saw that the forces were all actually 1.906 greater than measured.
How can the engineer change the equation of the line of best fit without calculating a new one?
- a. Increase the slope by 1.906.
- b. Increase the slope by 0.2563 x 1.906, and the y-intercept by 1.906.
- c. Increase the y-intercept by 1.906.
- d. Increase the slope by 1.906, and the y-intercept by 1.906.
Standard: S-ID.6a
Item Type: Multiple choice
- The number of congressional seats in the U.S. House of Representatives for each state is based on the population of the state. The U.S. population in 2010 was 3.09 million. A partial list of states and the relevant data is shown below.
State | Population
(millions) |
Number of Seats in Congress |
Alabama | 4.8 | 7 |
Arizona | 6.4 | 9 |
Connecticut | 3.6 | 5 |
Maryland | 5.8 | 8 |
New Hampshire | 1.3 | 2 |
New Jersey | 8.8 | 12 |
Let x million equal the state population and y equal the number of seats allocated to that state.
Which equation best represents the table?
- a. y = 3.1(x – 2.5)
- b. y = 0.69x
- c. y = 1.3x + 0.4
- d. y = 1.46x
Standard: S-ID.8, S-ID.3
Item Type: Multiple choice drop-down
- Select from the drop-down menus to correctly complete the sentences.
A statistician acquires the following data points:
(0,0), (0.2, 0.3), (0.5, 0.7), (0.6, 1.8), (0.7, 0.8), (1,1)
The line of best fit for these points is y = 0.18 + 1.17x, with correlation coefficient 0.674.
The correlation is therefore a {weak/moderate/strong} positive relationship.
The point {(0.2, 0.3)/(0.5, 0.7)/(0.6, 1.8)/(0.7, 0.8)} is an outlier.
If the outlier is dropped and a new line of best fit is found from the remaining 5 points, the correlation coefficient will {increase/decrease}.
Solutions and Explanations
- Key: The zeros of the function f(x) = x^{2} + 5x – 24 are –8 and 3 .
Solution:
The zeros are the solutions to f(x) = 0. Now f(x) = (x + 8)(x – 3) = 0, which implies (x + 8) = 0 or (x – 3) = 0. These in turn imply x = -8 or x = 3.
Commentary (formulas, ideas, mistakes, misunderstandings, advice):
An idea that is used repeatedly in mathematics is to reduce a new or complicated problem into older, simpler problems. Finding the zeros of this function means solving a quadratic equation. One way to do that is to use factoring to reduce this problem to solving two linear equations. The reason that is valid is the Zero Product Property: if A x B = 0, then A = 0 or B = 0. In particular, this is used with A and B equal to the linear factors of the quadratic expression.
- A ball is shot into the air from the ground with an initial velocity of 32 feet per second. The height, in feet, of the ball can be modeled by the equation h(t) = 32t – 16t^{2}, where t is the time in seconds after the ball is thrown.
Which inequality could be used to find the range of time when the ball is at least 10 feet above the ground?
Key: D
Solution:
The question asks for the time when the height h(t) > 10 feet, or when 32t – 16t^{2} > 10. Subtract 10 from each side of the inequality to get option D.
Commentary (formulas, ideas, mistakes, misunderstandings, advice):
The 32 feet per second is actually incorporated into the formula (the coefficient of t). In physics class, you would learn that the height of an object in feet, operating under the force of gravity, after t seconds is h(t) = h_{0} + v_{0}t – 16t^{2}, where h_{0} is the initial height and v_{0} is the initial velocity. In calculus class, you would derive this formula.
- For a school trip, the teacher bought 138 snacks for a total of $540. Some of the snacks are chocolate bars, which cost $3 each, and the rest are trail mix packages, which cost $5 each. What is the difference between the number of chocolate bars and the number of trail mix packages that the teacher bought?
Key: 12
Solution:
Let c = the number of chocolate bars and t = the number of trail mix packages.
Then the total and the money conditions yield: c + t = 138
3c + 5t = 540
Multiply the first equation by 3: 3c + 3t = 414
Subtract from the second: 2t = 126 → t = 63
Substitute into the first equation: c + 63 = 138 → c = 75
The difference is 12.
Commentary (formulas, ideas, mistakes, misunderstandings, advice):
The information in the problem is represented by two linear equations in two variables. Generally, such a system will have a unique solution, though it is possible that it could have no solutions (e.g., x + y = 0 & x + y = 1) or an infinite number of solutions (e.g., y = x & 2y = 2x). The “elimination” method was used in this solution (adding or subtracting equations to cancel one of the variables). “Substitution” is another method (e.g., Rewrite the first equation as c = 138 – t, and then substitute 138 – t for c in the second equation. Solve that new equation for t.)
- An equation is shown: 16x^{2} – 8x -3 = 0
- a. Factor the left side of the equation into a product of two binomials.
- b. Find the roots of the equation. Show or explain how you got your answer.
Robin wants to investigate how the roots of similar equations depend on the constant term. So he writes this equation: 16x^{2} – 8x + c = 0
- c. Find the value of c for which the equation will have exactly one root. Show or explain how you got your answer.
Key/Solution:
- a. 16x^{2} – 8x – 3 = (4x + 1)(4x – 3)
- b. (4x + 1)(4x – 3) = 0
4x + 1 = 0
4x = -1
x = -¼
or 4x – 3 = 0
4x = 3
x = ¾ - 1; 16x^{2} – 8x + 1 =(4x – 1)^{2} = 0 → 4x – 1 = 0 → x = ¼ only
Or: To get exactly one solution to a quadratic equation, the discriminant must be 0. So
b^{2} – 4ac = (-8)^{2} – 4 x 16c = 0 → 64 – 64c = 0 → c = 1
Commentary (formulas, ideas, mistakes, misunderstandings, advice):
See the commentary for item #1 about solving a quadratic equation by factoring. Another way to solve ax^{2} + bx + c = 0,is by using the quadratic formula: x = ^{-b + √b2 – 4ac} ⁄ _{2a} .
The part under the radical sign, b^{2} – 4ac, is called the discriminant. If this is positive, then it has two square roots, and the equation will have two solutions. If this is 0, then it has one square root, and the equation will have one solution. If it is negative, then it has no square roots, and the equation will have no solutions.
- The line y = mx + 10 is tangent to the graph of y = 1 – x^{2}.
What are the possible values of m?
Key: B, E
Solution:
There is only one solution to the system of equations. Using substitution yields mx + 10 = 1 – x^{2} → x^{2} + mx + 9 = 0. In order to have a unique solution, the discriminant must be 0. So m^{2} – 36 = 0 → m = +6.
Commentary (formulas, ideas, mistakes, misunderstandings, advice):
See the commentary for the previous two items regarding the substitution method, the quadratic formula, and the discriminant.
If you graph the two equations, you get a parabola (a curve) and a line. A line might intersect a parabola in 0, 1, or 2 points (try drawing the different cases). Saying that the line is tangent to the curve is saying that they intersect in 1 point. Intersection points are solutions to the system, so this system has one solution.
6.
- a. Marla has $10 to spend at the bead store. She needs to buy a total of at least 22 blue or green beads. Blue beads cost $0.50 each, and green beads cost $0.40 each. Write a system of inequalities to represent these conditions.
- b. Graph the solution to your system on the coordinate grid.
- c. Explain what g = 10 and g = 20.5 and their corresponding b-values within the solution mean in the context of the problem.
Key/Solution:
- a. Let b represent the number of blue beads and g represent the number of green beads. Then b > 0, g > 0, b + g > 22, 0.5b + 0.4g >< 10.
- b. The thin triangle shown below on the lower right is the solution set.
- c. There is one point where g = 10: the intersection of the two lines (where b = 12). This represents purchasing 10 green and 12 blue beads (22 total) for a cost of 0.5(12) + 0.4(10) = 6 + 4 + 10 dollars. There is a set of points where g = 20.5 (1.5 < b < 3.6), but this does not correspond to a real-world solution, as the numbers of beads must be whole numbers.
Commentary (formulas, ideas, mistakes, misunderstandings, advice):
The graph of a linear equality is a half-plane. You graph the line first, and then have to determine if the points above or below the line are the solutions. You include the line if the inequality uses < or >. If you have a system with two or more linear inequalities, then the solution is the intersection of all of the half-planes.
Remember that a mathematical solution to an inequality or an equation may not be a real-world solution for a problem in a given context. In this problem, only points with whole number coordinates are solutions. In some other problem, with weight as a variable, say, you could have any positive number as a solution, but not a negative number.
- George wants to know the time t, in seconds, when a ball thrown will reach a certain height. He uses the quadratic formula to solve the equation he set up and gets the following:
t = ^{-10 + √100+ 64(-2)} ⁄ _{-32}
Based on this information, what can George conclude?
Key: B
Solution:
The discriminant is 100 – 128 = -28 < 0. Therefore, the quadratic equation from whence it came has no real solutions. In terms of the context, this means that the ball will never be at that height (George considered a height that is greater than the maximum height).
Commentary (formulas, ideas, mistakes, misunderstandings, advice):
See the commentary for item #4.
- Which statements concerning the function f(x) = 4x^{2} + 16x + 5 are true?
Key: A, D
Solution:
f(x) = [(2x)^{2} + 2(4)(2x + 4^{2}] – 4^{2} + 5 = (2x +4)^{2} -11
The graph of this function is an upturned parabola.
The minimum occurs at the vertex, or when 2x + 4 = 0 → x = -2. At that point, f(x) = -11.
Commentary (formulas, ideas, mistakes, misunderstandings, advice):
The first line in the solution is the process of “completing the square.” This transforms the quadratic function from standard form to vertex form. Strictly speaking, it would be written like this: f(x) = 4 (x + 2)^{2} – 11, or in general, f(x) = a(x – h)^{2} + k. Written this way, it tells you that the vertex is (h ,k). If a < 0, it is a downturned parabola, so the vertex is at the top, and the function has a maximum value there.
- A numeric sequence is shown below.
1, 3, 9, 19, 33, 51, …
Which explicit formula represents the sequence, where n represents the position of a number in the sequence?
Key: D
Solution:
The differences between terms are 2, 6, 10, 14, 18. Because they are not constant, the sequence is not linear (cannot be A or B). The numbers are 1 more than 0, 2, 8, 18, 32, 50 or 1 more than 2(0), 2(1), 2(4), 2(9), 2(16), 2(25). So the numbers are each twice a square plus 1, which describes C and D. But the when n = 1 you obtain 3 and 1 for the first term in C and D, respectively, so it is D that exactly matches the sequence.
Commentary (formulas, ideas, mistakes, misunderstandings, advice):
If you start with a sequence and create a new one by taking the difference between each term and the previous term, this is called the first difference sequence (e.g., 2, 6, 10, 14, 18 above). If the first difference sequence is (a nonzero) constant, then the original sequence can be described by a linear function. If it is not constant, you can compute differences between its terms to create the second difference sequence (e.g., 4, 4, 4, 4, in the example above). If the second difference sequence is constant, then the original sequence can be described by a quadratic (2^{nd} degree) function. This was the case in this item. If the third difference sequence is constant, then the original sequence can be described by a cubic (3^{rd} degree) function.
- The formula shown represents a sequence: f(n) = 5n – 2
- Write the first six terms in the sequence.
- Show that the sequence is arithmetic.
- Write a recursive formula to model the sequence. Show or explain how you got your answer.
- Describe a scenario that can be modeled by the formula.
Key/Solution:
- a. 3, 8, 13, 18, 23, 28
- b. The difference of consecutive terms is the constant 5, so the sequence is arithmetic.
- c. A recursive formula for the sequence is f(n) = f(n – 1) + 5, f(1) = 3. I used the common difference between terms in the sequence from part b to find the operation to perform on each term to get the next term.
- d. Joe put $3 into his piggy bank in Week 1. Then he added $5 every week after that.
Commentary (formulas, ideas, mistakes, misunderstandings, advice):
An arithmetic sequence has constant differences between consecutive terms. This can be represented by a linear function. A geometric sequence has constant ratios between consecutive terms. This can be represented by an exponential function.
In general, a recursive formula is a formula that tells you how to find the value of one term of a sequence from the previous term (or terms). This is contrasted with an explicit formula (as given in the item), which directly allows you to compute the value.
- Patrick is graphing a function that has a domain that includes both positive and negative numbers. Which of the following could be the label of the x-axis? Select all that apply.
Key: A, D
Solution:
The x-axis shows the input, or the domain of a function. Temperature could be degrees above and below 0°. Altitude can be meters above or below sea level (0 meters). These numbers can be positive or negative. On the other hand, length of an object, amount of money earned, and the time from the start of something must be positive or 0.
Commentary (formulas, ideas, mistakes, misunderstandings, advice):
The domain of a function may be restricted by real-world considerations, as in this item, or by algebraic considerations. For example, the domain of f(x) = √x is x > 0, as you cannot take the square root of a negative number. The range, or output of a function, is subject to similar restrictions.
- A function is defined below.
Key:
Interval | Average Rate of Change |
15 < x < 18 | 0 |
–1 ≤ x ≤ 1 | 3 |
–20 < x < –10 | –2 |
Solution:
The first interval in the table is a subset of 1 < x, so corresponds to the third formula; the second interval corresponds to the second formula; the third interval corresponds to the first formula.
On the first interval in the table, the function is constantly 5, so the rate of change is 0.
Rate of change on –1 ≤ x ≤ 1 is ^{[½ (1 + 3)2 + 1] – [½(-1+3)2 + 1]} ⁄ _{1 – (-1)} = ^{9 – 3} ⁄ _{2} = 3.
On the third interval, the function is a ray with slope –2. Or, using the formula:
^{2|-10 + 3| – 1 – (2|-20 + 3| – 1)} ⁄ _{-10 – (-20)} = ^{2(7) – 1 -(2(17) – 1)} ⁄ _{-10 + 20} = ^{13 – (33)} ⁄ _{10} = ^{-20} ⁄ _{10} = -2
Commentary (formulas, ideas, mistakes, misunderstandings, advice):
The rate of change of a function y = f(x) over the interval a < x < b is defined to be ^{f(b) – f(a)} ⁄ _{b – a} . If the function is linear, then the rate of change equals the slope of its graph, regardless of which interval is chosen. If the function is not linear, then it is the slope of the “secant” line through the points (a, f(a)) and (b, f(b)) on the graph of the function. This slope will be different for different intervals. If the input is time, then this slope tells you how much the output changes during the given interval of time, or what one typically calls a rate of change.
- The parabolic graph of a function with x-intercepts 0 and 4 is shown.
Key: The equation representing the function is y = 4x – x^{2}.
This function has a maximum value. This value is 4 .
Solution:
y = a(x -0)(x – 4) and the graph passes through (2, 4).
Therefore, 4 = a(2)(-2) → a = -1. Hence, y = -x(x – 4).
The maximum occurs at the vertex, (2, 4), where the y-value (maximum) is 4.
Commentary (formulas, ideas, mistakes, misunderstandings, advice):
The graph of a function is a parabola if and only if the function is quadratic (y = ax^{2} + bx + c). The x-intercepts, or zeros of the function, are the solutions to the equation ax^{2} + bx + c = 0. If there are two x-intercepts, as with this graph, the function can be written with two linear factors. If there is one x-intercept (the graph is tangent to the x-axis), the function can be written with one linear factor raised to the power 2. If there are no x-intercepts, the function cannot be factored.
- Anya creates the following sequence of numbers: 7, 13, 31, 85, 247, …
Key: This sequence is a translated exponential function
and the formula for the nth term is 3^{n} + 4.
Solution:
The first differences are 6, 18, 54, and 162. They are not constant, so the sequence is not linear.
The second differences are 12, 36, and 108. They are not constant, so the sequence is not quadratic.
The differences are tripling, so compare to powers of three: 3, 9, 27, 81, 243.
These are all 4 less than the terms, so the nth term is 3^{n} + 4.
Commentary (formulas, ideas, mistakes, misunderstandings, advice):
A geometric sequence (a, ar, ar^{2}, ar^{3}…) has constant ratios between consecutive terms and can be represented by an exponential function (ar^{n – 1}. This problem is about a slight variation on this idea.
- Two input-output pairs are given for a function below.
x | f(x) |
5 | 16 |
6 | 32 |
Which formulas could describe this function?
Key: C, D
Solution:
Two points uniquely determine a linear function. The slope would be (32–16)/(6–5) = 16, so the function might be B or C. Now f(5) = 16 and option C is the unique linear function satisfying these conditions.
Two points uniquely determine an exponential function. This function has f(5) = 2^{4} = 2^{5 – 1} and f(6) = 2^{5} = 2^{6 – 1}. This matches option D.
Commentary (formulas, ideas, mistakes, misunderstandings, advice):
Two points do not uniquely determine a quadratic function (y = ax^{2} + bx + c) — well, they do if you insist that one of them is the vertex. But generally you need three points to uniquely determine a quadratic, because it has three parameters, a, b, and c. On the other hand, a linear function has two parameters (slope and intercept), and an exponential function has two parameters (initial value and the growth factor).
- The number of grams of a chemical compound is modeled by the function b(t) = 1850(0.997)^{t}
b(t) = 1850(0.997)^{t}, where t equals the time in years after the measurement starts. According to the model, which statement is correct?
Key: D
Solution:
This is a model of exponential decay. It approaches 0, but does not reach it. (B is false).
The original amount is b(0) = 1850•1 grams. (A is false.)
Each year, the amount is multiplied by 0.997, or 99.7%. (D is true.)
This is a decrease of 0.3% per year. (C is false.)
Commentary (formulas, ideas, mistakes, misunderstandings, advice):
In general, an exponential function has the form y = a x b^{t}, where the constant a is the “initial value,” and b is the “growth factor.” You can write b = 1 + r, where r is called the “growth rate” (it could be positive or negative). In this problem, r = -0.003. A negative growth rate means the amount is decreasing (decay) and a positive growth rate means the amount is increasing (growth).
- The actual length of a field is 300 feet long. Someone measured the field and reported that it is 309 feet long.
The formula for percent error is
Percent Error = ^{|actual – reported measurement} ⁄ _{actual measurement} x 100%
Which is the best approximation for percent of error for this measurement?
Key: C
Solution:
^{309 – 300} ⁄ _{300} x 100% = ^{9} ⁄ _{3}% = 3%
Commentary (formulas, ideas, mistakes, misunderstandings, advice):
Be careful not to divide by the reported measurement.
It is often a good idea to check your work “in reverse.” An error of 3% means you are off by 3% of 300, or 0.03 x 300 = 9. That could be or 300 + 9 = 309 – 9 = 291.
- The expression √x^{4}(x + y) * (x – y) * (x^{9})^{⅓} is evaluated at y = 0 and then multiplied by another expression. The final result is x^{10}.
Key: The expression that it was multiplied by is x^{3}.
Solution:
The expression simplifies to √x^{4}(x) * (x) * (x^{9})^{⅓} = x^{4*½ } * x^{2} * x^{9*⅓} = x^{2+2+3} = x^{7}. Secondly, x^{7} * x^{3} = x^{10}, so x^{3}is the expression it is multiplied by.
Commentary (formulas, ideas, mistakes, misunderstandings, advice):
This uses a combination of exponent rules and definitions
- x^{a} * x^{b} = x^{a+b}
- x^{a} + x^{b} = x^{a-b}
- (x^{a})^{b} = x^{a*b}
- x^{1} = x
- √u = u^{½}
- Which statement about the values of the function f(x) = Πx is true?
Key: D
Solution:
The number Π is irrational. If you multiply it by any rational number, other than 0, you will obtain an irrational number. This is equivalent to option D. (Option C completely contradicts this. Option B covers part of this, so the word “only” makes it false. Option A goes beyond this: it includes x = 0 and 1/Π and other numbers, for which the value of the function is rational.)
Commentary (formulas, ideas, mistakes, misunderstandings, advice):
In general, the product of two rational numbers is rational, and the product of a nonzero rational and irrational number is irrational. The product of two irrational numbers might be rational or irrational, depending on the numbers.
- A class took a 10-point quiz. The results were as follows:
- 1 student scored 5 points,
- 5 students scored 6 points,
- 3 students scored 7 points,
- 5 students scored 8 points, and
- 1 student scored 9 points.
What was the standard deviation of the scores?
Key: B
Solution:
Because the distribution is symmetric, the mean score is 7, the one in the middle. The standard deviation equals √^{1(2)2 + 5(1)2 + 3(0)2 + 5(1)2 + 1(2)2} ⁄ _{15} = √^{18} ⁄ _{15} = √1.2.
Commentary (formulas, ideas, mistakes, misunderstandings, advice):
When you have a set of data, you can describe it in brief by giving the mean (average). However, two sets of data with the same mean can still look very different; for example, one may have numbers clustered close to the mean and another could have numbers that are spread out. For example: 5 students scored 4, 1 student scored 5, 6, 7, 8, 9, and 5 students scored 10. As in the item, this describes 15 students with a mean score of 7, but most of the data is 3 points away from the mean rather than 1 point away.
In plain English, the standard deviation is some sort of average measurement of how far the data points are from the mean.
- An engineer measured the strength of metal bars and put the results on a scatterplot. The x-axis represented the thickness of each bar. The y-axis represented the force required to break each bar. A computer determined that the line of best fit is y = 0.2563x + 42.701. But the engineer discovered an error, and saw that the forces were all actually 1.906 greater than measured.
How can the engineer change the equation of the line of best fit without calculating a new one?
Key: C
Solution:
The forces are the y-values. Each of these should be 1.906 greater, so adding this amount to 0.2563x + 42.701 will make the correction. That is the same as adding 1.906 to 42.701, which is the y-intercept.
Commentary (formulas, ideas, mistakes, misunderstandings, advice):
Geometrically, you have a set of points and a line running through them to approximate their pattern. The points are translated up by a fixed amount, and so is the line.
- The number of congressional seats in the U.S. House of Representatives for each state is based on the population of the state. The U.S. population in 2010 was 309 million. A partial list of states and the relevant data is shown below.
State | Population
(millions) |
Number of Seats in Congress |
Alabama | 4.8 | 7 |
Arizona | 6.4 | 9 |
Connecticut | 3.6 | 5 |
Maryland | 5.8 | 8 |
New Hampshire | 1.3 | 2 |
New Jersey | 8.8 | 12 |
Let x million equal the state population and y equal the number of seats allocated to that state. Which equation best represents the table?
Key: C
Solution:
It is easy to see that each y-value is between 1 and 2 times the corresponding x-value, or x < y < 2x. So options A and B can be eliminated. The value 1.3x is slightly less than each value, and option C is fairly close (or plotting the points shows this line goes through the data). Option D is a line that is above 5 of the 6 points, so is not as good a fit.
Commentary (formulas, ideas, mistakes, misunderstandings, advice):
In real terms, the coefficient 1.3 means that for each increase of 1 million in the population, the state gets another 1.3 seats (roughly 1 new rep for each 770,000 people).
- A statistician acquires the following data points: (0,0), (0.2, 0.3), (0.5, 0.7), (0.6, 1.8), (0.7, 0.8), (1,1)
The line of best fit for these points is y = 0.18 + 1.17x, with correlation coefficient 0.674.
Key: The correlation is therefore a moderate positive relationship.
The point (0.6, 1.8) is an outlier.
If the outlier is dropped and a new line of best fit is found from the remaining 5 points, the correlation coefficient will increase.
Solution:
The correlation coefficient is between –1 and 1. When it is close to either extreme, the correlation is strong. When it is close to 0, it is weak. The number 0.674 is in between, so this is a moderate relationship.
The line of best fit has y-values slightly higher than x-values. The only point that does not fit that pattern is (0.6, 1.8), where the y-value is triple the x-value, so it is the outlier.
When the outlier is dropped, the new line will have a better fit, so the correlation coefficient will be closer to 1. That is, it will increase.
Commentary (formulas, ideas, mistakes, misunderstandings, advice):
In middle school, you learned to find a so-called “line of best fit” by eyeballing the data. Now you are probably plugging the data into a graphing calculator or an online linear regression calculator. What is the calculator doing? It is finding a line such that the sum of the squares of the vertical distances from the points to the line is a minimum.
To understand what this means, do this: graph the points and the line. Now draw little vertical segments from each point to the line. Some go up and some go down; the difference in the y-values is positive or negative. You are squaring all of those so that you only have positive amounts (if you directly added those, there would be cancelling; it would not truly measure how far the line is from the points). Then you are adding those. You are making that sum as small as possible.