Functions in Act Math
Presentation
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Mathematics
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10th - 12th Grade
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Hard
Joseph Anderson
FREE Resource
23 Slides • 28 Questions
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Functions & Statistics/Probability
Functions: The questions in this category test your ability to work with functions. Questions can ask you to manipulate and translate linear, radical, piecewise, polynomial, and logarithmic functions. They can also ask you to find and apply features of graphs.
Questions in the Statistics and Probability reporting category ask about probabilities of events and interpretation of data distributions, data collection methods, and data relationships. They also ask about averages and medians.
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Functions
1. Series, Sequences, and Consecutive Numbers
A series or sequence is the adding of many quantities, one after the other, to a given quantity, possibly in a repeating pattern. To discern a series or a sequence, look for the pattern or difference between each of the numbers. For example, to find the next number in series {1, 4,7,10,...}, the common difference is 3, so you know that the next number after 10 is 13.
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Arithmetic sequences have a common difference between terms, in other words, you add the same amount every time
Recursive Formula: an = an-1 + d; a1 = x
(add the common difference to the previous term, define a1)
Explicit Formula: an = a1 + d(n - 1)
(add the common difference n-1 times to the 1st term)
Geometric sequences have a common ratio between terms, in other words, you multiply by the same amount every time
Recursive Formula: an = r • an-1; a1 = x
(multiply the previous term by the common ratio, define a1)
Explicit Formula: an = a1 • r(n - 1)
(multiply the 1st term by common ratio n-1 times)
Functions | Sequences & Series
Example: {7,13,19, 25, 31, ...}
Recursive Formula:
an = an-1 + 6
a1 = 7
Explicit Formula:
an = 7 + 6(n - 1)
Example: {3,12, 48, 192, 768, ...}
Recursive Formula:
an = 4 • an-1
a1 = 3
Explicit Formula:
an = 3 • 4(n - 1)
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Consecutive numbers are of a specific series where numbers follow each other in order, typically with a common difference of 1 between each number: {5,6,7,8,9,...}.
Consecutive numbers specified as even or odd have a common difference of 2 between each number: {2,4,6,8,10,...} or {1,3,5,7,9,...}.
Consecutive numbers can also be negative: {−3,−2,−1,0,1,2,3,...}.
Functions | Sequences & Series
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Multiple Choice
What 2 numbers should be placed in the blanks below so that the difference between consecutive numbers is the same?
17, _ , _ , 41
A. 23, 29
B. 24, 34
C. 25, 33
D. 26, 35
E. 27, 31
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Multiple Choice
Which of the following statements describes the total number of dots in the first n rows of the triangular arrangement illustrated?
A. This total number is always equal to 25 regardless of the number of rows.
B. This total is equal to twice the number of rows.
C. This total is equal to 5 times the number of rows.
D. This total is equal to the square of the number of rows.
E. There is no consistent relationship between this total and the number of rows.
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Multiple Choice
Which of the following statements is NOT true about the arithmetic sequence 17,12,7,2,... ?
A. The fifth term is −3.
B. The sum of the first 5 terms is 35.
C. The eighth term is −18.
D. The common difference of consecutive terms is −5.
E. The common ratio of consecutive terms is −5.
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Functions
2. Slope-Intercept Form of a Linear Equation
The slope-intercept form of a linear equation is the form y = mx + b, where m is the slope and b is the y-intercept. With the line y = 3x + 5, the slope is 3 and the y-intercept is 5.
A linear equation can appear in a different form, such as 6x + 3y = 9. To find the slope and y-intercept, convert it to the slope-intercept form by setting the equation equal to y:
6x + 3y = 9
3y = −6x + 9
y = −2x + 3
With this line, the slope is −2 and the y-intercept is 3.
Note that parallel lines have identical slopes, and perpendicular lines have negative reciprocal slopes. For example, y = x + 4 is parallel to y = x + 2 but perpendicular to y = − x + 4.
Functions | Slope-Intercept Form of a Linear Equation
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Multiple Choice
What is the slope-intercept form of 8x − y − 6 = 0?
F. y = −8x − 6
G. y = −8x + 6
H. y = 8x − 6
J. y = 8x + 6
K. y = 6x − 8
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Multiple Choice
What is the slope of any line parallel to the line 7x + 9y = 6?
A. −7
B. − 7/9
C. 7/6
D. 6
E. 7
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Functions | Slope-Intercept Form of a Linear Equation
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Functions | Slope-Intercept Form of a Linear Equation
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Draw
Write the equation of the line:
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Draw
Write the equation of the line:
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Multiple Choice
What is the slope of the line given by the equation 14x − 11y + 16 = 0?
A. −11
B. −14/11
C. − 11/14
D. 14/11
E. 14
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Functions | Slope-Intercept Form of a Linear Equation
Pattern - In the form y = mx + b, recall that m=slope gives the rate and b=y-intercept gives the constant or initial value. We can then use a linear equation to model a real-life scenario. For example, sound travels at the constant speed of 1,125 feet per second, so to calculate the distance in feet, d, that a sound travels over a certain number of time in seconds, t, use the equation d = 1,125t. This is equivalent to a linear equation with a slope of 1,125 and an intercept of 0.
If the pattern is complicated, simply place the input or inputs into the equation. For example, to use the speed of sound equation d = 1,125t to find the distance that a sound travels in 200 seconds, place 200 for t:
d = 1,125(200) = 225,000 feet.
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Multiple Choice
When Jeff starts a math assignment, he spends 5 minutes getting out his book and a sheet of paper, sharpening his pencil, looking up the assignment in his assignment notebook, and turning to the correct page in his book. The equation t = 10p + 5 models the time, t minutes, Jeff budgets for a math assignment with p problems. Which of the following statements is necessarily true according to Jeff’s model?
F. He budgets 15 minutes per problem.
G. He budgets 10 minutes per problem.
H. He budgets 5 minutes per problem.
J. He budgets 10 minutes per problem for the hard problems and 5 minutes per problem for the easy problems.
K. He budgets a 5-minute break after each problem.
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Multiple Choice
What is sin12π given that 12π=3π−4π and that
(sin α −β) = (sin α)(cos β) − (cos α)(sin β)?
(Note: You may use the table of values.)
F. 1/4
G. 1/2
H. 43−2
J. 23−2
K. 46−2
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Functions | Slope-Intercept Form of a Linear Equation
2.2 Graphed Equations in the f(x) Form
A function can be a graphed equation where f(x) takes the place of y. For example, the functions y = 2x2 + 1 and f(x) = 2x2 + 1 are the same. On the coordinate system, x is the horizontal coordinate and f(x), like y, is the vertical coordinate. The value of x is placed into the f(x).
For example, if x = 3, the function would appear:
f(3) = 2(3)2 + 1 = 19.
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Multiple Choice
If f(x) = x2 − 2, then f(x + h) =?
x2 + h2
x2 − 2 + h
x2 + h2 − 2
x2 + 2xh + h2
x2 + 2xh + h2 − 2
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Functions | Slope-Intercept Form of a Linear Equation
A function can use letters other than f and x, such as g (h). Also, a function can be nested within another function. For example, if f(x) = x2 − 3 and g(h) = 2h − 1, the question can ask the value of the nested functions f(g(h)) when h = 5. To solve this:
1. Find g(h) by placing 5 for h: g(5) = 2(5) − 1 = 9.
2. Because g(5) = 9, place 9 for g(5): f(g(5)) = f (9).
3. Solve the final equation by placing 9 for x: f(9) = (9)2 − 3 = 78.
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Functions
3. Systems of Functions
A system of functions refers to two or more functions in a single instance. The number of solutions refers to the number of points where the functions cross. For example, this drawing of a system of functions shows two functions with two solutions, or two points where f(x) = g(x).
Finding the coordinates of the solutions was covered in "Algebra Review - Day 2."
Functions | Systems of Functions
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Functions
4. Trigonometric Functions
A trigonometric function is the graph of a continuous, smooth periodic oscillation. The simplest form is y = sin x or f (x) = sin x, also known as a sine wave, shown in the following:
Functions | Trigonometric Functions
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Functions
4. Trigonometric Functions
The terms describing a sine wave are period, amplitude, and frequency:
Period, also known as phase, is the distance along the x-axis for the function to complete one full cycle. The function in the drawing of y = sin x has a period of 2π. An x-coefficient reduces the period: for example, the graph of y = sin(2x) has a period of π.
Frequency is the number of cycles, or periods, in a given interval. The interval shown of 4π has a frequency of 2. An x-coefficient increases the frequency. For example, the graph of y = sin(2x) in the same interval of 4π has a frequency of 4.
Amplitude is the distance from the mean, in this case, the x-axis, to the maximum. The function in the drawing of y = sin x has an amplitude of 1. A coefficient on the sin x increases the amplitude. For example, the graph of y = 2sin x has an amplitude of 2.
Functions | Trigonometric Functions
Note that the graph of y = cos x produces a similar trigonometric function known as a cosine wave.
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Multiple Choice
A trigonometric function with equation y = a sin(bx + c), where a, b, and c are real numbers, is graphed in the standard (x, y) coordinate plane to the left. The period of this function f(x) is the smallest positive number p such that f(x + p) = f(x) for every real number x. One of the following is the period of this function. Which one is it?
2π
π
2π
4π
2
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Statistics & Probability
1. Probability
Statistics & Probability | Probability
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1. Probability
Complement: The probability that something will not occur is 1 minus the probability that it does occur.
~ Ex. The probability that you will not roll a three with a number cube is 1 - 1/6 = 5/6. This is also true because 5 of the six sides aren’t three.
General Addition Rule: The probabilities of one outcome or another outcome is the sum of each independent probability. If the events can occur simultaneously, subtract the overlap.
~ Ex. Mutually Exclusive Events: If the box has 2 red pens and 3 blue pens, the probability that you pull a red or a blue pen is 2/9 + 3/9 = 5/9.
~ Ex. Non-Mutually Exclusive Events: If you roll a number cube, the probability that you will roll an even or a prime number is 3/6 + 3/6 - 1/6 = 5/6. (three evens: 2, 4, 6; three primes: 2, 3, 5; one even prime: 2)
Statistics & Probability | Probability
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1. Probability
General Multiplication Rule: The probabilities of one outcome and another outcome is the product of each independent probability.
~ Ex. For Independent Events: The probability that you pull a red pen today is 2/9. If you place it back in the box, the probability that you pull a blue pen is 3/9. The probability that you pull a red pen then a blue pen is 2/9 × 3/9 = 6/81.
~ Ex. For Dependent Events:The probability that you pull a red pen today is 2/9. If you do NOT place it back in the box, the probability that you pull a blue pen is 3/8. The probability that you pull a red pen then a blue pen is 2/9 × 3/8 = 6/72.
Probability is always a number between 0 and 1. A probability of 0 means that the event will not occur, and a probability of 1 means that the event will definitely occur.
~ Ex. If a box of 10 sodas contains only diet sodas, then the probability that you pull a regular soda is 0/10, or 0.
~ Ex. From the same box, the probability that you pull a diet soda is 10/10, or 1.
Statistics & Probability | Probability
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Multiple Choice
If a marble is randomly chosen from a bag that contains exactly 8 red marbles, 6 blue marbles, and 6 white marbles, what is the probability that the marble will NOT be white?
F. 43
G. 53
H. 54
J. 103
K. 107
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2.1 Average/Arithmetic Mean
2. Sets of Numbers
The ACT mathematics test questions can ask for a simple analysis on a set of numbers, such as {2, 3, 6, 8, 11}.
Statistics & Probability | Sets of Numbers
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Statistics & Probability | Sets of Numbers
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Multiple Choice
A certain type of notebook costs $2.50 before sales tax is added. When you buy 9 of these notebooks you receive 1 additional notebook free. What is the average cost per notebook for the 10 notebooks before sales tax is added?
A. $2.78
B. $2.50
C. $2.30
D. $2.25
E. $2.15
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Multiple Choice
Kaya drove 200 miles in 5 hours of actual driving time. By driving an average of 10 miles per hour faster, Kaya could have saved how many hours of actual driving time?
A. 1/6
B. 2/3
C. 7/10
D. 1
E. 4
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Multiple Choice
The chart shows the current enrollment in all the mathematics classes offered by Eastside High School. What is the average number of students enrolled per section in Algebra I?
F. 24
G. 25
H. 26
J. 27
K. 29
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Multiple Choice
A company earned a profit of $8.0 million each year for 3 consecutive years. For each of the next 2 years the company earned a profit of $9.0 million. For this 5 year period, what was the company’s average yearly profit, in millions of dollars?
F. 8.2
G. 8.25
H. 8.4
J. 8.5
K. 8.6
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The mode of a set of numbers is the most commonly occurring value in the set. For example, the mode of {2, 5, 6, 8, 8} is 8. If the set of numbers has two values that occur the most number of times, it is called bimodal, meaning it has two modes. The modes of {2, 5, 5, 7, 9, 9} are 5 and 9.
Mode
The median of a set of numbers is the middle value in the set. For example, the median of the set {2, 3, 6, 8, 11} is 6. If the numbers are out of order, place them in order before taking the middle number. If there are two middle numbers, such as {3, 6, 8, 11}, take the average of the two middle numbers: 6+8/2 = 7
Median
Statistics & Probability | Sets of Numbers
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3. Charts
The ACT mathematics test features a few simple charts in the Statistics and Probability reporting category. The charts appearing in the ACT science test are far more extensive, but the following are the charts you’re likely to see in the ACT math test:
Statistics & Probability | Charts
The column chart shows dependent results per independent variables:
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3. Charts
The ACT mathematics test features a few simple charts in the Statistics and Probability reporting category. The charts appearing in the ACT science test are far more extensive, but the following are the charts you’re likely to see in the ACT math test:
Statistics & Probability | Charts
The line chart also shows dependent results per independent variables, but the line chart emphasizes a trend:
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3. Charts
The ACT mathematics test features a few simple charts in the Statistics and Probability reporting category. The charts appearing in the ACT science test are far more extensive, but the following are the charts you’re likely to see in the ACT math test:
Statistics & Probability | Charts
The bar chart is like the column chart, only the x- and y-axes are reversed:
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3. Charts
The ACT mathematics test features a few simple charts in the Statistics and Probability reporting category. The charts appearing in the ACT science test are far more extensive, but the following are the charts you’re likely to see in the ACT math test:
Statistics & Probability | Charts
The pie chart shows each value as a slice of the pie having either a number or a percent of the total. The entire chart is either the total number of items or 100%:
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Multiple Choice
The graph below shows the number of cars assembled last year in 4 cities, to the nearest 5,000 cars. According to the graph, what fraction of the cars assembled in all 4 cities were assembled in Coupeville?
A. 51
B. 41
C. 113
D. 103
E. 31
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Multiple Choice
Douglas wants to draw a circle graph showing the favorite colors of his friends. When he polled his friends asking each their favorite color, 25% of his friends said red; 30% of his friends said blue; 20% of his friends said green; 10% of his friends said purple; and the remaining friends said colors other than red, blue,green, and purple. The colors other than red, blue, green, and purple will be grouped together in an Other sector. What will be the degree measure of the other sector?
A. 108º
B. 54º
C. 27º
D. 15º
E. 10º
Functions & Statistics/Probability
Functions: The questions in this category test your ability to work with functions. Questions can ask you to manipulate and translate linear, radical, piecewise, polynomial, and logarithmic functions. They can also ask you to find and apply features of graphs.
Questions in the Statistics and Probability reporting category ask about probabilities of events and interpretation of data distributions, data collection methods, and data relationships. They also ask about averages and medians.
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