content covered by the act mathematics test
DESCRIPTION
Content Covered by the ACT Mathematics Test In the Mathematics Test, three subscores are based on six content areas: pre-algebra, elementary algebra, intermediate algebra, coordinate geometry, plane geometry, and trigonometry. Pre-Algebra - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/1.jpg)
Content Covered by the ACT Mathematics Test
In the Mathematics Test, three subscores are based on six content areas:
pre-algebra, elementary algebra, intermediate algebra, coordinate geometry, plane geometry, and trigonometry.
![Page 2: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/2.jpg)
Pre-AlgebraPre-Algebra (23%). Questions in this content area are based on basic operations using whole numbers, decimals, fractions, and integers; place value; square roots and approximations; the concept of exponents; scientific notation; factors; ratio, proportion, and percent; linear equations in one variable; absolute value and ordering numbers by value; elementary counting techniques and simple probability; data collection, representation, and interpretation; and understanding simple descriptive statistics.
![Page 3: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/3.jpg)
Elementary Algebra
Elementary Algebra (17%). Questions in this content area are based on properties of exponents and square roots, evaluation of algebraic expressions through substitution, using variables to express functional relationships, understanding algebraic operations, and the solution of quadratic equations by factoring.
![Page 4: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/4.jpg)
Intermediate Algebra
Intermediate Algebra (15%). Questions in this content area are based on an understanding of the quadratic formula, rational and radical expressions, absolute value equations and inequalities, sequences and patterns, systems of equations, quadratic inequalities, functions, modeling, matrices, roots of polynomials, and complex numbers.
![Page 5: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/5.jpg)
Coordinate GeometryCoordinate Geometry (15%). Questions in this content area are based on graphing and the relations between equations and graphs, including points, lines, polynomials, circles, and other curves; graphing inequalities; slope; parallel and perpendicular lines; distance; midpoints; and conics.
![Page 6: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/6.jpg)
Plane GeometryPlane Geometry (23%). Questions in this content area are based on the properties and relations of plane figures, including angles and relations among perpendicular and parallel lines; properties of circles, triangles, rectangles, parallelograms, and trapezoids; transformations; the concept of proof and proof techniques; volume; and applications of geometry to three dimensions.
![Page 7: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/7.jpg)
Trigonometry
Trigonometry (7%). Questions in this content area are based on understanding trigonometric relations in right triangles; values and properties of trigonometric functions; graphing trigonometric functions; modeling using trigonometric functions; use of trigonometric identities; and solving trigonometric equations.
![Page 8: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/8.jpg)
• Read each question carefully to make sure you understand the type of answer required.
• If you use a calculator, be sure it is working on test day and has reliable batteries.
• Solve the problem.• Locate your solution among the answer
choices.• Make sure you answer the question asked.• Make sure your answer is reasonable.• Check your work.
![Page 9: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/9.jpg)
Pre-Algebra
1. Number Problems2. Multiples, Factors, & Primes3. Divisibility and Remainders4. Percentages5. Ratios and Proportions6. Mean, Median, & Mode7. Probability8. Absolute Value9. Exponents and Roots10. Series
![Page 10: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/10.jpg)
Elementary Algebra
1. Substitution2. Simplifying Algebraic Expressions3. Writing Expressions & Equations4. Solving Linear Equations5. Multiplying Binomials6. Inequalities
![Page 11: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/11.jpg)
Intermediate Algebra
1. Solving & Factoring Quadratic Equations2. Solving Systems of Equations3. Relationship between Sides of an Equation4. Functions5. Matrices6. Logarithms
![Page 12: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/12.jpg)
Plane Geometry
1. Angles2. Triangles3. Polygons4. Circles5. Simple 3-D Geometry
![Page 13: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/13.jpg)
Coordinate Geometry
1. Number Lines & Inequalities2. The (x,y) Coordinate Plane3. Distance and Midpoints4. Slope5. Parallel & Perpendicular Lines6. Graphing Equations7. Conic Sections
![Page 14: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/14.jpg)
Trigonometry
1. SOHCAHTOA2. Solving Triangles3. Trigonometric Identities4. Trigonometric Graphs
![Page 15: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/15.jpg)
Whole Numbers
Natural or Counting Numbers
Whole Numbers
Integers
Rational Numbers Irrational Numbers
Real Numbers
![Page 16: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/16.jpg)
Terms and Operations
1. Basic Terms• sum (total)• difference• product• quotient• remainder
![Page 17: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/17.jpg)
Terms and Operations
2. Symbols of inclusion• parentheses• brackets• braces
(2 + 3) •4 = 20
2 + (3 • 4) = 14
![Page 18: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/18.jpg)
Terms and Operations
3. Order of Operation• Parentheses• Exponents, radicals• Multiplication/division• Addition/subtraction
![Page 19: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/19.jpg)
Terms and Operations
4. Factoring and Canceling• 5(2 + 3 + 4) = 5(9) = 45
![Page 20: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/20.jpg)
Factors, Multiples, and Primes
• Factor - A number that divides another number evenly.
• Multiple - A number that is evenly divisible by another number.
• Prime - A number that is only divisible by 1 and itself.
![Page 21: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/21.jpg)
Odd and Even Numbers
• Odd number - A number that is not evenly divisible by 2.
• Even number - A number that is divisible by 2.
1. Even + Even = Even 5. Even • Even = Even 2. Even + Odd = Odd 6. Even • Odd = Even 3. Odd + Even = Odd 7. Odd • Even = Even 4. Odd + Odd = Even 8. Odd • Odd = Odd
![Page 22: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/22.jpg)
Consecutive Integers
Consecutive integers immediately follow one another 1st 2nd 3rd n n + 1 n + 2
Consecutive even integers and consecutive odd integers
1st 2nd 3rd n n + 2 n + 4
![Page 23: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/23.jpg)
Fractions
• Convert Mixed Numbers to Improper Fractions
• Convert Improper Fractions to Mixed Numbers
• Reducing Fractions to Lowest Terms• Common Denominators• Operations of Fractions• Comparing Fractions
![Page 24: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/24.jpg)
Signed Numbers• Absolute Value• Adding Negative Numbers• Subtracting Negative Numbers• Multiplying Negative Numbers• Dividing Negative Numbers
1. Subtracting a negative is the same as adding a positive number.
2. Adding a negative number is the same as subtracting a positive number.
3. Multiplying or dividing an odd number of negative signs will always result in a negative number.
4. Multiplying or dividing an even number of negative signs will always result in a positive number.
![Page 25: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/25.jpg)
Decimals• A decimal is a special
way of writing fractions using a denominator that is a power of 10.
Millionths
Hundred T
housandthsTen T
housandthsT
housandthsH
undredthsTenths
0 . 1 2 3 4 5 6
![Page 26: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/26.jpg)
Decimals
• Converting Fractions to Decimals• Converting Mixed Numbers to Decimals• Converting Improper Fractions to
Decimals• Converting Decimals and Fractions to
Mixed Numbers• Operations of Decimals
![Page 27: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/27.jpg)
Percents
• Converting to and from Percents• Operations of Percents• Percent Story Problems
![Page 28: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/28.jpg)
Mean, Median, and Mode
• Mean - The average of a set data values
• Median– The median of an odd number of data
values is the middle value– The median of an even number of data
values is the average of the middle two• Mode - The value that occurs most
often
![Page 29: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/29.jpg)
Ratios and Proportions
• Ratio - A relationship between two quantities
• Proportion - An equation relating two proportions
1:2 = 2:4 1*4 = 2*2The product of the means equals the product of the extremes.
![Page 30: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/30.jpg)
Exponents
Rules For ExponentsIf a > 0 and b > 0, the following hold true for all real numbers x and y.
yxyx aaa 1.
yxy
x
aa
a 2.
xyyx aa 3.
xxx (ab)ba 4.
x
xx
b
a
b
a 5.
1a 6. 0
x
1a 7. x-
q pq
p
aa 8.
![Page 31: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/31.jpg)
Roots and Radicals
• Square Root
• Cube Root
• Rational Exponents
x b, or x b2
x3 b, or x b3
xn x1
n
![Page 32: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/32.jpg)
Algebraic Operations
• Elements of Algebra– Algebraic Terms
• Term• Coefficient• Variable• Exponent
3x 4 2y
Coefficients
Variables
Exponent
![Page 33: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/33.jpg)
Operations of Algebraic Terms
• Adding and Subtracting• Multiplying and Dividing• Algebraic Fractions• Factoring Algebraic Expressions• Absolute Value in Algebraic Expressions• Radicals in Algebraic Expressions
![Page 34: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/34.jpg)
Algebraic Equations and Inequalities
• Solving Algebraic Formulas• Solving Linear Equations• Solving Simultaneous Equations• Solving Quadratic Equations• Algebraic Inequalities• Exponents in Equations and Inequalities• Rational Equations and Inequalities• Radical Equations
![Page 35: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/35.jpg)
Geometry Notation
P QLine Segment PQ
Vertex A
B
C
Angle BAC <BAC
![Page 36: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/36.jpg)
Angle Properties
0°
270°
90°
180°
![Page 37: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/37.jpg)
Angle Properties
• Right Angle
• Acute Angle
• Obtuse Angle
• Vertical Angle xyw
z
x = zw = y
![Page 38: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/38.jpg)
Angle Properties
• Transversal - A line that intersects parallel line
1 2
5 6
3 4
7 8
– Alternate Interior– Alternate Exterior– Vertical Angles– Same Side Interior Angles
![Page 39: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/39.jpg)
Polygon Properties
• Polygon - A closed figure created by three or more lines.
• Triangle - A polygon with exactly three sides• Quadrilateral - A polygon with exactly four
sides• Pentagon - A polygon with exactly five sides• Hexagon - A polygon with exactly six sides
![Page 40: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/40.jpg)
Polygon Properties
• Regular Polygon - A polygon with equal sides and equal angles.
• The sum of the exterior angles of a polygon is 360°
• The sum of the interior angles of a polygon is 180(n - 2) where n is the number of sides in the polygon
![Page 41: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/41.jpg)
Triangle Properties
• Scalene Triangle - A triangle with no equal sides
• Isosceles Triangle - A triangle with exactly two sides equal, If the two sides are equal than the angles opposite the two sides are equal and vice versa.
• Equilateral Triangle - A triangle with three sides equal, If the three sides are equal than each angle equal 60° and vice versa
![Page 42: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/42.jpg)
![Page 43: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/43.jpg)
![Page 44: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/44.jpg)
Perimeter and Area
![Page 45: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/45.jpg)
Area and Volume
![Page 46: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/46.jpg)
Circle Properties
chord
radius
tangent
secant
diameter
![Page 47: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/47.jpg)
Circle Properties
O
B
CE
D
A
Central Angle - An angle withthe vertex at the center and with sides that are radii.
Inscribed Angle - An angle witha vertex on the circle and withsides that are chords.
![Page 48: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/48.jpg)
Coordinate Geometry
d(P,Q ) (x1 x2)2 (y1 y2)2
midpoint a c
2,b d
2
![Page 49: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/49.jpg)
Coordinate Geometry
m y2 y1x2 x1
y2 y1 m(x2 x1)
y mx b
Slope Equations of Lines
![Page 50: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/50.jpg)
Coordinate Geometry
Parallel lines m1 = m2
Perpendicular lines m1 = -1/m2
![Page 51: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/51.jpg)
Coordinate GeometryTransformationsh(x) = af(x) vertical stretch or shrink
h(x) = f(ax) horizontal stretch or shrink
h(x) = f(x) + k vertical shift
h(x) = f(x+h) horizontal shift
h(x) = -f(x) reflection in the y-axis
h(x) = f(-x) reflection in the x-axis
![Page 52: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/52.jpg)
1. log a (ax) = x for all x 2. a log ax = x for all x > 03. log a (xy) = log a x + log a y4. log a (x / y) = log a x – log a y5. log a xn = n log a x
Common Logarithm: log 10 x = log xNatural Logarithm: log e x = ln x
All the above properties hold.
Logarithms
![Page 53: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/53.jpg)
(1,0)
3
2,-
1
2
2
2,
2
2
1
2,
3
2
(-1,0)
(0,1)
(0,-1)
1
2,
3
2
1
2,
3
2
2
2,-
2
2
2
2,-
2
2
2
2,
2
2
3
2,1
2
3
2,1
2
3
2,-
1
2
Trigonometric Functions
60° 1
3
2
30°
45°
2
1
145°
A
CT
S
2
3,
2
1
![Page 54: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/54.jpg)
Trigonometric Functions
![Page 55: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/55.jpg)
Trigonometric Functions
![Page 56: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/56.jpg)
Trigonometric Functions
![Page 57: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/57.jpg)
Trigonometric Functions
![Page 58: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/58.jpg)
Trigonometric Functions
![Page 59: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/59.jpg)
Trigonometric Functions
![Page 60: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/60.jpg)
Coordinate Geometry
Quadratic functions f(x) = ax2+bx+cComplete the square f(x) = a(x-h)2+ kDiscriminant b2-4acVertex (h,k)
Quadratic Formula
b b2 4ac
2a
![Page 61: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/61.jpg)
Conic Sections
• Circles (x – h)2 + (y – k)2
• Parabolas (x – h)2 = 4a(y – k) (Up/Down) (y – k)2 = 4a(x – h) (Right/Left)
• Ellipses (x – h)2 /a2 + (y – k)2/ b2 = 1 (y – k)2 /a2 + (x – h)2/ b2 = 1
c 2 = a 2 - b 2
• Hyperbolas (x – h)2 /a2 - (y – k)2/ b2 = 1 (y – k)2 /a2 - (x – h)2/ b2 = 1 c 2 = a 2 + b 2
![Page 62: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/62.jpg)
Circles
Standard Equation of a Circle [Center at (h,k)]
The standard form of the equation of a circlecentered at (h,k) is
222 )()( rkyhx y
Radius rPoint on circle (x,y)
Center (h,k)
x
![Page 63: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/63.jpg)
Parabolas
x
y
F
a
V(h, k)V(h, k)
x
y
F
a
(x – h)2 = 4a(y – k)Vertex (h, k)Focus (h, k + a)a > 0 opens upa < 0 opens down
(y – k)2 = 4a(x – h)Vertex (h, k)Focus (h + a, k)a < 0 opens lefta > 0 opens right
![Page 64: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/64.jpg)
x
y
(h, k) b
a
y
x
ba
(h, k)
(x – h)2
a2 + (y – k)2
b2 = 1 a > b > 0 (x – h)2
b2 + (y – k)2
a2 = 1
Center (h, k) Center (h, k)Major axis 2a Major axis 2aMinor axis 2b Minor axis 2b
Ellipses
![Page 65: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/65.jpg)
a
b
y
x
(h, k)
y
x
ab(h, k)
Hyperbolas
(x – h)2
a2 – (y – k)2
b2 = 1 (y – k)2
a2 + (x – h)2
b2 = 1
Center (h, k) Center (h, k) Transverse axis 2a Transverse axis 2a Conjugate axis 2b Conjugate axis 2b
-
![Page 66: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/66.jpg)
DIRECTIONS: Solve each problem, choose the correct answer, and then fill in the corresponding oval on your answer document. Do not linger over problems that take too much time. Solve as many as you can; then return to the others in the time you have left for this test. You are permitted to use a calculator on this test. You may use your calculator for any problems you choose, but some of the problems may best be done without using a calculator. Note: Unless otherwise stated, all of the following should be assumed.
1 Illustrative figures are NOT necessarily drawn to scale.
2. Geometric figures lie in a plane. 3. The word line indicates a straight line. 4. The word average indicates arithmetic mean.
![Page 67: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/67.jpg)
1. Which of the following is equivalent to (x)(x)(x)(x), for all x ?
a. 4x b. x4 c. x + 4 d. 4 + x e. 2x2
![Page 68: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/68.jpg)
2. A rectangle is twice as long as it is wide. If the width of the rectangle is 3 inches, what is the rectangle's area, in square inches?
F. 6 G. 9 H. 12 J. 15 K. 18
![Page 69: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/69.jpg)
3. A vendor has 14 helium balloons for sale: 9 are yellow, 3 are red, and 2 are green. A balloon is selected at random and sold. If the balloon sold is yellow, what is the probability that the next balloon, selected at random, is also yellow?
A. 8
15 B.
9
15 C.
5
14 D.
8
13 E.
9
14
![Page 70: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/70.jpg)
4. 3 x 10 - 4 = ?
F. -30,000 G. -120 H. 0.00003 J. 0.0003 K. 0
![Page 71: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/71.jpg)
5. For all x > 0, simplifies to:
A. x + 3B. x + 4C. 2(x + 3)D. 2(x + 4)E. 2(x + 3)(x + 4)
424142 2
xxx
![Page 72: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/72.jpg)
6. If one leg of a right triangle is 8 inches long, and the other leg is 12 inches long, how many inches long is the triangle's hypotenuse?
F. G. H. J. K. 4
4 13
4 10
2 10
4 5
![Page 73: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/73.jpg)
7. In the standard (x,y) coordinate plane, the graph of (x - 2)2 + (y + 4)2 = 9 is a circle. What is the area enclosed by this circle, expressed in square coordinate units?
A. 3π B. 4π C. 6π D. 9π E. 16π
![Page 74: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/74.jpg)
8. How many solutions are there to the equation x2 - 15 = 0 ?
F. 0G. 1H. 2J. 4K. 15
![Page 75: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/75.jpg)
9. A circle with center (-3,4) is tangent to the x-axis in the standard (x,y) coordinate plane. What is the radius of this circle?
A. 3B. 4C. 5D. 9E. 16
![Page 76: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/76.jpg)
10. In ∆ABC, if <A and <B are acute angles, and sin A = what is the value of cos A ?
F.
G.
H.
J.
K
10
13
69
13
3
13
3
13
3 13
13
3
13
![Page 77: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/77.jpg)
11.What are the values of a and b, if any, where a|b - 2| > 0 ?
A. a < 0 and b < 2
B. a < 0 and b = 2
C. a < 0 and b > 2
D. a > 0 and b < 2
E. There are no such values of a and b.
![Page 78: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/78.jpg)
12. In a shipment of 1,000 light bulbs, of the bulbs were defective. What is the ratio of defective bulbs to nondefective bulbs?
F.
G.
H.
J.
K.
1
40
1
25
1
39
1
40
39
1
40
1
![Page 79: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/79.jpg)
13. Which of the following is divisible by 3 (with no remainder)?
A. 2,725B. 4,210C. 4,482D. 6,203E. 8,105
![Page 80: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/80.jpg)
14. A particle travels 1 x 108 centimeters per second in a straight line for 4 x 10 -6 seconds. How many centimeters has it traveled?
F 2.5 x 102 G. 2.5 x 10 13 H. 4 x 102 J. 4 x 10 -14 K. 4 x 10 -48
![Page 81: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/81.jpg)
15. The triangle below is isosceles with
<M = < N. What is the measure of <N ?
A. 22°
B. 68°
C. 78°
D. 79°
E. 89°
![Page 82: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/82.jpg)
16. In the figure, AB = AC and BC is 10 units long. What is the area, in square inches, of ABC ?
F. 12.5 G. 25 H. J. 50 K. Cannot be determined from the given information
![Page 83: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/83.jpg)
17. Which of the following statements
completely describes the solution set for
3(x - 4) = 3x - 12 ?
F. x = 3 only
G. x = 0 only
H. x = -12 only
J. There are no solutions for x.
K. All real numbers are solutions for x.
![Page 84: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/84.jpg)
18. When graphed in the (x,y) coordinate
plane, at what point do the lines
x + y = 5 and y = 7 intersect?
A. (-2,0)
B. (-2,7)
C. (0,7)
D. (2,5)
E. (5,7)
![Page 85: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/85.jpg)
19. The area of a trapezoid is h/2 (b1 + b2),
where h is the altitude, and b1 and b2 are the lengths of the parallel bases. If a trapezoid has an altitude of 5 inches, an area of 55 square inches, and one base 13 inches long, what is the length, in inches, of its other base?
F. 9.0 G. 16.8 H. 19.4 J. 45.0 K. 97.0
![Page 86: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/86.jpg)
20. If you have gone 4.8 miles in 24 minutes, what was your average speed, in miles per hour?
A. 5.0
B. 10.0
C. 12.0
D. 19.2
E. 50.0
![Page 87: Content Covered by the ACT Mathematics Test](https://reader035.vdocuments.mx/reader035/viewer/2022062517/5681386a550346895da01b07/html5/thumbnails/87.jpg)
21. If a and b are any real numbers such that 0 < a < 1 < b, which of the following must be true of the value of ab ?
F. 0 < ab < a G. 0 < ab < 1 H. a < ab < 1 J. a < ab < b
K. b < ab