8th Grade Math SEMESTER EXAM Study Guide
Rational & Irrational Numbers
Rational numbers are numbers that can be written as fractions in the form , where p and q are integers and q
0. All rational numbers can be written as terminating or repeating decimals.
Irrational numbers are non-terminating, non-repeating decimals and cannot be written in the form , where p
and q are integers and q 0. The square roots of whole numbers that are not perfect squares are irrational
numbers.
Examples of Rational Numbers
Examples of Irrational Numbers
Decimal Expansion
numerator by the denominator.
Example 1:
What is the decimal expansion of the following fraction?
Solution:
Divide the numerator by the denominator.
Since 1 divided by 25 is 0.04, the decimal expansion of
is 0.04.
Example 2:
What is the decimal expansion of the following fraction?
Solution:
Divide the numerator by the denominator.
Since 1 divided by 24 is 0.041666..., the decimal expansion of
is
Decimal Expansion
A terminating decimal expansion can be written as a fraction by using the value of its smallest decimal place.
Example 1:
What is the fraction represented by the following decimal expansion?
Solution:
To convert a terminating decimal to a fraction, first determine the value of its smallest decimal place.
The smallest decimal place of 0.42 is the hundredths place. So, write the number to the right of the decimal, 42,
as the numerator of a fraction with 100 as the denominator.
Then, reduce the fraction.
Therefore, the fraction
is represented by the decimal expansion 0.42.
A repeating decimal expansion can be written as a fraction by using the difference of two algebraic equations.
Example 2:
What is the fraction represented by the following decimal expansion?
Solution:
To convert a repeating decimal to a fraction, first let
. So,
also equals
.
Next, multiply both sides of the equation by 10 to generate an equivalent equation.
Then, subtract the equation
from the equation
Finally, use the equation above to solve for
.
Therefore, the fraction
Real Numbers
is represented by the decimal expansion
.
.
because
Natural numbers are the positive counting numbers.
{1, 2, 3, 4, 5, 6, 7, ...}
Whole numbers are the non-negative counting numbers.
{0, 1, 2, 3, 4, 5, 6, ...}
The natural numbers are a subset of the whole numbers.
Integers are the set of numbers consisting of
the whole numbers and their negatives.
{..., -3, -2, -1, 0, 1, 2, 3, ...}
The whole numbers are a subset of the integers.
Rational numbers are numbers which
can be written as a fraction of integers.
The integers are a subset of the rational numbers.
Irrational numbers are numbers which
cannot be written as a fraction of integers.
{x | x is a real number that is not rational}
Real numbers consist of the rational numbers and the irrational numbers.
Natural numbers, whole numbers, integers, rational numbers,
and irrational numbers are each a subset.
Natural Numbers
{x | x is a rational or irrational number}
Whole Numbers
Integers
Rational Numbers
Real Numbers
The Real Numbers can be split up into Rational and Irrational Numbers.
Example
The numbers shown above belong to which of the following subsets of the real numbers?
Natural Numbers
Whole Numbers
Integers
Rational Numbers
Irrational Numbers
Solution
All of the numbers can be written as a fraction of integers.
Therefore, all of the numbers only belong to subset IV, the rational numbers.
Example
The numbers shown above belong to which of the following subsets of the real numbers?
Natural Numbers
Whole Numbers
Integers
Rational Numbers
Irrational Numbers
Solution
Based on the definitions of the subsets of the real numbers, the numbers belong to subsets II, III, and IV.
Rational Approximations
Irrational numbers are numbers that cannot be written as , where p and q are integers.
Examples of irrational numbers are
and
.
When placing irrational numbers on the number line, it is important to know some of the squares and cubes of
numbers.
number
square
1
1
2
4
3
9
4
16
5
25
Example 1:
At what position on the number line is the red dot located?
A.
B.
C.
D.
Solution:
The square root of 5 is about 2.2.
The square root of 6 is about 2.4.
The square root of 7 is about 2.6.
The square root of 8 is about 2.8.
The red dot is closer to 2.5 than 3, so it is located at
.
Example 2:
At what position on the number line is the red dot located?
A.
B.
C.
D.
Solution:
The value of
is approximately 3.14. Therefore, the red dot is located at
Number Line - Irrational
.
On a number line, all numbers to the right of zero are positive, and all the numbers to the left of zero are
negative.
A number line with the points 0,
,
, and 5 graphed on it is shown below.
Example 1:
At what position on the number line is the red dot located?
A.
B.
C.
D.
Solution:
The dot is located between 1 and 2.
Therefore, the dot is located between
and
The only answer choice that falls in this range is
.
.
Example 2:
At what position on the number line is the red dot located?
A.
B.
C.
D.
Solution:
The dot is located between 4 and 5.
Therefore, the dot is located between
and
The only answer choice that falls in this range is
.
.
Rational Approximations
To compare the size of irrational numbers, calculate a rational approximation of the numbers and locate these on
a number line.
Example 1:
Use the number line above to correctly compare the following numbers.
Solution:
The numbers should be placed on the number line to determine the order, and then choose the correct
comparison. Change each number into a decimal for ease of placement.
Therefore, the correct comparison is shown below.
Example 2:
Use the number line above to order the following numbers from least to greatest.
Solution:
Place the numbers on a number line to determine the order. Change each number into a decimal for ease of
placement.
Therefore, the correct order is shown below.
Example 3:
Use the number line above to order the following numbers from least to greatest.
Solution:
Place the numbers on a number line to determine the order. Change each number into a decimal for ease of
placement.
Therefore, the correct order is shown below.
Independent & Dependent Variables
An independent quantity (variable) is a quantity whose value determines the value of other quantities, and it is
changed in an experiment to see its effect on other quantities.
A dependent quantity (variable) is a quantity that changes because of a change in the independent quantity.
Example 1:
Hector is working on a project in his history class. Right now there are 4 students in his group. He knows if
there were 8 students in his group, his group would finish the project twice as quickly. What are the
independent and dependent quantities in this situation?
Solution:
The independent and dependent quantities will both change. In this situation, the number of students in Hector's
group and the time it takes to finish the project both change, so one will be the independent quantity and one
will be the dependent quantity.
Hector knows that changing the number of students in his group will change the amount of time it takes to finish
the project, so the amount of time depends on the number of students.
Therefore, the amount of time is the dependent quantity and the number of students is the independent
quantity.
Example 2:
Allison knows it will take her less time to get to work if she leaves earlier in the morning. What are the
independent and dependent quantities in this situation?
Solution:
In this situation, the time Allison leaves and the time it takes her to get to work both change, so one will be the
independent quantity and one will be the dependent quantity.
Allison knows that changing the time she leaves will change the time it takes her to get to work, so the time it
takes her to get to work depends on the time she leaves.
Therefore, the time it takes her to get to work is the dependent quantity and the time she leaves is the
independent quantity.
Vertical Line Test
For points on the coordinate plane, the domain refers to the x-values and the range refers to the y-values.
A relation is a correspondence between a first set, the domain, and a second set, the range, such that each
element of the domain corresponds to at least one element of the range.
A function is a relation in which each element of the domain corresponds to exactly one element of the range.
Example 1:
Do the ordered pairs below represent a relation, a function, both a relation and a function, or neither a relation
nor a function?
(0, 1) , (2, -2) , (1, -1) , (2, 4) , (-3, 5)
Solution:
The set of ordered pairs is a relation, but is not a function because 2 is paired with both -2 and 4.
Example 2:
Do the ordered pairs below represent a relation, a function, both a relation and a function, or neither a relation
nor a function?
(-4, -1) , (-2, -2) , (0, -1) , (2, 5) , (3, 8)
Solution:
The set of ordered pairs is both a relation and a function because each x-coordinate is paired with exactly
one y-coordinate.
Notice that, even though the relation has two points with the same y-coordinate, (-4, -1) and (0, -1), it is still a
function.
The Vertical Line Test
Given the graph of a relation, if a vertical line can be drawn that intersects the graph at more than one point, then
the relation is not a function.
Understanding Functions
To find the mapping associated with a given function, substitute the input (domain or x-values) into the function to
calculate the corresponding output (range or y-values).
Example 1:
Create a mapping that represents the above function.
Solution:
Choose some x-values and find their corresponding y-values. In this case, the x-values -2, -1, 0, and 1 will be
used.
Now, create the mapping.
Example 2:
Determine the function that best represents the given mapping.
Solution:
Notice that the mapping maps -1 and 1 both to 0. Consider a function with x2 or x4.
Notice that (-2)2 = 4 while (-2)4 = 16. Thus, (-2)4 - 1 = 15. Since the given mapping maps -2 to 15, try the function
y = x4 - 1 with the other domain values.
Therefore, the function that best represents the mapping is shown below.
Compare Functions
• rate of change [slope]
• y-intercept
•
•
•
•
Characteristics of Linear Functions
Rate of Change [Slope]
If it is positive, the function is increasing.
If it is negative, the function is decreasing.
If it is zero, the function is a horizontal line.
If it is undefined, the function is a vertical line.
Example 1:
Compare and describe the rates of change of the following items.
Solution:
Item I is an equation in the form y = mx + b, where m is the slope, or rate of change.
The rate of change of item I is 3.
To determine the rate of change of the graph in item II, choose two points and use the formula for slope. This
calculation uses the points (-2,0) and (0,2).
The rate of change of item I is 3, and the rate of change of item II is 1.
Therefore, the rate of change of item I is greater than the rate of change of item II.
Example 2:
Compare and describe the rates of change of the following items.
Solution:
To determine the rate of change of the line described in item I, use the two given points and the formula for slope.
Since the rate of change is positive, the function is increasing.
To determine the rate of change of the function represented by the table, choose two points and use the
formula for slope. This calculation uses the points (0,-4) and (3,-3).
Since the rate of change is positive, the function is increasing.
Therefore, both functions are increasing and have different rates of change.
Example 3:
Compare and describe the rates of change of the following items.
Solution:
To determine the rate of change of the function represented by the mapping, choose two points and use the
formula for slope. This calculation uses the points (3,7) and (4,9).
To determine the rate of change of the function represented by the table, choose two points and use the
formula for slope. This calculation uses the points (-2,-10) and (-1,-6).
The rate of change of item I is 2, and the rate of change of item II is 4.
Therefore, the rate of change of item II is greater than the rate of change of item I.
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