Download 3 (subtract 3 from each side)

Math 8 Quick Study Guide
M8A1d: Solving Equations for one variable in terms of another
Given the equation d = rt, you can solve for any of the variables by using the properties of equality
M8A4a & b Slope & y-intercept
To solve for r: d = rt
d = rt (divide each side by t) so, r = d
t
t
t
slope – the ratio of rise over run, also known as the constant rate of change in a linear equation or
the coefficient of “x” in a linear equation or the “steepness” of a line. The slope is represented by
the variable “m” in a linear equation.
To solve for t: d = rt
d = rt (divide each side by r) so, t = d
r
r
r
y-intercept – the point on a line that crosses the y-axis, also known as the “fixed amount” in a
linear equation. The y-intercept is represented by the variable “b” in a linear equation.
Examples:
y = 5x
y=½x+1
y=9
y = 3x – 6
y= x
m = 5; b = 0
m = ½; b = 1
m = 0; b = 9
m = 3; b = -6
m = 1; b = 0
Parallel Lines
y = 2x and y = 2x = 1
are parallel lines since
they have the same slope
but different y-intercepts
Perpendicular Lines
y = 2x and y = -½x
are perpendicular lines
since they have opposite
reciprocal slopes
Slopes:
Negative
Slope
Zero
Slope
Positive
Slope
M8A4f: Writing & Graphing Linear Equations:
y = 5x
m = 5; b = 0
y=½x+1
m = ½; b = 1
y=9
m = 0; b = 9
y = 3x – 6
m = 3; b = -6
y= x
m = 1; b = 0
M8A2c & M8A4e Graphing Inequalities
> OPEN
< OPEN
(3, 5)
4.
M8N1g: Simplifying square roots
To simplify √50
1)
List all factors of 50 - they are
1, 2, 5, 10, 25, 50
2)
Identify the largest perfect square
it is 25
3) √50 = √25 ∙ √2 = 5 √2
M8N1g. Adding & Subtracting square roots
***To add square roots, the square root must be the
same on each term!
Ex#1 2√3 + √3 = 3 √3
Ex #2 5 √7 – 3 √7 = 2 √7
M8N1g: Multiplying & Dividing square roots
*** The square root do NOT have to have to be the
same!
√3 ∙ √5 = √15
√3 ∙ √3 = √9 = 3
2 √3 ∙ 6√5 = 12 √15
10 √3 = 2
5 √3
M8N1h: Rational & Irrational Numbers
0, ½ , -100, √4, √100, 0.5, 0.33 are all rational
√2, √5, √13, -√2, Π, - Π are all irrational
M8N1i: Simplifying exponents
Example: On the graph, the y-intercept is 3, so the point (0,3) is graphed
The slope is 3 so, we go up by 3 and over by 2 to get the point (3,5)
2
x > -5
Example: 4x + 2y = 16 (standard form)
You want to solve for y to get the equation in
slope-intercept form
4x + 2y = 16
-4x
-4x (subtract 4 from each side)
2y = -4x + 16
2y = -4x + 16 (divide each side by 2)
2
2 2
y = -2x + 8 (slope-intercept form)
To convert from slope-intercept to standard
form:
Example: y = 3x + 1(slope-intercept form)
You want to get x and y on the same side
y = 3x + 1
-3x -3x (subtract -3x from each side)
-3x + y = 1 (slope-intercept form)
10x13
5x7
= 2x13-7 or 2x6
23 ∙33 = (2∙3)3 or 63 or 216
(x2y3)2 is x4y6
Negative exponents
4-2 can be rewritten as 1 or 1
42 16
Zero exponents
ANYTHING raised to the power of zero is 1 !!!
10,000,0000 = 1,
a0 = 1, -190 = 1
M8N1j: Scientific Notation
When you convert from a decimal number to
scientific notation, the exponent will be negative
Ex. 10,000,000 = 1.0 x 107
When you convert from a whole number to scientific
notation, the exponent will be positive
Ex. 0.00032 = 3.2 x 10-4[
The number multiplied by the base of 10 raised to a
power must be between 1 and 9.999
M8A5: Systems of Equations
To solve a “system” of linear equations, you need to find the point of intersection for both lines.
(0, 3)
x
Identify the slope (m) and y-intercept (b)
Plot the y-intercept (0,b)
Use the slope in rise to find and plot the next point
run
Draw a line between the two points
To convert from standard to slope-intercept
form:
The “special rule”
When you multiply or divide by a negative
number, the inequality sign reverses!
Example: -2x < 10
To solve, divide each side by -2 -2x < 10
-2
-2
The inequality sign will reverse
run
M8A4c & d
Converting between slope-intercept & standard
forms
Standard form: Ax + By = C
For example, in the equation 2x + 4y = 10
A = 2, B = 4 and C = 10
Slope-intercept form: y = mx + b
m = slope and b = y-intercept
Graph x > 10
y
r
i
s
e
< CLOSED
> CLOSED
Graph x < 5
M8A3: Relations & Functions
A relation is a function if none of the x values repeat.
Function :
Not a function:
{(1,2), (2, 4), (3, 6)}
{(1, 2), (1, 3), (2,3)}
{(6, 5), (4, 10), (2,15 )}
{(½, ¾), (-½, ½), (0,0)}
M8A4c: Graphing linear equations
To graph a linear equation, follow these steps:
1.
2.
3.
M8A1 c & d Solving Equations & Inequalities (M8A2a&b)
7x + 2 = 9x + 3
-7x
-7x (subtract 7x from each side)
2 = 2x + 3 (subtract 3 from each side)
-3
-3
-1 = 2x (divide each side by 2)
2 2
-½ = x
x2∙x3 = x2 + 3 or x5
To write an equation given the slope of 2 and the point (1,2)
1st Use the coordinate to rewrite the equation substituting in 1 for x and 2 for y and m for the slope
y = mx + b
2 = 2(1) + b
2=2 +b
-2 -2 + b
0 =b
Equation y = 2x + 0 or y = 2x
Write an equation given the points (1,3) and (4, 6)
1st Plot the points
2nd Determine the slope by going from (1,3) to (4, 6)
slope is 3/3 or 1
3rd Substitute the slope into the equation
y = mx + b so, y = 1x +b
use either point and plug into the equation
3 = 1(1) + b, solve for b
2 =b
Equation y = 1x +2
M8N1f: Estimating square roots
To estimate √10
Find two consecutive integers √10 falls
between - It falls between √4 and √9
So, the solution to estimating √10 is about 3.2
2g + 3(g + 1) = 13 (distribute 3 to everything in parenthesis)
2g + 3(g + 1) = 13
2g + 3g + 3 = 13 (combine like terms)
5g + 3 = 13
-3
-3 (subtract 3 from each side)
5g
= 10 (divide each side by 5)
5
5
g =2
No
Slope
M8N1d-e: +/- roots Square Root of Zero
√4 = 4, -4 or ± 4
√0 = 0
Elimination (cancel out one of the variables)
To solve the system 4x – y = 6
3x + 2y = 21
1. Multiply one of the equations by a number so that one of the variable will cancel out
2. Add the equations together
3. Use what you found for x, to find y
1. 2(4x – y = 6)  8x – 2y = 12
Notice how the 2y and -2y CANCEL each other out
+ 3x + 2y = 21
11x = 33
x=3
Now we can use x =3 to find the value of y by substituting it back into either equation:
In 3x + 2y = 21  3(3) + 2y = 21
9 + 2y = 21
The solution to the system
4x – y = 6
-9
-9
3x – 2y = 6
2y = 12
y =6
is the point (3, 6)
M8D1a. Set theory.
A set is a collection of things. The "things" in the set are called the "elements", and are listed inside curly braces. {1, 2, 3, 4, 5,
6,} is a set. The numbers 1-6 are elements of the set. Sets can be related to each other. If one set is "inside" another set, it is
called a "subset". Suppose A = {1, 2, 3} and B = {1, 2, 3, 4, 5, 6}. Then A is a subset of B, since everything in A is also in B. A
union( U) of sets is the combining of two or more sets into one big set. An intersection(∩) is what two or more sets have in
common. A complement is an element not contained in the set. When sets have nothing in common, they are called
disjoint(Ø) sets. To show a disjoint set, you write.
Set notation symbols are shorthand, but mean the same thing.
Example: C = {1, 3, 5, 7, 9, 11, 13} and D = {2, 4, 6, 8, 10, 12, 14}. C U D(union of C and D) is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14}
C’ (the complement of C) is 2, 4, 6, 8, 10, 12, and 14. What is the intersection of sets C and D?
M8G1. Properties of parallel and perpendicular lines and understand the meaning of congruence.
Transversal and Parallel lines:
When a transversal line intersects parallel lines alternate interior angles are equal in measure an
alternate exterior angles are equal in measure. KL is transversal
Alternate Interior angles: two angle angles on opposite sides of transversal both between two lines.
Pair of alternate interior angles: <3 and <5, <6 and <4
Alternate exterior angles: A pair of non-adjacent exterior angles lying on opposite sides of a
transversal. Pair of alternate exterior angles: <2 and <8, <7 and < 1
Corresponding angles: two nonadjacent angles on the same side of a transversal, one between two
K
H
lines and outside the lines.
2
1
G
3
4
Pairs of corresponding angles
6
5
:<1 and <5, <2 and <6, <3 and <7, <4 and <8
I
M8D2a. Tree Diagrams & Outcomes
Tree diagram—a diagram that illustrates all the possible outcomes of an experiment containing 2 or more independent event.
Steps to make a tree diagram: 1. List all the outcomes of the first event. 2. Keep building until all outcomes are listed.
M8D2a. Addition and multiplication principles of counting
When dealing with the occurrence of more than one event, it is important to be able to quickly determine how many possible
outcomes exist.
Example: if ice cream sundaes come in 5 flavors with 4 possible toppings, how many different sundaes can be made with
one flavor of ice cream and one topping? Rather than list the entire sample space with all possible combinations of ice cream
and toppings, we may simply multiply 5 • 4 = 20 possible sundaes. This simple multiplication process is known as the
Counting Principle. The Counting Principle works for two or more activities.
7
Formula: a2 + b2 = c2
3 2 + 4 2 = c2
9 + 16 = c2
25 = c2
√25 = √c2
5= c
leg
M8D3a. Basic laws of probability.
a. Find the probability of simple independent events.
Probability - the desired outcome ÷ total number of outcomes.
A single event involves the use of ONE item such as:
•one person being chosen
•one card being drawn
•one coin being tossed
•one die being rolled
c2
c
M8D3b. Probability of compound independent events
A compound event involves the use of two or more items such as:
•two cards being drawn
•three coins being tossed
•two dice being rolled
•four people being chosen
If A and B are independent events, P(A and B) = P(A) x P(B).
Example: A drawer contains 3 red paperclips, 4 green paperclips, and 5 blue paperclips. One paperclip is taken from the
drawer and then replaced. Another paperclip is taken from the drawer. What is the probability that the first paperclip is red and
the second paperclip is blue?
Because the first paper clip is replaced, the sample space of 12 paperclips does not change from the first event to the
second event. The events are independent. P(red then blue) = P(red) x P(blue) = 3/12 • 5/12 = 15/144 = 5/48.
8
J
M8G2. Students will understand and use the Pythagorean Theorem.
Pythagorean Theorem states that in a right triangle, the sums of the squares of the legs of the
triangle are equal to the square of the hypotenuse.
Or a2 + b2 = c2 where a & b are the length of the legs and c is the length of the hypotenuse.
This relationship is only true for right triangles.
leg
Example: From a normal deck of 52 cards, what is the probability of choosing the queen of clubs?
The deck contains only one queen of clubs, so the probability will be 1/52.
L
a
a2
b2
b