Math Fundamentals Review #2

Math Fundamentals Review

In order to prepare for the NEXT unit, we are going to review:

• Plugging in BOTH x- and y-values

• Finding the slope of a line using the slope formula

• Finding the slope using RISE/RUN

• Graphing Coordinate Points

Plugging in x- and
y-values

y= 3x + 2

• y= 3(1) + 2

• y= 5

• y= 3(2) + 2

• y= 8

Solving one-step equations

• Use an inverse operation to get to the variable by itself

• Check your answer

Multiple Select

Check all that apply:

2.5 + A = 23.4

• the inverse operation of addition is subtraction

• + 2.5 on both sides

• -2.5 on both sides

• A = 21.1

• Check your answer by replacing the variable with its value

Solve two-step equations

• What are you solving for?

• + or - to isolate the variable term.

• × or ÷ to solve for the variable.

Multiple Select

Check all that apply:

5v + 4 = -26

• First step: +26 on both sides

• First step: -4 on both sides

• Second step: +26 on both sides

• Second step: -5 on both sides

• Second step: divide by 5 on both sides

Variables on both sides

Move all variable terms to one side

Multiple Choice

How to move all variables to one side?

2x - 4 = x + 5

• +4 on both sides

• -5 on both sides

• +2x on both sides

• -x on both sides

Multiple Select

How to move all variables to one side?

3a + 2 = 8a + 9

• -2 on both sides

• -9 on both sides

• -8a on both sides

• +8a on both sides

• -3a on both sides

Function Notation

• f(x) = 2x + 3, Find f(2)

• f(2) means you are replacing any x's with the number 2 and then evaluating

• 2(2)+3 f(2) = 7

• Always put parenthesis around the number you are subbing

Multiple Choice

g(x) = 3x + 1
g(2) = _______

• 6
• 7
• 8
• 9

Multiple Choice

Solve
4 = 2 + x

• x = 6
• x = 2
• x = 8
• x =-2

Multiple Choice

Solve
12 = y + 5

• y = 17
• y = -7
• y = 8
• y = 7

Multiple Choice

Solve

$$x-5=-8$$  

• x= -3

• x=13

• x= -13

• x= 3

Math Response

Given h(x) = 5 - 9x, find h(8). 

Type answer here

Correct Answer: -67

Deg^°

Rad

Multiple Choice

Evaluate f(-5) if f(x) = -2x - 12

• -22
• -2
• -19
• -5

Multiple Choice

If f(x) = 7, what is the value of x?

f(x) = 3x+1

• 2

• $$\frac{8}{3}$$

• 3

• 8

Slope Formula

Finding the Slope of a line given a graph or two points.

Slope & the
Coordinate Plane

Today we will be looking at types of slope and how to find slope from a graph, from a table and from two point using the slope formula, and from equations both standard and slope-intercept form.

​Dr. Tom Giles

You can use any two points on a line to find the slope

Types of slope

There are four main types of slope

=Positive Slope

=Negative Slope

=Zero Slope(y= line)

=Undefined Slope(x= line)

Multiple Choice

How many points do you need to find the slope of a line?

• 1

• 2

• 3

• 4

Multiple Choice

What type of slope is shown by the stairs in the image?

• positive

• negative

• undefined

• zero

Multiple Choice

What type of slope is shown in the image?

• Positive

• Negative

• Undefined

• Zero

Multiple Choice

What type of slope is Mario running in the picture?

• Positive

• Negative

• Undefined

• Zero

Step by Step:

• 1. Label the Points 1 and 2 (Slope is ALWAYS the same, no matter which point you pick to be the “first” or “second”.)

• 2. Label the X and Y for each point

• 3. Write the values in the slope formula

• 4. Simplify

Multiple Choice

What type of Slope?

• Positive

• Zero

• Negative

• Undefined

Practice

Find the slope of the line that goes through the points (5, 17) and (0,-3).

Step 1: Label the Points

• First Point ------> (X₁, Y₁) = (5, 17)

• Second Point ------> (X₂, Y₂) = (0, -3)

Multiple Choice

What type of Slope?

• positive slope

• negative slope

• undefined (no) slope

• zero slope

Step 2: Extract Values

• X₁ = 5 and Y₁ = 17

• X₂ = 0 and Y₂ = -3

Multiple Choice

What type of Slope?

• positive slope

• negative slope

• undefined (no) slope

• zero slope

Step 3: Plug the values into the slope formula.

-Remember that Y values go on top and X values go on bottom. 

-Make note of the MINUS in between each point for the top and bottom

-Don't mix the first (red) and second (blue) points. 

Multiple Choice

What is the slope in the picture?

• Positive
• Negative
• Zero
• Undefined

Step 4: Simplify

Solve for m using the Order of Operations (PEMDAS)

Make sure that you subtract the numbers correctly. Use a CALCULATOR!

**Remember- subtracting a negative number is the same as adding

Multiple Choice

What type of Slope?

• Negative

• Zero

• Undefined

• Positi

Multiple Choice

Find the slope of the line that goes through the points

(−11,−5) and (1,−12)

• $$-\frac{5}{8}$$

• $$\frac{16}{11}$$

• $$\frac{1}{6}$$

• $$-\frac{7}{12}$$

What does slope measure?

Multiple Choice

Find the slope of the line that goes through the points

(−1,−2) and (−3,−4).

• 1

• 2

• 3

• 4

Multiple Choice

When using a graph to find slope, which do you do first?

• Rise

• Run

Multiple Choice

The 'rise' part in rise/run tells us which way to go on which axis?

• y-axis

• x-axis

• coordinate plane

• y-intercept

Multiple Choice

What direction do you go when you are finding the rise?

• left and right

• up and down

Multiple Choice

The 'run' part in rise/run tells us which directions to go on the x-axis?

• up and down

• diagonal

• left and right

Multiple Choice

What is the slope of this line?

• 5/2

• 2/5

• -2/5

• -5/2

Multiple Choice

What is the slope of the line?

• 2/3

• -3/2

• 3/2

• -2/3

To find the slope of a line from a table

Multiple Choice

What is the slope?

• 3
• -3
• 10
• -6

Multiple Choice

Find the rate of change shown in the table.

• -1.5 cm/week

• 1.5 cm/week

• -1/5 cm/week

• 1/5 cm/week

Multiple Choice

Find the rate of change shown in the table.

• 3/125

• 125/3

• 500/12

• -3/125

Multiple Choice

Find the average rate of change shown in the table above.

• -3 degrees F every 2 hours.

• -1.5 degrees F every hour.

• 3 degrees F every 2 hours.

• 1.5 degrees F every hour.

Finding slope from two point:

Multiple Choice

Determine the slope.

• m=1/7
• m=-1/7
• m=-7
• m=7

Multiple Choice

Find the slope of the line that passes through:
(-3, -4) and (7, 6)

• 0
• Undefined
• -1
• 1

Multiple Choice

Find the slope of the line that passes through:
(-12, -5) and (0, -8)

• 1/4
• -1/4
• 4
• -4

Multiple Choice

Find the slope of the line that passes through:
(10, 1) and (5, 2)

• 1/5
• -1/5
• 5
• -5

Identifying Slope from an Equation

Slope intercept form of an equation: y=mx+b

m=slope

b=y-intercept

Slope is always attached to the x value when in slope intercept form of a line

Multiple Choice

What does "m" represent in y=mx+b?

• y-intercept
• slope
• macaroni
• nothing

Multiple Choice

What is the slope in the equation:  y=4x+3

• 3
• 4x
• 4
• x

Multiple Choice

Which of these represents slope-intercept form?

• y - y₁ = m(x-x₁)

• Ax + By = C

• y = mx + b

Multiple Choice

What is the slope of the line with the equation
$$y\ =\ -\frac{3}{2}x\ +\ 4$$  

• $$-\frac{3}{2}$$  

• $$\frac{3}{2}$$  

• $$4$$  

• $$-4$$  

Multiple Choice

What is the slope in the equation y=2x + 3?

• 2
• 3
• 5
• Cannot be Determined 

Parallel Lines

Parallel lines are lines that have the exact same slope but different y-intercepts.

If the y-intercepts were the same, you would have the exact same line not parallel lines.

Multiple Choice

The slopes of parallel lines are...

• the same

• opposite

• always different

Multiple Choice

What slope is parallel to:  
m = ³/₄

• ⁴/₃
• ³/₄
• ^-3/₄
• ^-4/₃

Multiple Choice

Which line is parallel to
y = 2x - 7  ?

• y = -2x + 8
• y = ¹/₂x - 7 
• y = ^-1/₂x + 2
• y = 2x - 3 

Multiple Choice

y = 3x +2
y = 3x - 3
These lines are...

• Parallel
• Perpendicular
• Neither
• The same line

Perpendicular Lines

Perpendicular lines are lines that have slopes that are opposite recipricols of each other.

- opposite - signs change

- recipriols - position of the numbers change

ex: 3 and -1/3 are perpendicular slopes

Multiple Choice

Slopes of perpendicular lines are...

• opposite reciprocals.
• opposites.
• the same.
• always 7.

Multiple Choice

What is the slope of a line perpendicular to a line with slope ^-6/₅ ?

• ^-6/₅
• ^-5/6
• ⁵/₆
• ^-4/₃

Multiple Choice

Which line is perpendicular to
y = 2x - 7  ?

• y = -2x + 8
• y = ¹/₂x - 7 
• y = ^-1/₂x + 2
• y = 2x - 3 

Multiple Choice

y = ^{- 2}/₃x + 2
y = ³/₂x + 9
These lines are...

• Parallel
• Perpendicular
• Neither

Graphing Coordinate Points

​Coordinate Plane

​The coordinate plane is a two - dimensional plane formed by the intersection of the x-axis (horizontal axis) and the y-axis (vertical axis)

​

​The point where the the x-axis and y-axis intersect is referred to as the origin.

​Quadrants

​When the x-axis and y-axis intersect, they form 4 quadrants.

​

​

​Points in the Coordinate Plane

​The points in each of the 4 quadrants are identified using and ordered pair consisting of an x-coordinate and y - coordinate (x,y).

​

​Points in the Coordinate Plane

The x-coordinate represents the horizontal distance from the origin and the y-coordinate represents vertical distance from the origin.

​

Hotspot

Which line is the X-AXIS?

Multiple Choice

What is the axis of symmetry for

2x² - 20x + 7 = 0

hint: x = -b/2a

• x = -5

• x = 7/2

• x = 5

• x = -20

Hotspot

Which line is the Y-AXIS?

Graphing a Point

Always written in the form (x,y)

​

The x-value tells us to

move left or right

​

The y-value tells us to

move up or down

​

Graph (4,6)​

​

Graphing a Point

Coordinate pairs are listed (x, y). X tells you how far left (negative) or right (positive) you go from the origin. Y tells you how far up (positive) or down (negative) you go from the origin.

(-2, 3) - move 2 left and up 3

(0, -3.5) - don't move over, and go down 3.5

Cartesian Coordinates

We represent points on a cartesian plane with coordinates. The x-axis coordinate is always the first number listed.

Graphing on a Coordinate Plane

Labelling

Label the quadrants in the grid shown.

Drag labels to their correct position on the image

I

II

IV

III

Labelling

Identify the general form of coordinates found in each quadrant.

Drag labels to their correct position on the image

(+,-)

(+,+)

(-,-)

(-,+)

Fill in the Blanks

Labelling

Identify the coordinates of each point plotted.

Drag labels to their correct position on the image

(-1,6)

(-2,7)

(-5,-8)

(3,4)

(4,3)

(6,-1)

(7,-2)

(-8,-5)

Graphing

Plot the following points on the grid shown.

(-6,4)
(5,1)
(0,-7)
(2,-3)

Graphing

Plot the points on the grid shown.

(-3,5)
(9,-5)

(-10,0)

(7,2)

Labelling

Identify the coordinates of each point graphed.

Drag labels to their correct position on the image

(5,-6)

(8,0)

(-6,5)

(-5,4)

(0,8)

(4,-5)

(-3,-1)

(-1,-3)

Labelling

Identify the coordinates of each point graphed.

Drag labels to their correct position on the image

(9,7)

(-7,9)

(7,9)

(2,5)

(9,-1)

(-9,-1)

(-2,-5)

(6,-8)

(6,8)

Fill in the Blanks

Fill in the Blanks

Fill in the Blanks

Fill in the Blanks

Fill in the Blanks

Draw

Plot a point at (3,8) on the grid. Label the point A.

But for our purposes for the ACT, when graphing Inequalities -

Use Desmos

Draw

Plot a point at (-7,-2) on the grid. Label the point B.

Draw

Plot a point at (3,-5) on the grid. Label the point C.

Draw

Plot a point at (-4,9) on the grid. Label the point D.