Everything Slopes & Their Angles

Everything Slopes & Their Angles

Today (and maybe tomorrow), we'll be looking at types of slopes and how to find slopes from graphs, tables, and two points using the slope formula and from equations in both standard and slope-intercept forms.

​Dr. Tom Giles

Types of slope

There are four main types of slope

=Positive Slope

=Negative Slope

=Zero Slope(y= line)

=Undefined Slope(x= line)

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

Multiple Choice

What type of Slope?

• Positive

• Zero

• Negative

• Undefined

Multiple Choice

What type of Slope?

• positive slope

• negative slope

• undefined (no) slope

• zero slope

Multiple Choice

What type of Slope?

• positive slope

• negative slope

• undefined (no) slope

• zero slope

Multiple Choice

What is the slope in the picture?

• Positive
• Negative
• Zero
• Undefined

Multiple Choice

What type of Slope?

• Negative

• Zero

• Undefined

• Positi

What does slope measure?

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...

• always 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

That's Great if you can Graph Linear Equations, but what about:
Graphing: Quadratic Eq?
Graphing: Inequalities?

Graphing a quadratic equation in standard form

ax² + bx + c = 0

Steps to graph a quadratic equation

• Be sure it’s in the form ax² + bx + c = 0

• Identify a, b and c

• Use x = -b/2a to find the axis of symmetry

• Plug the x you found into the equation to find y, which is your vertex

• Select 2 more x’s close to one side of the vertex to plug into the equation.

• Reflect those 2 points to the other side of the axis of symmetry.

Poll

Did you read the previous slide? Be honest.

• Yes

• No

Multiple Choice

Is this equation in standard form? If not, what should you do to get it in standard form?

2x² - 20x = 7

• Yes it is in standard form

• No, move the 20x to the other side.

• No, move the 7 to the other side.

• I don't know what standard form is.

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

Fill in the Blanks

Since the axis of symmetry for

2x² - 20x -7 is x = 5

Find the vertex (5, ?)

y =

Multiple Choice

This is what you should have found so far. Click on the image to enlarge. The next step is to pick 2 x’s close to the axis of symmetry and on the same side of the axis of symmetry. What would you pick? The axis of symmetry is x = 5

• 10 & 20

• 4 & 6

• 3 & 4

• 5 & 6

Poll

If you plug in 3 & 4 for x you’ll get the two points which are now graphed. When a graph has an axis of symmetry, it means the graph is the same on both sides. Do you see that you could reflect the new points across the axis of symmetry to get two more points on the other side? Click on image to make it larger.

• Yes

• No

Open Ended

Here’s what your graph should look like. What questions do you have? Click on the image to view the graph.

Or, Use Desmos

When graphing Inequalities, be sure to know where to shade the inequalities.

What the lines look like

Where to shade

Multiple Choice

Select the correct graph for each inequality.

• A
• B
• C
• D

Multiple Choice

Do I need to flip my sign?
-3x  ≥ 12

• Yes
• No

Multiple Choice

How can I tell if an inequality will have a solid line?

• It has an equal sign.
• The symbol is greater than.
• The symbol has "or equal to."
• You just guess.

Multiple Choice

Which of the following equations match the given graph?

• y > 2x - 5
• y ≥ 7/4x + 2
• x < -5
• x > -5

Multiple Choice

Identify the inequality.

• y > 2

• y ≤ 1

• x ≥ 2

• y ≥ 1

Multiple Choice

Select the graph for y >  -2x - 3

• A
• B
• C
• D

Multiple Choice

Which is true about this graph??

• it is shaded above and has a dashed line

• it is shaded below and has a solid line

• it is shaded above and has a solid line

• it is shaded below and has a dashed line

Multiple Choice

Select the correct graph for each inequality.

• A
• B
• C
• D

Multiple Choice

Select the correct graph for each inequality.

• A
• B
• C
• _D

Multiple Choice

which equation best represents this graph?

• y < 1/2x + 1

• y < 1/2x + 2

• y >1/2x + 2

• y > 2x

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

Use Desmos