FIRE UP!!

Muddiest Point….

What concept(s) on over the last 2 classes and your last quiz were you unsure about?

On the ½ sheet of paper, finish this sentence….

I was clear on everything from the past lessons, except…

FRIDAY

Linear Function Review

Sections 2-2, 2-3, 2-4Pages 82-103

Objectives

I can find slope by these methods: Between 2 points From a graph (rise over run)

I can graph a linear line using slope and y-interceptI can manually graph pointsI can write an equation for a linear line

Slope – The BIG Picture

The slope of a line is the steepness of the line. Slope can be positive, negative, zero, or undefined.We use the little letter (m) to represent slope in an equation y= mx + b

4 Types of Slope

PositiveNegativeNo SlopeUndefined Slope

See examples next slides

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4

3

2

7

56

8

9

x-axis

y-axis

0

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-4

-3

-2

-1

-7

-5

-6

-8

-9

0

-1

Positive Slope

m > 0

Goes UP a Mountain

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2

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9

x-axis

y-axis

0

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-4

-3

-2

-1

-7

-5

-6

-8

-9

0

-1

Negative Slope

m < 0

Goes DOWN a Mountain

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y-axis

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-3

-2

-1

-7

-5

-6

-8

-9

0

-1

Zero Slope

m = 0

Horizontal Line

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y-axis

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-3

-2

-1

-7

-5

-6

-8

-9

0

-1

Undefined Slope

m = #/0

Vertical Line

Real World Slope Problems

What REAL World things would not work correctly without slope?

Ramps for Various Purposes

The Roof on Buildings

Decking & Roads

Slope

Slope can really be defined as the vertical change divided by the horizontal change. (Rise over Run)The slope of a line passing through two points (x1, y1), and (x2, y2) can be found using the following formula”

m = = 12

12

xx

yy

Run

Rise

GUIDED PRACTICE

Let (x1, y1) = (0, 3) and (x2, y2) = (4, 8).

m =y2 – y1

x2 – x1=

8 – 3 4 – 0 =

54

Find the slope of the line passing through the given points.

SOLUTION

3. (0, 3), (4, 8)

for Examples 1 and 2

ANSWER 54

GUIDED PRACTICE

Let (x1, y1) = (– 5, 1) and (x2, y2) = (5, – 4)

m =y2 – y1

x2 – x1=

4. (– 5, 1), (5, – 4)

SOLUTION

– 4 – 1 5 – (–5) =

12

–

for Examples 1 and 2

ANSWER 12

–

GUIDED PRACTICE

Let (x1, y1) = (– 3, – 2) and (x2, y2) = (6, -2).

m =y2 – y1

x2 – x1=

-2 –( – 2) 6 – (–3) =

09

5. (– 3, – 2), (6, -2)

SOLUTION

for Examples 1 and 2

ANSWER = 0

GUIDED PRACTICE

Let (x1, y1) = (7, 3) and (x2, y2) = (7, -1).

m =y2 – y1

x2 – x1=

6. (7, 3), (7, -1)

SOLUTION

-1 – 3 7 – 7 = 4

0–

for Examples 1 and 2

ANSWER = Undefined

Finding Slope on a Graph or Real Object

You can also find the slope of a graphed line or real object by using rise/runPick two points on the graph or object and then look at the rise and run between the pointsThe units must be the same for rise and run

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-1

-7

13

m = -7/13

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-1

Rise

9

Run

10

m = 9/10

Slope Intercept Format

Recall from Algebra-1

y = mx + bm is the slopeb is the y-intercept value

Looking at X and Y-Intercept

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

-7

-6-5

-4

-3

-2-1

12

3

4

56

7

0,4

3,0

Finding x and y Intercepts

The x-intercept is the x-coordinate of the point it crosses the x-axis.Likewise, the y-intercept is the y-coordinate of the point crossing the y-axis.The x-intercept is the value of x when y=0The y-intercept is the value of y when x=0

Example 15x – 3y = 15 (Find x & y intercepts)x intercept is when y=05x – 3(0) = 155x =15x = 3 (So the x intercept is (3,0)y intercept is when x=05(0) – 3y = 15-3y = 15y = -5 (So the y intercept is (0,-5)

Graphing Example 1

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

-7

-6-5

-4

-3

-2-1

12

3

4

56

7

0,-5

3,0

Graphing with slopeIf we know the slope of a line (m) and at least one ordered pair on the line (x1, y1), then we can graph the line.First: Plot the known pointSecond: Use the slope (rise over run) to find more pointsLast: Connect the points with a straight line

Graph: m = -4, (3,4)

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

-7

-6-5

-4

-3

-2-1

12

3

4

56

7

Graph line thru Point (-2,-5) with slope of 3/5

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

-7

-6-5

-4

-3

-2-1

12

3

4

56

7

Graphing with Slope-Intercept

Another method to graph quickly is to get the equation in Slope-Intercept FormatThis gives us the slope (m) and Y-Intercept (b)

Slope Intercept Formy = mx + bm is the slope of the lineb is the y-intercept point

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-1

y=-3x + 2

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-1

y=2/3x - 3

Constant Linear Lines

What do these look like???

4y 3x

Getting Slope-Intercept Format

Many times the equation is not in Slope-Intercept Format y = mx + bThe goal is to get y all by itself on the left side of the equation.Lets do some examples.

2x + 3y = 9

2x + 3y = 9 (Write equation)3y = -2x + 9 (Move 2x to the right)y = -2/3x + 3 (Divide by 3)

3x = 4y + 12

3x = 4y + 12 (Write equation)4y + 12 = 3x (Flip equation)4y = 3x - 12 (Move 12 to the right)y = 3/4x – 3 (Divide by 4)

Parallel Lines Review

Parallel Lines have the SAME SLOPE

m1 = m2.

Perpendicular Lines Review

Perpendicular Lines have NEGATIVE RECIPROCAL SLOPES

21

1

mm

Finding New Slopes

Given y = -3x + 4

What is the parallel slope?-3What is the perpendicular slope?1/3

Writing an Equation

Need slope (m)Need y-intercept (b)

Then just plug in:

y = mx + b

ExampleFind the slope intercept form of the line with a slope of –2/3 and passes through point (-6,1)y = mx + b1 = -2/3(-6) + b1 = 4 + bb = -3y = -2/3x -3

Homework

WS 1-3