slope & y-intercept on the coordinate plane

Slope & y-intercepton the Coordinate Plane

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Day 2

Imagine trying to tell a person how to draw a line on the Cartesian Plane.

Consider this graph of the Cartesian Plane, also called a Coordinate Plane or XY-Plane.

y

x246810

-2-4-6-8-10

2 4 6 8 10-2-4-6-8-10 0

You only need a few facts about a line to completely describe it:

· Its y-intercept (where it crosses the y-axis)

"b"

· Its slope (how much it rises or falls)

"m"

· y = mx + b

The Equation of a Line

The y-intercept ("b")of a line is the point where the line intercepts the y-axis.

In this case, the y-intercept of the line is +4.

The y-intercept

This is the ordered pair (0,4).

y

x246810

-2-4-6-8-10

2 4 6 8 10-2-4-6-8-10 0

11 What is the y-intercept of this line?

b = 6 Answer(click me)

b = -4


Answer(click me)


b = 8 Answer(click me)

y

x246810

-2-4-6-8-10

2 4 6 8 10-2-4-6-8-10 0

14 What is the x-intercept of this line?

(-4,0) Answer(click me)

(3,0)


Answer(click me)


(-6,0) Answer(click me)

17 The graph of the equation x + 3y = 6 intersects the y-axis at the point whose coordinates are

A (0,2)B (0,6)C (0,18) D (6,0)

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Defining Slope on the Coordinate Plane

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"Steepness" and "Position" of a Line

y

x

246810

-2-4-6-8-10

2 4 6 8 10-2-4-6-8-10 0

An infinite number of lines can pass through the same location on the y-axis...they all have the same y-intercept.

Examples of lines with a y-intercept of ____ are shown on this graph. What's the difference between them (other than their color)?

Consider this...

y

x246810

-2-4-6-8-10

2 4 6 8 10-2-4-6-8-10 0

The lines all have a different slope.

Slope is the steepness of a line.

Compare the steepness of the lines on the right.

Slope can also be thought of as the rate of change.

The Slope of a Line

y

x246810

-2-4-6-8-10

2 4 6 8 10-2-4-6-8-10 0

The red line has a positive slope, since the line rises from left to the right.

The Slope of a Linerun

rise

y

x

246810

-2-4-6-8-10

2 4 6 8 10-2-4-6-8-10 0

The orange line has a negative slope, since the line falls down from left to the right.

The Slope of a Line

riserun

y

x

246810

-2-4-6-8-10

2 4 6 8 10-2-4-6-8-10 0

The purple line has a slope of zero, since it doesn't rise at all as you go from left to right on the x-axis.

The Slope of a Line

The Slope of a Line

The black line is a vertical line. It has an undefined slope, since it doesn't run at all as you go from the bottom to the top on the y-axis.

rise 0

= undefined

18 Is the slope of the following graph positive, negative, zero, or undefined?

A positiveB negativeC zeroD undefined

22 Is the slope of the following graph positive, negative, zero,or undefined


23


Is the slope of the following graph positive, negative, zero, or undefined?

25


Is the slope of the following graph positive, negative, zero, or undefined?

While we can quickly see if the slope of a line is positive, negative or zero...we also need to determine how much slope it has...we have to measure the slope of a line.

Measuring the Slope of a Line

The slope of the line is just the ratio of its rise over its run.

The symbol for slope is "m".

So the formula for slope is:


slope = riserun

The slope is the same anywhere on a line, so it can be measured anywhere on the line.


slope = riserun

Keep in mind the direction:· Up (+) Down (-)· Right (+) Left (-)

For instance, in this case we measure the slope by using a run from x = 0 to x = +6: a run of 6.

During that run, the line rises from y = 0 to y = 8: a rise of 8.


slope = riserunm = 8

6m = 4

3

But we get the same result with a run from x = 0 to x = +3: a run of 3.




3

But we can also start at x = 3 and run to x = 6 : a run of 3.



3


But we can also start at x = -6 and run to x = 0: a run of 6.

During that run, the line rises from y = -8 to y = 0: a rise of 8.


slope = riserun

m = 86

m = 43

How is the slope different on this coordinate plane?

The line rises 8, however the run goes left 6 (negative). Therefore, it is said to have a negative slope


slope = riserun

m = 8-6

m = -4 3

*most often the negative sign is placed in the numerator

26 What's the slope of this line?

A -20B 0C 20D undefined


32 In the diagram below, what is the slope of the line passing throughpoints A and B?

A -2B 2C -1/2D 1/2

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011

A

B

.

33 A straight line with slope 5 contains the points (1,2) and (3,K). Find the value of K. [The use of the accompanying grid is optional.]

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Tables and Slope

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x y-3 -10 53 11

How can slope and the y-intercept be found within the table?

· Look for the change in the y-values· Look for the change in the x-values· Write as a ratio (simpified) - this will be the "slope"· Determine the corresponding y-value to the x-value of 0 - this will be the "y-intercept"

x y-3 -10 53 11

+6

+6

+3

+3

63 = 2 is the slope 5 is the y-intercept

-4 is the y-intercept is the slope

x y5 -50 -4-5 -3

Determine the slope and y-intercept from this table.

click to reveal answer

34

x y-2 50 32 1

What is the slope?

35

x y-1 -121 -43 4

What is the y-intercept? You may have to do a little extra work for this one...

36

x y-1 -121 -43 4

What is the slope?

37

x y-3 10-2 8-1 6

What is the y-intercept? You may have to do a little extra work for this one...

38

x y-3 10-2 8-1 6

What is the slope?

slope & y-intercept on the coordinate plane

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