Transcript Slope St AIR

Hitting the Slopes
An adventure on the bunny hill
of linear equations!
Start
What would you like to learn about?
1
Calculating the
slope of the line,
given two points
2
Solving for y
(slope-intercept form)
and graphing
the line
3
Determining the
equation of the line
from the graph
The definition of slope
Given two points, (x1,y1) and (x2,y2),
the slope of a line is
determined by this equation:
y 2  y1
m
x 2  x1
where m = slope.
We think of slope as the change in y
divided by the change in x.
Look at an
example
Example of finding slope.
Find the slope of the line between the points (0,6) and (5,10).
y 2  y1
1. Remember the formula. m  x  x
2. Identify which numbers represent the given variables.
2
x1  0
y1  6
x2  5
1
y2  10
3. Substitute into the formula and simplify.
m
10  6
50
m
4
5
The slope of the line extending through
4
the points (0,6) and (5,10) is .
5
Take a Quiz
Definition of Slope Quiz
Click on the correct answer below:
Slope is a number that represents the sum of y divided by the sum of x
Slope is a number that represents the change in x divided by the change in y
Slope is a number that represents the change in y divided by the change in x
Hooray!
You are correct, congratulations!
Slope is a number that represents the change in y
divided by the change in x.
rise
We can also think of it as:
run
Since the y-axis is the vertical axis on the
coordinate grid, we think of the change in y as
“rising” and since the x-axis is the horizontal axis,
we think of the change in x as “running”.
Try
calculating
slope
Oops!
Remember, slope is defined as the change
in y divided by the change in x. It is
important to understand that change
means difference. In math, the term
difference tells us to subtract. The answer
you chose contained the word “sum”,
which means to add.
Still a bit confused,
take me back
to the definition.
I got it!
Take the quiz
Again.
Oops!
You are close, but let’s recall the definition:
Slope is the change in y divided by the change
rise
in x. We think of slope as run . Since the y-axis is
the vertical axis on the coordinate grid, we think
of the change in y as “rising”. Since the x-axis is
the horizontal axis on the coordinate grid, we
think of the change in x as “running”.
So, it’s important to rise first and then run!
Still a bit confused,
take me back
to the definition.
I got it!
Take the quiz
Again.
Calculating Slope
What is the slope of the line
between (5,2) and (10,1)?
1

5
5
2
18
*Be sure to click on the blue part of the button, not in the white box*
Congratulations!
Yes, the correct slope is
1
 .
5
In this case, the slope was in fraction form, but
sometimes this won’t always be the case.
If, for example, the slope of a line was 4. We
should think of it as a fraction:
4
1
Uh Oh!
Let’s recall the steps to finding slope:
y y
m

1. Remember the formula.
x x
2. Identify which numbers represent the given
variables. x1  5 y1  2 x 2  10 y2  1
3. Finally, substitute the numbers into the formula
and simplify!
Still confused,
take me back
to the beginning.
2
1
2
1
I got it!
Return to the quiz.
The Graphs of Slopes
Let’s graph the
previous examples.
Example 1:
The slope between (0,6) and
4
(5,10) is .
5
Example 2:
The slope between (5,2) and
(10,1) is  1 .
Example 3: 5
The slope between (-1,2) and
(-2,-2) is 4.
Plotting Points & Graphing Lines
3.
run 5
run 1
run -5
Take a quiz
rise 1
rise 4
2.
In order to graph the
lines, we must first plot
each of the points.
Then draw a line
through the points,
adding arrows at each
end to represent a line
(rather than a line
segment).
Now, start at the bottom
of the two points. If you
use the slope to “rise
and run”, you should
end at the second point.
rise 4
1.
Graph Quiz
Given the slope, click on the correct graph:
3
m
7
*Be sure to click on the blue button, not the graph when choosing an answer*
You’re Right!
Yes, if you start at the bottom point, then rise 3
and run 7 you will end at the other point!
Think about this…
You could also start at the top point, run -3 and
rise -7 (which means go down three and left 7).
3 3
This works because 7   7 . Both fractions simplify
3
to 7 .
Not Quite!
Since all of the points were already plotted and
the lines were graphed, we just need to focus
on how to get from one point to another.
Remember to rise from the bottom point and
then run to the other point. The number of
places you “rise” is the numerator of the slope.
The number of places you “run” is the
denominator. In the quiz, remember to run 3
and rise 7.
I’m still a bit confused,
return to the example.
I got it!
Take the quiz again.
Where you see Slope
outside of the Classroom
Engineers & carpenters consider slope
when determining the pitch of a roof (or how
steep it is). A pitch of 8/12 means that the
roof rises 8” for every 12” (or 1’) it runs.
run 12’’
See more
examples
Where you see Slope
outside of the Classroom
Ski hills have a variety of slopes. The
incline of slopes vary from gentle, like a
bunny hill, to steep, like a black diamond.
Congratulations, you have successfully
completed Part 1! What is your next step?
Stop
(I will complete the next section(s) another time)
Continue to next section
(Part 2-solving for y and graphing)
Equations in Slope-Intercept Form
y  3x  3
slope
3
y  x2
4
y-intercept
y-intercept
y  2x  4
slope y-intercept
slope
The three equations above are all in slope-intercept form.
The slope is the number in front of the x-variable,
and the y-intercept is the number after the x-variable.
Slope-Intercept Form
An equation is written in slope-intecept
form if it is of the form:
y=mx+b
• The equation must be solved for y.
• m is the slope of the line, we think of slope as a
rise
m

fraction,
.
run
• The y-intercept is represented by the variable, b.
◦ b can be positive (y=mx+b), negative (y=mx-b or y=mx+-b), or
zero (y=mx)
• The y-intercept is the point (0,b), where the graph of
the line crosses the y-axis.
Solving for y.
If an equation is not in y-intercept form, follow
these steps to solve for y:
1.
2.
3.
4.
Is y positive?
Is y by itself?
Is y on the left side of the equal sign?
Is the equation exactly in y=mx+b form?
If you answered no to any of the questions,
manipulate the equation so the answer
becomes yes, and then move on to the next
question.
Take a Quiz
Slope-Intercept Form Quiz
Which of the following equations are in slopeintercept form?
a)
y
5
x2
4
b) y  2x  4
c) y  7x  2
a
a and b
d)
a and c
2
a and d
5
xy0
3
All of the above
Absolutely!!
5
y  x2
4
and
y  7x  2
Correct, in both equations y is:
• positive
• by itself
• on the left of the equal sign
• and, in the form of y=mx+b
You’re So
Close!
5
You are correct that y  4 x  2 is in slopeintercept form. However, this is not the only
equation that is. Ask yourself the 4 key
questions again:
1.
2.
3.
4.
Is y positive?
Is y by itself?
Is y on the left side of the equal sign?
Is the equation exactly in y=mx+b
form?
What other equation(s) can you answer yes to
all four questions?
Still confused, return to
examples & explanation.
I got it!
Take the quiz again.
Not Quite
Let’s examine the four key questions again:
1.
Is y positive?
Yes, y is positive in each equation.
2.
Is y by itself?
5
Take a look at each equation again y  4 x  2 and y  2x  4. Is y all alone on
one side of the equal sign? NO! In which equation is this false?
3.
Is y on the left side of the equal sign?
Yes, in both equations y is on the left side of the equal sign.
4.
Is the equation exactly in y=mx+b form?
No, both equations are not in slope-intercept form. Which one isn’t?
Still confused, return to
examples & explanation.
I got it!
Take the quiz again.
Not Quite
Let’s examine the four key questions again:
1.
Is y positive?
Yes, y is positive in each equation.
2.
Is y by itself?
5
5
Take a look at each equation again y  x  2 and 2  x  y  0. Is y all alone
3
4
on one side of the equal sign? NO! In which equation is this false?
3.
Is y on the left side of the equal sign?
Yes, in both equations y is on the left side of the equal sign.
4.
Is the equation exactly in y=mx+b form?
No, both equations are not in slope-intercept form. Which one isn’t?
Still confused, return to
examples & explanation.
I got it!
Take the quiz again.
Uh Oh!
Although more than one equation is in
slope-intercept form, not all of the
equations are. Ask yourself the four key
questions of determining if an equation is
in slope-intercept form:
1.
2.
3.
4.
Is y positive?
Is y by itself?
Is y on the left side of the equal sign?
Is the equation exactly in y=mx+b
form?
Still confused, return to
examples & explanation.
I got it!
Take the quiz again.
Slope-Intercept Form
Remember, an equation is written in slope-intecept
form if it is of the form:
y=mx+b
• The equation must be solved for y.
• m is the slope of the line, we think of slope as a
rise
m

fraction,
.
run
• The y-intercept is represented by the variable, b.
◦ b can be positive (y=mx+b), negative (y=mx-b or y=mx+-b), or
zero (y=mx)
• The y-intercept is the point (0,b), where the graph of
the line crosses the y-axis.
Graphing Lines from equations in
Slope-Intercept Form
Once an equation is in slope-intercept form,
graphing the line is a breeze!
y=mx+b
First, use the y-intercept to plot
the point (0,b). This shows where
the line crosses the y-axis (where
the x-value is zero).
Next, use the slope to “rise and
run” to another point on the graph.
See an
example
Example of Graphing
y  3x  3
1. Identify the y-intercept.
The number after the x
is +3, so the y-intercept
is a positive 3. The
graph intersects the yaxis at (0,3).
3. Plot the y-intercept, then use the slope
to find another point.
Plot (0,3) on the graph. Then, from (0,3)
rise -3 (go down 3) and run 1 (go right 1).
Next, connect the points to draw a line.
2. Identify the slope and
think of it as a fraction.
The number before the
x is -3, so the slope of
the line is -3. We think
of it as:
3
1
Take a Quiz
Graph Quiz
Which of the following is the correct process for
3
graphing the equation y  x  2 ?
4
Oh So Smart!
Absolutely! You chose the correct sequence of
3
y

x2 .
steps to plot the line
4
1. First, plot the y-intercept (0,-2)
2. Second, use the slope to “rise 3 and run 4”
3. Finally, connect the points with a line.
(Remember a line extends forever in both
directions, so arrow heads are required!)
Not Quite!
It is very tempting to start at the origin
(0,0) and plot the slope from there. But
remember, the origin isn’t always a point
on a graph. In this case, the line does not
pass through the origin so you cannot use
it as your starting point! The y-intercept
(0,-2) should be your starting point.
Still confused, return to
examples & explanation.
I got it!
Take the quiz again.
Congratulations, you have successfully
completed Part 2! What is your next step?
Stop
(I will complete the next section(s) another time)
Continue to next section
(Part 3- finding the equation from the graph)
Review the previous section
(Part 1- calculating slope between two points)
Review
So far you have learned…
• how to calculate the slope between two points
y y
m
using x  x .
• that the y-intercept is the point, (0,b) where a
graph of a line crosses the y-axis.
• using the slope to “rise & run” from the yintercept will give another point on the graph,
and by connecting the two points the line can be
drawn.
2
1
2
1
Determining the Equation
from the Graph of the Line
Now, using what you have learned
previously, you will learn how to determine
an equation from a graph.
Finding the Equation in
Slope-Intercept Form
1. Identify the two points plotted on the
graph.
(-4,-1) and (2,8)
2. Using the formula from Part 1Calculating Slope from Two Points,
find the slope of the line.
m
y 2  y1
x 2  x1
m
8  1
2  4
m
8 1
24
m
9
6
m
3
2
3. Look at the graph to determine where
the line intersects the y-axis.
The line & y-axis intersect at (0,5)
4. Substitute into the slope-intercept
formula
y  mx  b
y
3
x5
2
Take a Quiz
Equation Quiz
Which equation matches the graph of the line:
y  3x  5
y  3x  2
y  2x  1
1
y   x 1
2
You’re Great!
Yes, the two points plotted on the graph are (-2,2)
and (6, -2). By calculating the slope, you find:
y 2  y1
m
x 2  x1
m
22
6  2
m
4
8
m
1
2
and the y-intercept is (0,1). So, by substituting the
slope and y-intercept, you find the equation:
1
y   x 1
2
Uh Oh!
Review the steps necessary to find the
equation of the line:
1. Identify the two points plotted on the graph.
2. Using the formula from Part 1-Calculating Slope from Two Points, find the
slope of the line.
3. Look at the graph to determine where the line intersects the y-axis.
4. Substitute into the slope-intercept formula
Still confused, return to
examples & explanation.
I got it!
Take the quiz again.
Congratulations!
You have successfully completed this section. Think back to the barn
example. Let’s determine the equation for the slope of the left pitch of
the barn roof, assuming the peak of the barn roof is the point (0,0).
Since the roof peaks at (0,0) we can substitute the y-intercept is
y=mx+0. From the peak, for every 8” down (rise of -8), there is a run of
12” back (run -12). Let’s substitute the slope: y   8 x  0
 12
Now simplify the equation:
y
8
x0
12
y
2
x0
3
x-axis
y
2
x
3
y-axis
y
2
x
3
Congratulations, you have successfully
completed Part 3! What is your next step?
Stop-I’m the Slope Master!
(I have completed all three sections)
Review the previous section
(Part 2- solving for y and graphing the line)
Review the first section
(Part 1- calculating slope between two points)