Chapter 4
Graphing Linear Equations and Functions
4.1 Coordinates and Scatter plots on the calculator:
On the graph paper below please put the following items: x and y axis,
origin,quadrant numbering system, and graph an ordered pair in each quadrant using
(x,y).
Who discovered the coordinate system?_____________________________
The graph of a scatter plot can have three kinds of correlation. Name them, and
describe that each one looks like.( Or draw a sketch of each one)
________________ __________________
________________
What is an outliers within a set of data ?____________________________
Now turn to page 205 in your text book and construct a scatter plot with your
graphing calculator using the data in the “snowmobile” problem at the top of the
page.
Remember there are several things you have to do:
1) Go to “list” and input your data
2) Go to”Stat Plot” and set up your information for a scatter plot
3) Go to the “window” and set both the x and y ranges
4) Press “Graph”
Pg. 209 in your textbook will help you set up your scatterplot if you have forgotten
how to do it from last year even with the directions given above.
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Quick Review:
Name this form of a linear function 2x +4y=12_______________________________
Name this form of a linear function y= 2x –5_________________________________
Can you describe what a solution , to either one of the above equations, looks like?
___________________________________________________________________
Can you explain why these equations have 2 different variables?_______________
___________________________________________________________________
What would you put the solutions, to either one of the equations above,on?_______
4.2 Graphing Linear Equations:
Linear Equations in one variable
(3,3) , ( 3.-2), ( 3,-1) , ( 3,0) , ( 3,1) , (3, 2)
Graph these ordered pairs here!
AND write the coordinate next to each
point you graph.
Label with x and Y axes
1) What is unique about these ordered
pairs?______________________________
2)The graph of these ordered pairs is what
kind of line?_______________________
3) What is the equation of this line?_______
4) How would you describe this line?________________________________
5) This line is parallel to which axis?________________
6) Where do you find the graph of an ordered pair when the x coordinate is
zero?_____________________________
7) Where do you find the graph of an ordered pair when the y coordinate is
zero?_____________________________________________________
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( 2, -3), ( 4, -3), ( 0, -3), ( 1, -3), ( -2,-3), (-3, -3)
Graph these ordered pairs here!
AND write the coordinate next to each
point you graph. Label with x and Y axes
1) What is unique about these ordered
pairs?______________________________
2)The graph of these ordered pairs is what
kind of line?_______________________
3) What is the equation of this line?_______
4) How would you describe this line?______
___________________________________
5) This line is parallel to which axis?______
Now graph the following equations of a line on the graph provided, and
give me the coordinates of the point of intersection.
x = -4, y = 3
x=0, y=1
Point of intersection__________
Describe the graph of the line x=– 4
___________________________
Point of intersection___________
Describe the graph of the line y = 1
_______________________________
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4.2 Graphing Linear Equations in Two Variables:
One way to find solutions to a linear equation so you can graph it is to make a
table of values. You can use it from either standard form or slope-intercept form.
You can select ____________values for “x” .
Y = 2x + 1
4x + 2y =1
X
X
Y
Y
You can check to see if an ordered pair is a solution of a linear equation by
substitution the X and Y coordinates in the equation. Then check to see if the
equality is true.
Which of the points is on the graph of x + 3y =6 ?
see!
(1,2) or ( -2,
8
)
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Check and
Remember the graph of an equation is a visual model of the equation.
a) Sketch the graph of y = x and y = –x on the same coordinate plane.
What do you notice about the two graphs?____________________________
Describe the direction of the graph of each line.________________________
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b) Complete the table of values for Y = –2x +5
X
-3
2
–1
0
1
2
3
Y
4.3 Quick graphs Using intercepts
Intercepts are ?_______________________________________________
What is unique about the ordered pair of each intercept?
____________________________________________________________
Name the two intercepts for a linear equation.________________________
and _____________________________
How do you find the x-intercept of the line 2x – y =4 ? Show your work.!!!!!!!
How do you find the y-intercept of the line 2x –y =4 ? Show your work!!!!!
Is –8 the x-intercept or the y-intercept of the line y = x + 4 ? Prove it.
Is the y-intercept of the line y = 6x –4 equal to 6 or 4 ? Justify your answer with
work.
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4.4 Slope of the line
The slope of the line is a ratio of the rise of the line .
the run of the line
The formula for slope of a line is :______________________________
What do the x1, x2 , y1, y2 mean
___________________________________________________________________
There are four types of slope:
1) Find the slope of ( 2 , 0) and ( 3, 1)
2) Find the slope of ( -1, 2) and ( 2 ,2)
Describe the slope ________________
Describe the slope ________________
3) Find the slope of ( 0, 0 ) and ( 1, – 1)
4) Find the slope of ( 2 , 1) and ( 2 , 4)
Describe the slope ________________
Describe the slope ________________
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• Slope is represented by the letter “m”. What word do you think it stands for?
(Hint: It is French.)_________________________________
• Turn to page 229 and look at example 6. A ___________ rate of change is also
called slope. Find the constant rate of change for that example without LOOKING
at the answer. No peeking!!!!
• On page 231 find the slope of #47 and #48. Then check with your neighbor to
see if they got the same answer but with different ordered pairs than you did.
4.5 Direct Variation:
Two variable quantities that have the same rate of _____, regardless of the
values of the variables , have a _________ variation. For example if you get $5
per hour, then your total_____ varies directly with the number of ________
you work. Check it out below.
Total pay (y)
Hours worked(x)
$30
$50
$100
6
10
20
$350
70
Rate of pay?(k)
A MODEL for all Direct Variations:
“y varies directly as x” =
y
x
This variation is always equal to a “k” constant.
y
=k
x
or “ y = kx ”. So in our example above “ Your total pay varies directly as your
number of hours worked. (Hint: The more hours you work the more money you get
paid.) But the rate you are paid per hour NEVER CHANGES. ( Unless you get a
raise .)
So
Total − Pay
=K
# of − hours − worked
K is the “constant” value ( your hourly rate of pay) that
does not change as the”y” and “x” values change.
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Decide if these statements are direct variations.
a) “You are riding your bike at an average speed of 12 miles per hour. The number
of miles you ride, d, during, t , hours is given by d = 12t. __________________
b) The time, t, spent waiting in line for the Super Looper roller coaster, and the
number of people, n , in the line are related by the equation t = Kn .___________
c) If “ y varies directly as x ” and y = 15 and x = 2 find the constant,k.________
Now we will apply this to the Slope of a line. Where the constant, k , is _______
of the line no matter __________ two ordered pairs you ______ from the line.
Y= the change in y 2 − y1
x = the change in x 2 − x1
• The variables x and y vary directly. When x = 15 and y = 105 write an equation
that relates x and y,________________ Find the value of the constant._____
Now find the value of y when x =12.
y=______
• Graph the equation. Find the constant of variation and the slope of the line for
y=–x
Turn to page 238 and look at the Violin Family problem.
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4.6 Quick Graphs Using Slope-Intercept Form:
To make a quick graph of a linear equation:
– x + 3y = 6
First_______________________________________________________________
Next pluck out _______________________and ____________________________
Now graph _______________________ and then use ___________over
_________ to graph the line.
Now! Do it again with the two examples below.
3x – 6y = 9
x – 2y = –3
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Give the slope and y-intercept for each of the following equations
EQUATION
SLOPE
Y- INTERCEPT
1) y = – 2x + 3
2) y = – 2x
3) y = – 2x – 1
Now graph all three on the graph below
What is unique about all three
graphs? _______________________
What does the slope have to do with this?
_______________________________
Summarize what you have learned about the slope and the graphs of these lines.
EQUATION
SLOPE
Y- INTERCEPT
1
2
1) y = x + 3
1
2
2) y = x – 1
3) y =
–1
x+2
2
Now graph all three on the graph below
What is unique about all three
graphs? _______________________
What does the slope have to do with this?
_______________________________
Summarize what you have learned about the slope and the graphs of these lines.
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4.7 Solving Linear Equations Using Graphs:
One variable equation:
0 = 2x – 8 , solve for x
Two variable equation
y = 2x – 8
Now substitute the value for “ x “ into the “ Two variable equation” and what are you
going to solve for ?_______
What intercept are we using the one variable equation to solve for? ____________
Forms of linear equations:
a) Slope intercept form ____________________
b) Y intercept form________________________
c) Solve for the x-intercept. Write the slope- intercept form of the equation as it would
look after you’ve replaced the “y” with a zero _______________. This should look
like the One Variable Equation at the top of the page.
TRUE or FALSE ? The solution of 0 = 9x - 36 is the x-intercept of y = 9x –36?
Explain your reasoning and show your work?
Now try another one.
2x –5 = 1 Solve for x. Don’t forget to check your solution back in the equation.
Write the equation in the form ax + b = c
_x+_=_
Replace the “zero” with a Y.
Y=_x+_
Now check the function by graphing it and checking the intercepts!!!
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Now do it again with
4
6
x + 3 = x and follow the steps from the last example.
3
3
Don’t forget to graph and check those intercepts,
4.8 Functions and Relations:
A relation is defined as a
set of _________ pairs (x ,y).
A function is a relation in which
NO two ordered pairs have the
same x-value.
Input
Output
1
5
1
2
2
7
2
4
3
9
3
5
4
Input
Output
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• Write the set of ordered pairs that is shown by each” Input-Output “above.
• Check the ordered pairs to see if the criteria for the relation or the function
are met?
• Graph each set of ordered pairs on the graphs on the next page.
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Now we will use the drop -line test to see if the graphs are a relation or a
function. Get out your small ruler out to use as a “vertical line tester”. Drag your
ruler across each graph, keeping it parallel to the Y axis. The question is does it
intersect only one point or more than one point at any one time??????
Vertical line test:
If the vertical line intersects with only ____ point it is a ____________
If the vertical line intersects with more than ____ point it is not a _________
but it is a ______________.
Now copy the graphs, below, in the same order I put them on the board .
Function or Relation?
___________
___________
___________
___________
X-values are called the “____________of the function( input).
Y-values are called the “ ____________ of the function (output).