Algebra Honors ( Period 4)

Chapter 3-6 Proportional and Nonproportional Relationships

This is just real world review of concepts we’ve already covered… comparing and contrasting the two types of linear relationships

SAME: both are linear—meaning they graph as lines
both are diagonal
both have a constant rate of change or slope that can be found by finding the rise/run or the difference of the y’s over the difference of the x’s

DIFFERENT:
proportional relationships go through the origin (0,0) and nonproportional do NOT

Nonproportional have a y-intercept other than 0

Proportional relationships: YOU can take any point and divide the y/x and it will equal the same value as diving any other y/x. This value  is the slope—which is the constant rate of change  VS Nonproportional relationships when you divide the y/x of a point it will NOT equal the y/x of another point. This value is NOT the slope and is NOT the CONSTANT RATE OF CHANGE

TO find the equation for anonproportional relationship
This isn’t as easy as f(x) = kx because it does not go through (0,0)

You will need to find the y intercept (0,y)

Say you find the rate of change or slope is 3 for the following 2 points

(2, 12) and (4, 18)

You can graph these two points and count the slope down to the y intercept

You can find a missing number that will make the equation work

y = 3x + ?  will make ( 2, 12) work in the equation

Plug in( 2, 12) to find b

12 = 3(2) + b

12 = 6 +b

 b = 6

You can try it with other points as well…

 It still works b = 6

So the non-proportional equation is y = 3x + 6

6 is the y intercept on the graph

Algebra ( Period 5)

Chapter 3-4 Direct Variation

We’ve learned that the unit rate is the constant rate of change in a linear relationship and that it’s the slope of a line when it’s graphed. We’ve also learned that if a graph of an equation goes through the origin (0,0)  it’s proportional  and the ratio of any y value to it’s x value is a constant (which turns out to be the unit rate or constant rate of change or slope of the line)

When the linear relationship is proportional, we say it’s a DIRECT VARIATION. Now the constant rate of change, the slope, the unit rate, is called the CONSTANT OF VARIATION or the CONSTANT OF PROPORTIONALITY

This is not a new concept. IT IS  just NEW VOCAB!

We also say: y varies directly (constantly) with x.

The slope is now replaced by the letter k instead of m

Finding the equation of a line that is proportional

Find k (the slope) by counting the rise/run of the graph

Write the equation using the format  y = kx

Notice: if you always pick the origin as the point to count rise/run from—the slope (k) is always just y/x

In a word problem, if it says one amount VARIES DIRECTLY with another, you know that the origin is one of the points!!

You also know that the equation is y = kx 
YOU just need to find k
and k is y/x of any point OTHER THAN THE ORIGIN

A babysitting example

The amount of money earned  VARIES DIRECTLY with the time worked.

THINK: the graph and equation go through (0,0)

THINK: Any other point will give you the slope, or constant of proportionality, or unit rate ( all the same thing) SO you only need one additional point.

We are given that she earns $30 for 4 hours. Find the equation.

Rise/Run = y/x
BECAUSE THEY SAID IT VARIED DIRECTLY!!

k = 30/4
Simplify
k = 7.5
So the equation is y = 7.5x

What does the 7.5 represent?
The unit rate of $7.50/ hour of babysitting!

A bicycling example 
The distance the cyclist bikes in miles VARIES DIRECTLY with the time in hours that he bikes.

THINK: The graph and equation go through the origin (0,0).THINK: Any other point will give you the slope, or constant of proportionality, or unit rate (all the same thing) SO you only need one additional point.

He bikes 3 miles in ¼ hour. Find the equation.

Rise/run = y/x
BECAUSE THEY SAID IT VARIES DIRECTLY

k = 3/¼  or 3/.25 Now the hardest part is doing this 3/.25

If you kept it as 3/¼  you could read this as 3 divided by ¼
THINK: instead of dividing, multiply by the reciprocal of ¼

or 3 (4/1) = 12 (Wait, wasn’t that much easier than dividing 3 by .25!!
k = 12

The equation is y = 12x

What does the 12 represent?

The unit rate of 12 miles/ hour – that’s the cyclist’s speed 12mph

  Determining whether a Table of Values is Direct Variation If you are given a table of values, you can determine if the relationship is direct variation by dividing 3 y’s by their x values and making sure that you get the SAME value. If you do, it is proportional, goes through the origin (0,0) and the slope of y/x is the unit rate ( which is now called the constant of variation)!

Example

Given 3 points (5, 20) , (6, 24), and (7, 28):

Divide each y/x

20/5 = 4

24/6 = 4

28/7 = 4

Since all the ratios simplify to the same value (4), it is a direct variation. The slope of 4 is the unit rate, which is the constant rate of change and is now also called the constant of variation.

Finding Additional Values for the Direct Variation once you have the Equation
Once you have the equation y = kx, you can find infinite additional values (points) that will work.
For example, in the first babysitting example, the equation is y = $7.50x, which we write as y = 7.5x  If she babysits for 20 hours, how much did she earn?
x = 20

so y = 7.5(20) = 150 so She earns $150.
If she earns $750, how many hours did she need to work?

Now y = 750  so  750 = 7.5x 
It is a one-step equation and we get
x = 100 or 100 hours!

 Finding the Equation if you know 1 point and then Finding Additional Values

y varies directly with x. Write an equation for the direct variation. Then find each value

If y = 8 when x = 3, find y when x = 45

FIRST you need to find k

y = kx… In this case we have 8 = k(3) or 8 = 3k

Solve this 1 step equation—leaving it in fraction form!

8/3= k

so

y = (8/3)x

Now, find y when x = 45

y = (8/3)(45)

solve

y = 120

Applying direct variation to the Distance Formula d = rt

A jet’s distance varies directly as the hours it flies

If it traveled 3420 miles in 6 hours, how long will it take to fly 6500 miles?

k = 3420/6 = 570mph ( its speed)

6500 = 570t

t ≈11.4

about 11.4 hours

Algebra Honors ( Period 4)

Chapter 3-4 Direct Variation

We’ve learned that the unit rate is the constant rate of change in a linear relationship and that it’s the slope of a line when it’s graphed. We’ve also learned that if a graph of an equation goes through the origin (0,0)  it’s proportional  and the ratio of any y value to it’s x value is a constant (which turns out to be the unit rate or constant rate of change or slope of the line)

When the linear relationship is proportional, we say it’s a DIRECT VARIATION. Now the constant rate of change, the slope, the unit rate, is called the CONSTANT OF VARIATION or the CONSTANT OF PROPORTIONALITY

This is not a new concept. IT IS  just NEW VOCAB!

We also say: y varies directly (constantly) with x.

The slope is now replaced by the letter k instead of m

Finding the equation of a line that is proportional

Find k (the slope) by counting the rise/run of the graph

Write the equation using the format  y = kx

Notice: if you always pick the origin as the point to count rise/run from—the slope (k) is always just y/x

In a word problem, if it says one amount VARIES DIRECTLY with another, you know that the origin is one of the points!!

You also know that the equation is y = kx 
YOU just need to find k
and k is y/x of any point OTHER THAN THE ORIGIN

A babysitting example

The amount of money earned  VARIES DIRECTLY with the time worked.

THINK: the graph and equation go through (0,0)

THINK: Any other point will give you the slope, or constant of proportionality, or unit rate ( all the same thing) SO you only need one additional point.

We are given that she earns $30 for 4 hours. Find the equation.

Rise/Run = y/x
BECAUSE THEY SAID IT VARIED DIRECTLY!!

k = 30/4
Simplify
k = 7.5
So the equation is y = 7.5x

What does the 7.5 represent?
The unit rate of $7.50/ hour of babysitting!

A bicycling example
The distance the cyclist bikes in miles VARIES DIRECTLY with the time in hours that he bikes.

THINK: The graph and equation go through the origin (0,0).
THINK: Any other point will give you the slope, or constant of proportionality, or unit rate (all the same thing) SO you only need one additional point.

He bikes 3 miles in ¼ hour. Find the equation.

Rise/run = y/x
BECAUSE THEY SAID IT VARIES DIRECTLY

k = 3/¼  or 3/.25 Now the hardest part is doing this 3/.25

If you kept it as 3/¼  you could read this as 3 divided by ¼
THINK: instead of dividing, multiply by the reciprocal of ¼

or 3 (4/1) = 12 (Wait, wasn’t that much easier than dividing 3 by .25!!
k = 12

The equation is y = 12x

What does the 12 represent?

The unit rate of 12 miles/ hour – that’s the cyclist’s speed 12mph

  Determining whether a Table of Values is Direct Variation If you are given a table of values, you can determine if the relationship is direct variation by dividing 3 y’s by their x values and making sure that you get the SAME value. If you do, it is proportional, goes through the origin (0,0) and the slope of y/x is the unit rate ( which is now called the constant of variation)!

Example

Given 3 points (5, 20) , (6, 24), and (7, 28):

Divide each y/x

20/5 = 4

24/6 = 4

28/7 = 4

Since all the ratios simplify to the same value (4), it is a direct variation. The slope of 4 is the unit rate, which is the constant rate of change and is now also called the constant of variation.

Finding Additional Values for the Direct Variation once you have the Equation

Once you have the equation y = kx, you can find infinite additional values (points) that will work.
For example, in the first babysitting example, the equation is y = $7.50x, which we write as y = 7.5x  If she babysits for 20 hours, how much did she earn?
x = 20

so y = 7.5(20) = 150 so She earns $150.
If she earns $750, how many hours did she need to work?

Now y = 750  so  750 = 7.5x 
It is a one-step equation and we get
x = 100 or 100 hours!

 Finding the Equation if you know 1 point and then Finding Additional Values

y varies directly with x. Write an equation for the direct variation. Then find each value

If y = 8 when x = 3, find y when x = 45

FIRST you need to find k

y = kx… In this case we have 8 = k(3) or 8 = 3k

Solve this 1 step equation—leaving it in fraction form!

8/3= k

so

y = (8/3)x

Now, find y when x = 45

y = (8/3)(45)

solve

y = 120

Applying direct variation to the Distance Formula d = rt

A jet’s distance varies directly as the hours it flies

If it traveled 3420 miles in 6 hours, how long will it take to fly 6500 miles?

k = 3420/6 = 570mph ( its speed)

6500 = 570t

t ≈11.4

about 11.4 hours

Algebra ( Period 5)

Chapter 3-3 Rate of Change & Slope

We’ve already looked at the slope (m) of lines—today we will connect slope to the RATE of the CHANGE of the linear function (the line). the rate of change for a line is a CONSTANT… it is the same value EVERYWHERE on the line

This change, also know as the slope, is found by  finding the rise over the run between ANY 2 points.  rise/run

The rise is the change in y and the run is the change in x.

In a real world example, the rate of change is the UNIT RATE

If you are buying video games that are all the same price on BLACK FRIDAY, two data points might be

# of computer           Total
games                          cost

4                                  $156
6                                  $234

The slope or rate of change  is the  change in y/ the change in x

(234- 156)/ 6-4

78/2

or $39/ video game

Again, as long as the function is linear, or one straight line, it has a constant rate of change, or slope between ANY TWO POINTS

The constant rate of change, or slope, is the rise over the run—or the change in y over the change in x

or
y₂ – y₁/ x₂-x₁ 

Slope = rise/run ( rise over run)
=change in the y values/ change in the x values =
Difference of the y values/ Difference of the x values

Mrs Sobieraj uses “Be y’s first!” Be wise first!  meaning always start with the y vales on top (in the numerator)

TWO WAYS OF CALCULATING on a graph:

       1) Pick 2 points and use the following formula
Difference of the 2 y –values/ Difference of the 2 x-values
The formal is restated with SUBSCRIPTS on the x’s and y’s below: (memorize this) y₂ – y₁/ x₂-x₁  The subscripts just differentiate between point one and point two. You get to decide which point is point one or two. I usually try to keep the difference positive, if I can—but often, one of them will be negative and the other will be positive.

EXAMPLE:   ( 3, 6)  and (2, 4)    y₂ – y₁/ x₂-x₁       6-4/3-2 = 2/1 = 2

    2)Count the slope on the GRAPH using rise over run.
From the point (2,4) count the steps UP ( vertically) to (3,6): I get 2 steps
Now count how many steps over to the right (horizontally): 1 step
Rise = 2 and Run = 1 or 2/1 = 2

HORIZONTAL LINES  have only a y intercept (unless it’s the line y = 0 and then that is the x-axis) The equation of a horizontal line is y = b where b is a constant. Notice that there is NO X in the equation. For example y = 4 is a horizontal line parallel to the x-axis where the y value is always 4 What is the x value? All real numbers! Your points could be ( (3, 4) or ( 0, 4) or ( -10, 4)
Notice y is always 4! The constant rate of change  or slope is 0

If you take any 2 points on a horizontal line the y values will always be the same so the change ( or difference) in the numerator = 0.

EXAMPLE  y = 4

Pick any two points Let’s us ( 3,4) and (-10, 4)

(4 - 4)/ (3 - ^-10) becomes ( 4-4)/ 3 + 10 = 0/13 = 0

VERTICAL LINES ( which are NOT functions)  have only an x intercept ( unless it is the line x = 0 and then it is the y-axis) The equation of a vertical line is x = a, where a is a constant. Notice that there is NO Y in this equation.

EXAMPLE: x = 4

This is a vertical line parallel to the y axis 4 steps to the right of it. Pick any two points on this line Let’s use ( 4, -1) and (4, 7)

This time the change in y is -1  - 7 = -8

and the change in x is 4 -4 = 0

BUT -8/0 is UNDEFINED

Make sure you write undefined for the slope!

Finding a Missing Coordinate if you know 3 out of 4 values and the Slope

Say you know the following:
(1,4) and (-5, y) and the slope is given as 1/3

Find the missing y value

Use the slope formula

Change in y/ change in x

(y – 4)/- 5 – 1  and you know that the slope is 1/3

That means

(y – 4)/- 5 – 1   = 1/3

(y – 4)/-6 = 1/3

Solve

3(y -4)= -6

3y – 12 = -6

 y = 2

Or you could have divide both sides by 3 FIRST

y - 4 = -2

y = 2

Algebra Honors ( Period 4)

Chapter 3-3 Rate of Change & Slope

We’ve already looked at the slope (m) of lines—today we will connect slope to the RATE of the CHANGE of the linear function (the line). the rate of change for a line is a CONSTANT… it is the same value EVERYWHERE on the line

This change, also know as the slope, is found by  finding the rise over the run between ANY 2 points.  rise/run

The rise is the change in y and the run is the change in x.

In a real world example, the rate of change is the UNIT RATE

If you are buying video games that are all the same price on BLACK FRIDAY, two data points might be

# of computer           Total
games                          cost

4                                  $156
6                                  $234

The slope or rate of change  is the  change in y/ the change in x

(234- 156)/ 6-4

78/2

or $39/ video game

Again, as long as the function is linear, or one straight line, it has a constant rate of change, or slope between ANY TWO POINTS

The constant rate of change, or slope, is the rise over the run—or the change in y over the change in x

or
y₂ – y₁/ x₂-x₁ 

Slope = rise/run ( rise over run)
=change in the y values/ change in the x values =
Difference of the y values/ Difference of the x values

Mrs Sobieraj uses “Be y’s first!” Be wise first!  meaning always start with the y vales on top (in the numerator)

TWO WAYS OF CALCULATING on a graph:

       1) Pick 2 points and use the following formula
Difference of the 2 y –values/ Difference of the 2 x-values
The formal is restated with SUBSCRIPTS on the x’s and y’s below: (memorize this) y₂ – y₁/ x₂-x₁ 
The subscripts just differentiate between point one and point two. You get to decide which point is point one or two. I usually try to keep the difference positive, if I can—but often, one of them will be negative and the other will be positive.

EXAMPLE:   ( 3, 6)  and (2, 4)    y₂ – y₁/ x₂-x₁       6-4/3-2 = 2/1 = 2

    2)Count the slope on the GRAPH using rise over run.
From the point (2,4) count the steps UP ( vertically) to (3,6): I get 2 steps
Now count how many steps over to the right (horizontally): 1 step
Rise = 2 and Run = 1 or 2/1 = 2

HORIZONTAL LINES  have only a y intercept (unless it’s the line y = 0 and then that is the x-axis) The equation of a horizontal line is y = b where b is a constant. Notice that there is NO X in the equation. For example y = 4 is a horizontal line parallel to the x-axis where the y value is always 4 What is the x value? All real numbers! Your points could be ( (3, 4) or ( 0, 4) or ( -10, 4)
Notice y is always 4! The constant rate of change  or slope is 0

If you take any 2 points on a horizontal line the y values will always be the same so the change ( or difference) in the numerator = 0.

EXAMPLE  y = 4

Pick any two points Let’s us ( 3,4) and (-10, 4)

(4 - 4)/ (3 - ^-10) becomes ( 4-4)/ 3 + 10 = 0/13 = 0

VERTICAL LINES ( which are NOT functions)  have only an x intercept ( unless it is the line x = 0 and then it is the y-axis) The equation of a vertical line is x = a, where a is a constant. Notice that there is NO Y in this equation.

EXAMPLE: x = 4

This is a vertical line parallel to the y axis 4 steps to the right of it. Pick any two points on this line Let’s use ( 4, -1) and (4, 7)

This time the change in y is -1  - 7 = -8

and the change in x is 4 -4 = 0

BUT -8/0 is UNDEFINED

Make sure you write undefined for the slope!

Finding a Missing Coordinate if you know 3 out of 4 values and the Slope

Say you know the following:
(1,4) and (-5, y) and the slope is given as 1/3

Find the missing y value

Use the slope formula

Change in y/ change in x

(y – 4)/- 5 – 1  and you know that the slope is 1/3

That means

(y – 4)/- 5 – 1   = 1/3

(y – 4)/-6 = 1/3

Solve

3(y -4)= -6

3y – 12 = -6

 y = 2

Or you could have divide both sides by 3 FIRST

y - 4 = -2

y = 2

Math 8 ( Period 1)

Chapter 3-4 Slope-intercept Form

In a proportional linear relationship, the line ( and equation) go through the origin. When linear relationships are NOT PROPORTIONAL, the equation and graph go through a point on the y axis OTHER THAN (0,0)

The x-value will always be 0 because you are on the y-axis. The y value can be any value other than 0.

We call this the y-intercept and the variable we give this is b.
Mrs Sobieraj calls b your “Home Base” because  you will graph this point first before using the slope ( rise/run) to find additional points on the line.

y = mx + b  the most used form of a linear equation

The Slope- Intercept Form

y = mx + b

 where m = slope and b = y intercept (where this line hits the y-axis)

All you do is solve the equation for “y”  meaning isolate the y on one side of the equal sign.

For example:

-3y = -2x – 6 is NOT in slope –intercept form because y is NOT by itself.

Restate

-3y = -2x – 6 into slope intercept form
divide both sides by -3

y = (2/3)x + 2
So m = 2/3
and b = 2

If you are looking to find a point by  plugging into the equation:
Look at the coefficient for x

What x values will give you INTEGER ANSWERS for y? ( That will be easier to graph) In this example, they need to be multiples of 3. Now look at the graph of y = (2/3)x + 2
Notice that the   + 2     at the end is the y-intercept
(without having to do ANY WORK).
Use the counting method for slope on your graph.
You should have counted : UP 2 and then to the RIGHT 3

The slope therefore is 2/3 Look at the equation.
It told you 2/3 WITHOUT ANY WORK!!!

Graph when line is in SLOPE INTERCEPT form
So if you have the slope- intercept form of the equation, it is really easy to graph the line.

           1)Graph the intercept on the y-axis
    (That is the+/- constant at the end of the equation y= mx + b)

 “   2)Count” the next point by using the slope of the x-coefficient as a fraction.

For the equation
 y = (2/3)x + 2

1   1)Put a dot at (0,2)

    2)From (0,2) count UP 2 and over to the RIGHT 3 to find the next coordinate ( 3,4)

Remember slope is y over x  so the numerator is the change in y and the denominator is the change in x.

If it is positive you count UP (positive)  and over to the RIGHT (positive)
OR you can count DOWN (negative) AND to the LEFT (negative) because 2 negatives make a positive

If it is negative you count DOWN (negative) and over to the RIGHT (positive)
OR you can count UP (positive) AND to the LEFT (negative) because a positive and a negative make a negative

If you are given the slope and the y intercept, you can write the equation of any line.

Just use: y = mx + b

So if m = ¾ and b = -9

the equation is

y = (3/4)x – 9

It is easy to create the equation if you are either given BOTH m and b or you can find them by looking at the graph

 EXAMPLE:
m = -2/3 and b = -12

 The line would be y = (-2/3)x – 12

If you are given the graph of this line

Read the y- intercept by looking at the  y-axis ( 0, -12)
Count the slope from there to get another point that you can read on the graph using rise/run.

Another example:
Given 3x + 4y = 10

Solve for y
subtract 3x from both  sides
4y =-3x + 10
Now divide both sides by 4

y = (-3/4)x + 10/4
or
y = (-3/4)x + 5/2

Remember: The slope is the coefficient of the x
m = -3/4 ( so you are sliding DOWN at a little less than a 45 degree angle)

the y- intercept ( b) is the constant
b = 5/2
So the line crosses the y axis at  2½   

Notice that in this case the “b” is a fraction
That makes it harder to graph the slope from this point!
When this happens, the slope intercept form may not be the best from to graph the line

You must start at 2 ½ on the y axis and count down 3 from there and right 4. That is hard to do accurately!
Neither 3 nor 4 is a factor of the constant 10 so the intercepts will be fractions!

We need the x term to end up with ½ so that when we add that to the b ( 5/2) we will get an integer. so let’s make x = 2 and plug in

That will cross cancel with -3/4 slope to get

y = (-3/4)(2) + 5/2 = -3/2 + 5/2 carefully use your additional rules to get 2/2 = 1

So we found a coordinate that has just integers  ( 2, 1)
Now count the slope ( -3/4) from this point  instead of from the y-intercept

Real world meanings of the y intercept (b) and SLOPE (m)

Remember that the slope is still the constant rate of change, which means it is the unit rate.
They y-intercept is the initial value of the real world problem.

Make sure to watch the T-shirt video example in your McGraw Portal
In the example, the initial design of the shirt costs $20 and then each shirt costs $5 to print. Even before printing one shirt, you need to pay $20 so this is the initial value of the y-intercept (b)
The unit rate of constant change is the slope of $5/shirt.

Algebra ( Period 5)

Chapter 3-2 Solving Linear Equations by Graphing

We need LOTS of graph paper. YOU must graph on graph paper—using a ruler or a straight edge! Make sure to label your x and y axes!  Put arrows on them!

Linear function: A line in the format:

f(x) = x or y = x

This is called the PARENT GRAPH

This parent graph has a FAMILY of GRAPHS related to it that has similar characteristics but is in someway different 
The Slope is Different
The y intercept is Different

The ROOT of the function or line is the X – intercept of the graph  It is called a ZERO on the graphing calculators. To find the root, find the value of x that makes the equation true when the y value is 0.

Linear functions ( equations) have at most 1  ROOT ( solution) because once the line intercepts the  x-axis it cannot curve around and intercept it again

Again, the FUNCTION is the ENTIRE GRAPH … and is in the form f(x) = x   or y = x

The linear EQUATION related to the function is only concerned with one value, the x- intercept.  The y value would be ZERO on the x-axis so you set the function = 0 and solve!

Here are the synonyms:
x-intercept= the solution = the root = the zero

 If you graph the function
f(x) = 2x – 8, you will see the x intercept is ( 4, 0)

Therefore, the solution, the root, the zero of the function is 4.

set f(x)  or y = 0
2x – 8 = 0
2x = 8
x = 4

You can solve a linear function 2 ways:

Graphically- Graph the function and read the x-intercept
Algebraically- Set y or f(x) equal to 0 and solve for x

Notice you are solving multi-step equation but now the value means that you found the root of the function. How often will you be able to read the exact answer from a graph? NOT OFTEN!

Therefore we usually solve for the root Algebraically!

Example:
Find the root or zero of y = 20- .75x

set y = 0

0 = 20 - .75x

x = 26 2/3

Algebra Honors (Period 4)

Chapter 3-2 Solving Linear Equations by Graphing

We need LOTS of graph paper. YOU must graph on graph paper—using a ruler or a straight edge! Make sure to label your x and y axes!  Put arrows on them!

Linear function: A line in the format:

f(x) = x or y = x

This is called the PARENT GRAPH

This parent graph has a FAMILY of GRAPHS related to it that has similar characteristics but is in someway different
The Slope is Different
The y intercept is Different

The ROOT of the function or line is the X – intercept of the graph  It is called a ZERO on the graphing calculators. To find the root, find the value of x that makes the equation true when the y value is 0.

Linear functions ( equations) have at most 1  ROOT ( solution) because once the line intercepts the  x-axis it cannot curve around and intercept it again

Again, the FUNCTION is the ENTIRE GRAPH … and is in the form f(x) = x   or y = x

The linear EQUATION related to the function is only concerned with one value, the x- intercept.  The y value would be ZERO on the x-axis so you set the function = 0 and solve!

Here are the synonyms:
x-intercept= the solution = the root = the zero

 If you graph the function
f(x) = 2x – 8, you will see the x intercept is ( 4, 0)

Therefore, the solution, the root, the zero of the function is 4.

set f(x)  or y = 0
2x – 8 = 0
2x = 8
x = 4

You can solve a linear function 2 ways:

Graphically- Graph the function and read the x-intercept
Algebraically- Set y or f(x) equal to 0 and solve for x

Notice you are solving multi-step equation but now the value means that you found the root of the function. How often will you be able to read the exact answer from a graph?
NOT OFTEN!

Therefore we usually solve for the root Algebraically!

Example:
Find the root or zero of y = 20- .75x

set y = 0

0 = 20 - .75x

x = 26 2/3

Math 8 ( Period 1)

Chapter 3-3 Equations in the form of y = mx +b

We’ve learned that the unit rate is the constant rate of change in a linear relationship and that it’s the slope of a line when it’s graphed. We’ve also learned that if a graph of an equation goes through the origin (0,0)  it’s proportional  and the ratio of any y value to it’s x value is a constant (which turns out to be the unit rate or constant rate of change or slope of the line)

When the linear relationship is proportional, we say it’s a DIRECT VARIATION. Now the constant rate of change, the slope, the unit rate, is called the CONSTANT OF VARIATION or the CONSTANT OF PROPORTIONALITY This is not a new concept. IT is just NEW VOCAB!

We also say: y varies directly (constantly) with x.

Finding the equation of a line that is proportional

Find m (the slope) by counting the rise/run of the graph

Write the equation using the format  y = mx

Notice: if you always pick the origin as the point to count rise/run from—the slope (m) is always just y/x

In a word problem, if it says one amount VARIES DIRECTLY with another, you know that the origin is one of the points!!

You also know that the equation is y = mx 
YOU just need to find m
and m is y/x of any point OTHER THAN THE ORIGIN

A babysitting example from Page 190 in your textbook

The amount of money earned  VARIES DIRECTLY with the time worked.

THINK: the graph and equation go through (0,0)

THINK: Any other point will give you the slope, or constant of proportionality, or unit rate ( all the same thing) SO you only need one additional point.

We are given that she earns $30 for 4 hours. Find the equation.

Rise/Run = y/x
BECAUSE THEY SAID IT VARIED DIRECTLY!!

m = 30/4
Simplify
m = 7.5
So the equation is y = 7.5x

What does the 7.5 represent?
The unit rate of $7.50/ hour of babysitting!

A bicycling example from Page 191 in your textbook
The distance the cyclist bikes in miles VARIES DIRECTLY with the time in hours that he bikes.

THINK: The graph and equation go through the origin (0,0).
THINK: Any other point will give you the slope, or constant of proportionality, or unit rate (all the same thing) SO you only need one additional point.

He bikes 3 miles in ¼ hour. Find the equation.

Rise/run = y/x
BECAUSE THEY SAID IT VARIES DIRECTLY

m = 3/¼  or 3/.25 Now the hardest part is doing this 3/.25

If you kept it as 3/¼  you could read this as 3 divided by ¼
THINK: instead of dividing, multiply by the reciprocal of ¼

or 3 (4/1) = 12 (Wait, wasn’t that much easier than dividing 3 by .25!!
m = 12

The equation is y = 12x

What does the 12 represent?

The unit rate of 12 miles/ hour – that’s the cyclist’s speed 12mph

  Determining whether a Table of Values is Direct VariationIf you are given a table of values, you can determine if the relationship is direct variation by dividing 3 y’s by their x values and making sure that you get the SAME value. If you do, it is proportional, goes through the origin (0,0) and the slope of y/x is the unit rate ( which is now called the constant of variation)!

Example 5 on Page 195

Given 3 points (5, 20) , (6, 24), and (7, 28):

Divide each y/x

20/5 = 4

24/6 = 4

28/7 = 4

Since all the ratios simplify to the same value (4), it is a direct variation. The slope of 4 is the unit rate, which is the constant rate of change and is now also called the constant of variation.

Finding Additional Values for the Direct Variation once you have the Equation
Once you have the equation y = mx., you can find infinite additional values ( points) that will work.
For example, in the first babysitting example, the equation is y = $7.50x, which we write as y = 7.5x  If she babysits for 20 hours, how much did she earn? x = 20

so y = 7.5(20) = 150 so She earns $150.
If she earns $750, how many hours did she need to work?

Now y = 750  so  750 = 7.5x  It is a one step equation and we get
x = 100 or 100 hours!

 Finding the Equation if you know 1 point and then Finding Additional Values

y varies directly with x. Write an equation for the direct variation.
Then find each value

If y = 8 when x = 3, find y when x = 45

FIRST you need to find m

y = mx… In this case we have 8 = m(3) or 8 = 3m

Solve this 1 step equation—leaving it in fraction form!

8/3= m

so

y = (8/3)x

Now, find y when x = 45

y = (8/3)(45)

solve

y = 120

Comparing direct variations:

By observing the slopes of different direct variations, you can determine which is greater or lower.

For example if a rabbit moves d distance in t time and has an equation of d = 35t and a bear’s equation is d = 30t, the bear is slower because the slope is lower. The bear covers less distance in the same about of time.