Algebra Honors ( period 6)

Chapter 3 -5 Arithmetic Sequences as Linear Functions

An arithmetic sequence is an ordered list of numbers ( called terms) where there is a common difference (d) between consecutive terms. the common difference can either be positive (increasing)  or negative ( decreasing)

Because an arithmetic sequence has a constant difference, it is a linear function. There is a formula using this common difference to find the equation of any arithmetic sequence:

a_n = a₁ + ( n-1)d

Notice that it is saying that any term in the sequence, a_n,  can be found by adding 1 less than the number of terms of the common difference to the first term, a_1.   Why 1 less than the number of the terms you want?

Take the following arithmetic sequence:

…17, 21, 25,29, 33….

If it’s the 2^nd term, there is only 1 common difference of 4 between the 2 terms. If it is the 3^rd term, there would be 2 common differences of 4 between the 3  terms, etc.

n is always positive because it represents the number of terms and that can’t be negative.

Where  a_n  represents any term in the sequence and a₁ represents the first term in a sequence, n represents the number of the terms in a sequence and d represents the common difference between consecutive terms in a sequence.

This is called an EXPLICIT FORMULA You can explicitly find any term number in the sequence as long as you know the 1^st term and the common difference. For example if the the common difference is 4 and the first term is 1 and you are trying to find the 27^th term

a_n = a₁ + ( n-1)d

a₂₇ = 1 + ( 27-1)4

a₂₇ = 1 + ( 26)4

a₂₇ = 1 + ( 104

a₂₇ = 105

You can also find the next term in the sequence if you know the
RECURSIVE FORUMULA

This is a Function Rule that tells you what the relationship between consecutive terms is. For example if the common difference is 4 and your last term is 101 the next term is 105—without knowing any of the other preceding terms. The terms in the sequence are shown as a list with 3 periods (called an ellipsis) at the end showing that it continues infinitely.

For example, the arithmetic sequences for the above example was

1, 5,9, 13, …

Graphing the terms of an arithmetic sequence shows that it is a linear function

a_n = 1 + ( n-1)4

Simplify

a_n = 1 + ( n-1)4

a_n = 1 + 4n -4

a_n = 4n -3

Now just substitute y for a_n   and x for n

y = 4x – 3

Notice that d is now the slope and also notice that the domain of the sequence is the natural numbers (the counting numbers) because you can’t have a negative term number!

Graph 3 points using 1, 2, 3 for the first 3 terms

(1, 1) ( 2,5) (3, 9)

Algebra ( Periods 1 & 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 EquationOnce 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 6)

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 EquationOnce 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 ( Periods 1 & 4)

Chapter 4-1 Graphing Equations in Slope-Intercept Form

The most used form of a linear equation: Slope Intercept form

You must restate the equation to get it into the following format:

y = mx + b

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

The m and b are called the PARAMETERS of the equation.

Solve the equation for “y” means isolate the y on one side of the equal signs

Example:  -3y = -2x  - 6

If you tried graphing randomly, (setting up a small table or t chart) you would problem select the x points of 0, 1, 2.  Only when x = 0 will give you an integer value for y. All the other y values are fractions/ decimals à which makes it difficult to graph accurately!
 

The slope- intercept form provides the explanation for this.

RESTATE -3y = -2x – 6 INTO SLOPE-INTERCEPT FORM:

 Divide BOTH sides by -3:

y = (2/3)(x) + 2

Look at the coefficient for x?
What  x values will give you integer answers for y?

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 doing any work!) YAY!!!

Use the counting method for slope on your graph, you should have counted:
UP  2 and RIGHT 3. The slope therefore is 2/3
Look at the equation—it told you the slope was 2/3 without any work (YAY!!)

Graphing when the 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 positive or negative constant at the end of your equation. Your HOME BASE)

2) Count the next point by using the slope of x coefficient as a fraction (so if you have an integer—place it “over” 1)

For the equation y = (2/3)(x) + 2

1) graph  a point 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” or “Rise over Run”
The numerator is the change in y  and the denominator is the change in x.

If it is positive you are counting up (positive) and to the right (positive) OR
you can count down (negative) and to the left (negative) because when multiplying 2 negatives become positive

If it is negative you are counting down (negative) and to the right (positive) OR
you can count up (positive) and to the left (negative) because then you would have a positive ∙ negative = negative

Example: m = -2/3 and b = -12
The line would be y = (-2/3)x - 12

Restate Standard Form to Slope- Intercept Form

Another example: 3x + 4y = 10 is the STANDARD FORM of a line
(Notice; x and y are on the same side of the equal sign, x is positive, there are NO fractions, and the constant is alone)
This equation is NOT easy to graph in this form because your y intercept will not be an integer ( In fact it will not be that easy to graph in slope intercept either—but we will get to that later)

Restate into slope intercept

Solve for y
First 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

The slope is the coefficient of the x so

m = -3/4 (so you are sliding down at a little less than a 45 degree angle. Remember a slope of 1 or -1 is  45 degrees)

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

Notice that the ‘b’ is a fraction.
When this happens the slope- intercept form MAY NOT be the best form to graph the line. You MUST start at 2½ on the y axis and count down 3 and to the right 4. That is actually HARD to get accurate!

Neither 3 nor 4 is a factor of the constant 10 ( 3x + 4y = 10) so the intercepts will also be fractions.

We need the x term to end up with ½ so that when we add that to the b (which is 5/2) we will get an integer.
So try letting x be 2 because that will cross cancel with the -3/4 slope

y = (-3/4)(2) + 5/2 =  -3/2 + 5/2 = 1
so we just found a coordinate that has just integers ( 2, 1)
Graph that point. Now count the slope from THAT POINT ( instead of the y intercept)

HORIZONTAL LINES
You can think of these lines in Slope- Intercept form as y = mx + b IF your REMEMBER that the slope of a horizontal line = 0 àthe equation is y = (0)x + b
We have already learned that the equation of any horizontal line is y = a constant
Except for the horizontal line y = 0 ( which is the x axis) horizontal lines have no x intercept and therefore no intercepts ( no roots, no solutions, no zeros)

Finding the equation of a line by looking at its graph is easy if you can read the y intercept!

Simply plug in the y intercept as  b and then count the rise over run as the slope!

Algebra Honors ( Period 6)

Chapter 4-1 Graphing Equations in Slope-Intercept Form

The most used form of a linear equation: Slope Intercept form

You must restate the equation to get it into the following format:

y = mx + b

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

The m and b are called the PARAMETERS of the equation.

Solve the equation for “y” means isolate the y on one side of the equal signs

Example:  -3y = -2x  - 6

If you tried graphing randomly, (setting up a small table or t chart) you would problem select the x points of 0, 1, 2.  Only when x = 0 will give you an integer value for y. All the other y values are fractions/ decimals à which makes it difficult to graph accurately!
 

The slope- intercept form provides the explanation for this.

RESTATE -3y = -2x – 6 INTO SLOPE-INTERCEPT FORM:

 Divide BOTH sides by -3:

y = (2/3)(x) + 2

Look at the coefficient for x?
What  x values will give you integer answers for y?

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 doing any work!) YAY!!!

Use the counting method for slope on your graph, you should have counted:
UP  2 and RIGHT 3. The slope therefore is 2/3
Look at the equation—it told you the slope was 2/3 without any work (YAY!!)

Graphing when the 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 positive or negative constant at the end of your equation. Your HOME BASE)

2) Count the next point by using the slope of x coefficient as a fraction (so if you have an integer—place it “over” 1)

For the equation y = (2/3)(x) + 2

1) graph  a point 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” or “Rise over Run”
The numerator is the change in y  and the denominator is the change in x.

If it is positive you are counting up (positive) and to the right (positive) OR
you can count down (negative) and to the left (negative) because when multiplying 2 negatives become positive

If it is negative you are counting down (negative) and to the right (positive) OR
you can count up (positive) and to the left (negative) because then you would have a positive ∙ negative = negative

Example: m = -2/3 and b = -12
The line would be y = (-2/3)x - 12

Restate Standard Form to Slope- Intercept Form

Another example: 3x + 4y = 10 is the STANDARD FORM of a line
(Notice; x and y are on the same side of the equal sign, x is positive, there are NO fractions, and the constant is alone)
This equation is NOT easy to graph in this form because your y intercept will not be an integer ( In fact it will not be that easy to graph in slope intercept either—but we will get to that later)

Restate into slope intercept

Solve for y
First 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

The slope is the coefficient of the x so

m = -3/4 (so you are sliding down at a little less than a 45 degree angle. Remember a slope of 1 or -1 is  45 degrees)

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

Notice that the ‘b’ is a fraction.
When this happens the slope- intercept form MAY NOT be the best form to graph the line. You MUST start at 2½ on the y axis and count down 3 and to the right 4. That is actually HARD to get accurate!

Neither 3 nor 4 is a factor of the constant 10 ( 3x + 4y = 10) so the intercepts will also be fractions.

We need the x term to end up with ½ so that when we add that to the b (which is 5/2) we will get an integer.
So try letting x be 2 because that will cross cancel with the -3/4 slope

y = (-3/4)(2) + 5/2 =  -3/2 + 5/2 = 1
so we just found a coordinate that has just integers ( 2, 1)
Graph that point. Now count the slope from THAT POINT ( instead of the y intercept)

HORIZONTAL LINES
You can think of these lines in Slope- Intercept form as y = mx + b IF your REMEMBER that the slope of a horizontal line = 0 àthe equation is y = (0)x + b
We have already learned that the equation of any horizontal line is y = a constant
Except for the horizontal line y = 0 ( which is the x axis) horizontal lines have no x intercept and therefore no intercepts ( no roots, no solutions, no zeros)

Finding the equation of a line by looking at its graph is easy if you can read the y intercept!

Simply plug in the y intercept as  b and then count the rise over run as the slope!