Algebra Honors ( Period 6)

Chapter 3-3 Rate of Change and 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 (Periods 1 & 4)

Chapter 3-3 Rate of Change and 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 6)

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 6)

Chapter 3-1 Graphing Linear Equations

There are FOUR types of linear graphs and this chapter begins with an  OVERALL, BIG  picture

Positive Slope- slants up from left to right
Negative Slope- slants down from left to right
Horizontal line- stays flat from left to right ( constant function)
Vertical Line- stays straight up and down ( Not a function—why??)

Somethings to look for:
Domain
Range
End behavior
Intercepts
Extrema
Positive/Negative
Increasing/Decreasing
Symmetry

A Linear Equation is an equation that forms a line when it is graphed. Linear equations are often written in the form Ax + By = C
This is called standard form. In this equation C is called  a constant  Ax and By are variable terms.

A ≥ 0
A and B BOTH cannot be 0
A, B,and C are ALL integers with a GCF= 1

If you see a term such as xy attached to together it cannot be a linear equation. If the exponent on a variable is different than the understood 1,  it is not a linear equation

in 3x + 2y = 5

A = 3
B = 2
C = 5

In x = -7 ( Yes that is in Standard Form)

A = 1
B = 0
C = -7

Identify Linear Equations

Determine whether each equation is  a linear equation. Write the equation in Standard form
y = 4 – 3x   YES

To put this equation in standard form, we need to move the -3x term to the other side, using the  Addition Property of Equality and the Additive Inverse Property.  So that the x and y values are on the SAME side and the constant is always on the other side to the right of the equal sign.
3x + y = 4
A = 3
B = 1
C = 4

6x –xy = 4   NO

the term xy has two variables the equation cannot be written in AX + By = C . It is not a linear equation

(1/3)y = -1  Yes

It becomes y = -3

A= 0
B = 1
C = -3

A linear equation can be represented on a coordinate graph. The x- coordinate of the point at which the graph of the equation crosses the x-axis is called the x-intercept. The y- coordinate of the point at which the graph of the equation crosses the y-axis is called the y-intercept.

The graph of  linear equation has AT MOST one x- intercept and ONE y-intercept ( unless it is the equation x = 0, which is the y-axis or y = 0, which is the x-axis. In those two special cases every number is a y-intercept or an x-intercept, respectively)

Real World Example  Swimming Pool Page 157 in your textbook

A swimming pool is being drained at a rate of 720 gallons per hour. The table on Page 157 shows the function relating the volume of water in a pool and the time in hours that the pool has been draining.

Find the x- and y- intercepts on the graph of the function.

Looking at the table we see that the x intercept is 14 ( that is when y is 0)
and the y-intercept is 10,080 ( that is the value of y, when x = 0)

Describe what the intercepts mean in this situation: This should remind you of our unit at the beginning of the year!

The x intercept 14 means that after 14 hours the pool is completed drained because it has a volume of 0 gallons!

The y- intercept of 10,080 means that the pool contained 10,080 gallons of water at time 0 ( or before it started to drain)

Graph by Using Intercepts
Graph 2x + 4y = 16 using just the x-intercept and y-intercept

2x + 4(0) = 16   replace y with 0 (or as taught in class cover over the y value and solve)

2x = 16 so x = 8 ( when y = 0) ( 8,0)

This means the graph intersects the x-axis at (8,0)

Now

2(0) + 4y = 16  replace x with 0 ( or as taught in class- cover over the x value and solve)

4y = 16
y = 4  ( when x = 0)  ( 0, 4)

This means the graph intersect the x-axis at (0, 4)

Plot these two point and draw a line through them

Notice that this has both an x- intercept and  y-intercept

Some lines have only an x- intercept and NO y-intercept  or vice versa

y = b is a horizontal line that has only a y- intercept (unless b=0)

The graph of x = a is a vertical line that has only an x- intercept (unless a = 0)

Lines that are neither vertical or horizontal cannot have more than one x- and/or y-intercept.

Graphing Using an XY Table

Another way to graph is choosing random x values , plugging those into the equation to find the corresponding y values, and graphing those points you found.

Although 2 points determine a line, it is always best to find 3 points so that you are sure you did not make a mistake on either of the first two points.

If the coefficient of x is a fraction, select a value that is  multiple of the denominator so hopefully you won’t end up with fractions to graph! 

Algebra (Periods 1 & 4)

Chapter 3-1 Graphing Linear Equations

There are FOUR types of linear graphs and this chapter begins with an  OVERALL, BIG  picture

Positive Slope- slants up from left to right
Negative Slope- slants down from left to right
Horizontal line- stays flat from left to right ( constant function)
Vertical Line- stays straight up and down ( Not a function—why??)

Somethings to look for:
Domain
Range
End behavior
Intercepts
Extrema
Positive/Negative
Increasing/Decreasing
Symmetry

A Linear Equation is an equation that forms a line when it is graphed. Linear equations are often written in the form Ax + By = C
This is called standard form. In this equation C is called  a constant  Ax and By are variable terms.

A ≥ 0
A and B BOTH cannot be 0
A, B,and C are ALL integers with a GCF= 1

If you see a term such as xy attached to together it cannot be a linear equation. If the exponent on a variable is different than the understood 1,  it is not a linear equation

in 3x + 2y = 5

A = 3
B = 2
C = 5

In x = -7 ( Yes that is in Standard Form)

A = 1
B = 0
C = -7

Identify Linear Equations

Determine whether each equation is  a linear equation. Write the equation in Standard form
y = 4 – 3x   YES

To put this equation in standard form, we need to move the -3x term to the other side, using the  Addition Property of Equality and the Additive Inverse Property.  So that the x and y values are on the SAME side and the constant is always on the other side to the right of the equal sign.
3x + y = 4
A = 3
B = 1
C = 4

6x –xy = 4   NO

the term xy has two variables the equation cannot be written in AX + By = C . It is not a linear equation

(1/3)y = -1  Yes

It becomes y = -3

A= 0
B = 1
C = -3

A linear equation can be represented on a coordinate graph. The x- coordinate of the point at which the graph of the equation crosses the x-axis is called the x-intercept. The y- coordinate of the point at which the graph of the equation crosses the y-axis is called the y-intercept.

The graph of  linear equation has AT MOST one x- intercept and ONE y-intercept ( unless it is the equation x = 0, which is the y-axis or y = 0, which is the x-axis. In those two special cases every number is a y-intercept or an x-intercept, respectively)

Real World Example  Swimming Pool Page 157 in your textbook

A swimming pool is being drained at a rate of 720 gallons per hour. The table on Page 157 shows the function relating the volume of water in a pool and the time in hours that the pool has been draining.

Find the x- and y- intercepts on the graph of the function.

Looking at the table we see that the x intercept is 14 ( that is when y is 0)
and the y-intercept is 10,080 ( that is the value of y, when x = 0)

Describe what the intercepts mean in this situation: This should remind you of our unit at the beginning of the year!

The x intercept 14 means that after 14 hours the pool is completed drained because it has a volume of 0 gallons!

The y- intercept of 10,080 means that the pool contained 10,080 gallons of water at time 0 ( or before it started to drain)

Graph by Using Intercepts
Graph 2x + 4y = 16 using just the x-intercept and y-intercept

2x + 4(0) = 16   replace y with 0 (or as taught in class cover over the y value and solve)

2x = 16 so x = 8 ( when y = 0) ( 8,0)

This means the graph intersects the x-axis at (8,0)

Now

2(0) + 4y = 16  replace x with 0 ( or as taught in class- cover over the x value and solve)

4y = 16
y = 4  ( when x = 0)  ( 0, 4)

This means the graph intersect the x-axis at (0, 4)

Plot these two point and draw a line through them

Notice that this has both an x- intercept and  y-intercept

Some lines have only an x- intercept and NO y-intercept  or vice versa

y = b is a horizontal line that has only a y- intercept (unless b=0)

The graph of x = a is a vertical line that has only an x- intercept (unless a = 0)

Lines that are neither vertical or horizontal cannot have more than one x- and/or y-intercept.

Graphing Using an XY Table

Another way to graph is choosing random x values , plugging those into the equation to find the corresponding y values, and graphing those points you found.

Although 2 points determine a line, it is always best to find 3 points so that you are sure you did not make a mistake on either of the first two points.

If the coefficient of x is a fraction, select a value that is  multiple of the denominator so hopefully you won’t end up with fractions to graph!