Linear Equations and Functions

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Linear Equations and Functions. Chapter 2 Mr. Hardy. DO NOW. Complete the Personal Identity Worksheet- ANONYMOUSLY Take out the Chapter 1.4 Practice B Assignment. DO NOW. Skill Review on Page 66 in the textbook!. Journal Topics.

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### Linear Equations and Functions

Chapter 2

Mr. Hardy

DO NOW
• Complete the Personal Identity Worksheet- ANONYMOUSLY
• Take out the Chapter 1.4 Practice B Assignment
DO NOW
• Skill Review on Page 66 in the textbook!
Journal Topics
• If 1/3 of a number is 2 more than 1/5 of the number, then what is an equation that can be used to find the number x?
• If a six-sided polygon has 2 sides of length x – 2y each and 4 sides of length 2x + y, what is its perimeter?
ACT Practice
• A book contains 10 photographs, some in color and some in black-and-white. Each of the following could be the ratio of color to black-and-white photographs EXCEPT:
• A) 9:1
• B) 4:1
• C) 5:2
• D) 3:2
• E) 1:1
Chapter 2 Intro
• Topics
• Functions and their Graphs
• Slope and Rate of Change
• Quick Graphs of Linear Equations
• Writing Equations of Lines
• Correlation and Best-Fitting Lines
• Linear Inequalities and Two Variables
• Absolute Value Functions
Relations vs Functions
• A relation is a mapping, or pairing, of input values with output values. Simply, it is any ordered pair that shows a relationship between an input and output value
• Domain – input values
• Range – output values
• (D, R)
• A relation is a FUNCTION provided there is exactly one output for each input (one x-value yields one and only one y-value)
Domain and Range
• Find the Domain and Range of the ordered pairs
• (0, 9.1)
• (10, 6.7)
• (20, 10.7)
• (30, 13.2)
• (34, 15.5)
Functions in Context
• What is the domain and range?
• Based on what we know, could this chart represent a function?
Take a Better Look

This IS NOT a Function

This IS a Function

More Examples

This IS a Function

This IS a Function

Observe the Following Number Sets

THIS IS A FUNCTION

THIS IS NOT A FUNCTION

Relations Using Ordered Pairs
• Ordered Pairs come in the form of (x, y)
• Ordered Pairs can be plotted as points on a coordinate plane
• Describe the x and y values in each quadrant
Vertical Line Test
• The vertical line test is a trick used to determine whether or not a graph represents a function
• Simply draw a vertical line down the graph. If it goes through more than one (two or more) points on the graph, it does NOT represent a function
Vertical Line Test

BY THE VERTICAL LINE TEST, THIS IS NOT A FUNCTION

Graphing a Function
• Graph the Function y = ½ x + 1 (linear function) or in function notation f(x) = ½x + 1
• We can begin by constructing a table
• Plot the Points
• Draw a line
Example Evaluating f(x)
• F(x)= 2x + 1
• 2 (0) + 1 = 1 (0, 1)
• 2 (1) + 1 = 3 (1, 3)
• 2 (2) + 1 = 5 (2, 5)
• 2 (6) + 1 = 13 (6, 13)
• Every input will result in one output; therefore, this is a function
Evaluating F(x)
• F(x) = √x
• √1 = ±1
• √4 = ±2
• √9 = ±3
• √100 = ±10
• √144 = ±12
• Every input will result in two outputs (± the squares); consequently, this is NOT a function!!!!!!!
Linear Functions
• y = mx + b
• Function notation
• f(x) = mx + b
• The symbol f(x) is read as “f of x,” meaning the value of the function at x.
• This is another name for y!
• Think in terms of independent and dependent variables.
GO TO THE BOARD
• Draw or write an example of a function and an example that is NOT a function.
• A table
• A map
• Coordinates
• Or Graph
Wrapping Up
• Is a function always a relation? Is a relation always a function? Explain your reasoning
• Homework: Chapters 2.1 Numbers 20 - 42 even, 49 and 50, and 53 - 58 all.
• Read chapter 2.2 and take notes!!!
DO NOW: Homework Quiz
• Chapter 1 Review (pg 58 – 60)
• 6, 15, 18, 22, 32
• Section 2.1 (71 – 73)
• 30
Recall
• Solve the equation
• -⅓(x – 15) = -48
• 6x + 5 = 0.5(x + 6) – 4
• Simplify, when x = 6, and y = 4
Evaluate the Function
• Evaluate f(x) = -3x2 – 2x + 8
• when x = -2
• Evaluate f(5) = -½x - 9
Introduction
• Complete the Chart and Analyze a trend or pattern based on gas prices. Graph the ordered pairs if necessary
WHAT DID YOU NOTICE?
• Graph the ordered pairs- Label the axes.
• What does the horizontal axis represent?
• What does the vertical axis represent?
WHAT ELSE DID YOU NOTICE?
• What is the pattern that you noticed?
• What is the ratio of the vertical change to the horizontal change? Simplified.
• What does this represent?
• Does it remain constant based on the gas prices?
• Check other ratios to find out.

### Slope

A line has a positive slope if it is going uphill (increasing) from left to right.

A line has a negative slope if it is going downhill (decreasing) from left to right.

Slope
• The steepness of a line that compares the RATE of CHANGE (The change in y per unit x). The larger the slope the steeper the line
• Random information:
• Slope is usually denoted by
• The m comes from the French verb monter, meaning “to rise” or “to ascend.”
Classification of Slope
• A line with a positive slope rises (increases) from left to right (m > 0).
• A line with a negative slope falls (decreases) from left to right (m < 0).
• A line with a slope of zero is horizontal (m = 0) or y = b
• A line with an undefined slope is vertical (m is undefined) or x = a
Parallel and Perpendicular Lines
• Parallel lines do not intersect, and have the same slope
• Perpendicular lines intersect, and form a right angle. The lines are perpendicular if AND ONLY if their slopes are negative reciprocals of one another
• Ex. m = ½ and m = -2
• Give another example!
• Try guided practice 12 – 15
What’s the Slope?

When given the graph, it is easier to apply “rise over run”.

Determine the slope of the line.

Start with the lower point and count how much you rise and run to get to the other point!

rise

3

=

=

run

6

6

3

Notice the slope is positive AND the line increases!

Slope and Rate of Change
• In 2008, 23% of students at RHS were not proficient in mathematics. In 2012, 14% were not proficient. Find the rate of change.
Savings Account
• Michael started a savings account with \$300. After 4 weeks, he had \$350 dollars, and after 8 weeks, he had \$400. What is the rate of change of money in his savings account per week?
• Find out how much money he would have after a year and a half.
Find the Slope
• (-1, 4) and (1, -2)
• (-5, 3) and (-6, -1)
• (0, 5) and (4, ½)
Pop Quiz
• Draw a map that represents function and a graph that represents a non-function. SHOW AND EXPLAIN WHY they represent functions/non-functions
Homework
• Chapter 2.2
• 18 - 30 all, 38-46 all, 55-58 all
• Read 2.3 and take notes!

### Chapter 2.3

Mr. Hardy

Algebra 2

DO NOW
• Solve the equation
• |x – 10| =17
• Solve for h
• S = 2πrh + 2πr2
• Find the slope of a line given points:
• (-⅕, ⅝) and (¾, -⅔)
Recall: Slope as a Rate of Change
• You are driving from Grand Rapids, MI to Detroit, MI. You leave Grand Rapids at 4:00pm. At 5:10 pm you pass through Lansing, MI, a distance of 65 miles.
• Approximately what time will you arrive in Detroit if it is 150 miles from Grand Rapids?
Discussion: Illegal Drug Use
• Graph shows illegal drug use by age group. Find the slope of the line segment for ages 12-17. Describe what is means in practical terms.
Chapter 2.3- Quick Graphs
• y-intercept
• (0, b)
• To find the y-intercept of a line, let x = 0 in the equation, then solve for y.
• x-intercept
• (a, 0)
• To find the x-intercept of a line, let y = 0 in the equation, then solve for x.
Slope-Intercept Form
• The slope-intercept form of a linear equation is y= mx + b, where m is the slope and b is the y-intercept
• In order to sketch a quick graph
• Write an equation in slope-intercept form
• Find the y intercept and plot it
• Find the slope and use it to plot a second point
• Draw a line through the two points
Using the Slope
• Using the slope for the second point
• Start from the y intercept
• If the slope is positive rise up, the given amount, and run right the given amount.
• If the slope is negative rise up, the given amount, and run left. (or go down, and right)
• When plotting a negative slope, NEVER go down and left! That would make a positive slope.
Standard Form
• Ax + By = C,
• where A and B are not both 0.
• The quickest way to graph from standard form is to plot its intercepts (when they exist)
• The x-intercept
• Point where the line intersects the x-axis
Graphing from Standard Form
• Write the equation in Standard Form
• Find the x intercept by letting y=0 and solving for x.
• Find the y intercept by letting x=0 and solving for y.
• Draw a line through the two points
Horizontal and Vertical Lines
• You can graph x = a as a vertical line
• x = n is a horizontal line that passes through the point (n, 0). Every point will have a x- coordinate of n. For example x = -1 (-1, 0)
• You can graph y =b as a horizontal line
• y = n is a horizontal line that passes through the point (0, n). Every point will have a y- coordinate of n. For example y = 5 (0, 5)
Slope Intercept
• Graph:
• y = -2x + 3
• y = ¾x - 2
• y = -4x + ¼
Standard Form
• Graph:
• 2x + 3y = 12
• -3x – 8y = -24
• 15x + 3y = -15
GRAPH
• x = -6
• y = ¼
• x = 0
• y = x
Car Wash
• A car wash charges \$8 per wash and \$12 per wash and wax. After a busy day sales totaled \$3464. Use the verbal model to write an equation that shows the different numbers of washes and wash and waxes that could have been down. Then graph the equation
Teeter Totter
• The center post on a teeter-totter is 2 feet high. When one side rests on the ground, each end of the teeter-totter is 7 feet from the center of the post.
• Find the slope of the teeter-totter
• Find the y-intercept
• Write an equation of a line that models the tetter-totter
Saving Moeny
• Each time you get dimes or quarters for change, you throw them into a jar. You are hoping to save fifty dollars. Write a model that shows the different numbers of dimes and quarters you could accumulate to reach your goal.
• Graph the equation
Closure Questions
• When is it easier to graph using standard form?
• When is it easier to graph using slope intercept form?
• How are the two methods different?
Homework
• Chapter 2.3 #16 – 62 even
• Quiz on Wednesday/Thursday!
Do Now: Teeter Totter
• The center post on a teeter-totter is 2 feet high. When one side rests on the ground, each end of the teeter-totter is 7 feet from the center of the post.
• Find the slope of the teeter-totter
• Find the y-intercept
• Write an equation of a line that models the tetter-totter
Application: Car Wash
• A car wash charges \$8 per wash and \$12 per wash and wax. After a busy day sales totaled \$3464. Use the verbal model to write an equation that shows the different numbers of washes and wash and waxes that could have been down. Then graph the equation
Saving Moeny
• Each time you get dimes or quarters for change, you throw them into a jar. You are hoping to save fifty dollars. Write a model that shows the different numbers of dimes and quarters you could accumulate to reach your goal. Graph the equation
Writing Equations of Lines
• Slope - Intercept Form (given the slope and y intercept)
• y = mx + b
• Point – Slope Form (given the slope and a point)
• y – y1 = m(x – x1)
• Two Points (x1, y1) and (x2, y2)
• Use the slope formula to find the slope, then the point-slope form to find
Given Slope and Y Intercept
• Find the equation of a line with the given information m = 3/2, b = -1
• We need to use the slope intercept form
• y = mx + b
• Here, all we need to do is plug in the numbers for m and b
• Our equation of the line is:
• y = 3/2x - 1
Try These
• Slope: ⅞, y intercept: 2
• Slope: ⅗, y intercept: -½
• Slope: -4, y intercept: 0
• Slope: 0, y intercept: -3.789
Equation Given Slope and a Point
• Write an equation of a line that passes through (2, 3) and has a slope of -½
• Since we have a slope and a point, we need to use the point slope form!
• y – y1 = m(x – x1)
• The point will be plugged in/substituted for x1 and y1
• y – 3 = -½(x – 2)
• Next, Distribute
• y – 3 = -½x + 1
• Add 3 on both sides to isolate y
• y = - ½x + 4 is the equation of a line
Try These
• Slope: 0, passes through (3, 7)
• Slope: 2, passes through (0, -4)
• Slope: -¾, passes through (-7, 0)
• Slope: -3, passes through (5, 2)
Parallel and Perpendicular Lines
• Write an equation of the line that passes through (3, 2) and is PERPENDICULAR to the line y = -3x + 2
• Information given
• A Point (3, 2)
• And a line with slope -3
• What is the slope of the PERPENDICULAR LINE?
• m = ⅓ (positive ⅓)
NEXT
• Now we can use the point slope form with the given point and the slope m = ⅓ to find the equation of the line.
• We need the point slope form
• y – y1 = m(x – x1)
• y – 2 = ⅓(x – 3)
• y – 2 = ⅓x – 1
• y = ⅓x + 1
Try These
• Write an equation of a line that passes through (3, 2); it’s parallel to the line y = -3x + 2
• Write an equation of a line that passes through (1, -6) and is perpendicular to y = 3x + 7
• Write an equation of a line that is parallel to the above line.
Equation Given Two Points
• Write an equation of a line that passes through (-2, -1) and (3, 4)
• First, use the slope formula
• m = 4 - - 1

3 - - 2

• m = 4 + 1

3 + 2

• m = 5/5 = 1
• NOW, use the point slope form (either point is fine)
Continued
• y – y1 = m(x – x1)
• y - -1 = 1( x – -2)
• y + 1 = x + 2
• y = x + 1
Try These
• Line passes through (4, 8) and (1, 2)
• Line passes through (0, 2) and (-5, 0)
Deeper Thinking
• Write an equation of a line under the given conditions
• The line passes through (—2, 6) and is perpendicular to the line whose equation is x = -4.
• The line passes through (—6,4) and is perpendicular to the line that has an x-intercept of 2 and a y-intercept of - 4 .
News Outlets
• Find a linear function in slope-intercept form that models the given description. Each function should model the percentage of Americans, P(x), who regularly used the news outlet, x years after 2000
• In 2000, 47% of Americans regularly used newspapers for getting news and this has decreased at an average rate of approximately 1.2% per year since then.
• In 2000, 23% of Americans regularly used online news for getting news and this has increased at an average rate of approximately 1.3% per year since then.
Bookstore Sales
• In 1990 retail sales at bookstores were about \$7.4 billion. In 1997, retail sales at bookstores were about \$11.8 billion. Write a linear model for retail sales s(in billions of dollars) at bookstores from 1990 through 1997. Let t represent the number of years since 1990. Then estimate what the retail sales at bookstores were in 2012.
Life Expectancy

Write an equation for the life expectancy of your sex!

Direct Variation
• Two variables x and y show direct variation provided y = kx, and k≠ 0. The nonzero constant k is called the constant of variation and y is said to vary directly with x. The graph of y=kx is a line through the origin.
• Ex.
• The variables x and y vary directly, and y = 12 when x = 4
• Use the given values of x and y to find k
• y = kx
• 12 = k(4)
• 3 = k
• The equation is y = 3x
• What is y, when x = 5?
Homework
• Chapters 2.4. #14-42 even, 59 – 62 all
• Read 2.6 and Take notes!
Do Now: Bookstore Sales
• In 1990 retail sales at bookstores were about \$7.4 billion. In 1997, retail sales at bookstores were about \$11.8 billion. Write a linear model for retail sales s(in billions of dollars) at bookstores from 1990 through 1997. Let t represent the number of years since 1990. Then estimate what the retail sales at bookstores were in 2012.
Life Expectancy

Write an equation for the life expectancy of your sex!

2.6- Linear Inequalities in Two Variables
• A linear inequality in two variables is an inequality that can be written in the following forms
• Ax + By < C
• Ax + By ≤ C
• Ax + By > C
• Ax + By ≥ C
• Using ordered pairs, we can find the solution of a linear inequality, when the values for x and y are substituted.
Example 1
• Think of Linear Inequalities as being cousins to Linear Equations.
• The majority of the concepts are similar
• Check whether the given ordered pair is a solution of 2x + 3y ≥ 5
• (0, 1)
• (4, -1)
• (2, 1)

2 (0) + 3 (1) = 3 ≥ 5

• (0, 1) IS NOT A SOLUTION
• 2 (4) + 3 (-1) = 5 ≥ 5
• (4, -1) IS A SOLUTION
• 2 (2) + 3 (1) = 7 ≥ 5
• (2, 1) IS A SOLUTION
Graphing Linear Inequalities
• The graph of a linear inequality in two variables is a half-plane.
• A half-plane is a region consisting of all points on one side of an infinite straight line, and no points on the other side.
• Step 1.
• Graph the boundary line of the inequality (same as graphing a linear equation EXCEPT:
• Dashed line for < or >
• Solid line for ≤ or ≥
• Solid lines represent the “or equal to” part of the inequality
Graphing Linear Inequalities
• To decide which side of the boundary line to shade, test a point that is NOT on the boundary line to see whether it is a solution of the inequality. Then shade the appropriate half-plane
• The shaded region contains all of the points that are solutions of the inequality.
• The unshaded region contains the points that ARE not solutions of the inequality.
Graphing Linear Inequalities in Two Variables
• Graph
• 9x – 2y < 18 ( graph in standard form)
• 2x - 5y ≥ -10 (graph in slope intercept form)
• First, graph the boundary line
• Next, choose a point outside of the boundary line to test the inequality
• Finally, shade the half-plane that represents the points in the inequality!
Babysitting
• You earn \$3 per hour when you babysit the Thompson children. You earn \$3.50 per hour when you babysit the Stewart children. You would like to buy a \$47.50 ticket for a concert that is coming to town in 5 weeks.
• Write and graph an inequality that represents the number of hours you need to babysit for the Thompson’s and Stewart’s to earn enough money to buy your concert ticket.
Homework
• Chapters 2.6 Practice Worksheet
Do Now: Homework Quiz
• Take out 2.6 Worksheet and Pass it up
• FIVE MINUTES
• Chapter 2.2
• #18, 28, 30,
• Chapter 2.4
• #14, 18, 26, 40
Recall
• Write an equation of the line that passes through (1, -2) and is perpendicular to the line that passes through (4, 2) and (0, 4).
• Graph the Inequalities
• y≤ -3x + 11
• 9x – 9y > -36
Application: Babysitting
• You earn \$3 per hour when you babysit the Thompson children. You earn \$3.50 per hour when you babysit the Stewart children. You would like to buy a \$47.50 ticket for a concert that is coming to town in 5 weeks.
• Write and graph an inequality that represents the number of hours you need to babysit for the Thompson’s and Stewart’s to earn enough money to buy your concert ticket.
DO NOW: Pop Quiz 
• Graph the following inequalities
• 1) y < ½
• 2) 3x + 4y ≥ 12
• 3) y > -⅔x – 5
• 4) y ≤ 5 - ⅘x
• 5) An exam included multiple choice questions that were worth 4 points each and true/false question worth 2 points each. The highest score earned by a person was 92. Write and graph the inequality that represents the possible combination of questions answered correctly
Lesson Opener
• You bought a bottle of Rihanna’s Rogue on sale for \$35. Macy’s is running an engraving promotion! The first six letters are engraved free. Each additional letter costs \$.20.
• Think about, and write a model that price of thebottlewith x engraved letters
• Graph the function
• What is the price of the bottle with the name Sidney Lynn Arrington engraved?
Another View: Group Rate Discounts
• A company provides bus tours of historical cities. The given function describes the rate for small groups and the discounted rate for larger groups, where x is the number of people in your group.
Piecewise Functions 2.7
• Piecewise functions are represented by a combination of equations, each corresponding to a part of the domain
• For Example:
What Do They Look Like?
• You can EVALUATE and GRAPH piecewise functions!
• What do you notice about the domain?

x2 + 1 , x  0

x – 1 , x  0

f(x) =

Evaluating Piecewise Functions:

Evaluating piecewise functions is just like evaluating functions that you are already familiar with.

x2 + 1 , x  0

x – 1 , x  0

f(x) =

Let’s calculate f(2).

You are being asked to find y when

x = 2. Since 2 is  0, you will only substitute into the second part of the function.

f(2) = 2 – 1 = 1

Let’s calculate f(-2).

x2 + 1 , x  0

x – 1 , x  0

f(x) =

You are being asked to find y when

x = -2. Since -2 is  0, you will only substitute into the first part of the function.

f(-2) = (-2)2 + 1 = 5

2x + 1, x  0

2x + 2, x  0

f(x) =

Evaluate the following:

f(-2) =

-3

?

f(5) =

12

?

f(1) =

4

?

f(0) =

?

2

One more:

3x - 2, x  -2

-x , -2  x  1

x2 – 7x, x  1

f(x) =

Evaluate the following:

f(-2) =

2

?

?

f(3) =

-12

?

f(-4) =

-14

?

f(1) =

-6

DO NOW
• Evaluate the piecewise function
• f (x) = 2x -1, if x ≤ 1

3x + 1, if x > 1

f(-4)

f(0)

f(5)

f(-10)

f(10)

GRAPHING PIECEWISE FUNCTIONS
• Be sure to pay attention to the given domain
• The domain determines the endpoints- either an open or closed circle
• Graph the line using the given equation; however, be sure that it adheres to the rules of the given domain.
Piecewise Function

A piecewise function is any function that is in, well, pieces!

Piecewise functions indicate intervals for each part of the function

Graph f(x) =

y

f(x) =

f(x) = 1

x

y

f(x) =

f(x) = {1 x < 3

x

3

y

f(x) = x + 1

f(x) =

x

y

f(x) =

f(x) = {x+1 x > 3

x

3

Graph:
• For all x’s < 1, use the top graph (to the left of 1)
• For all x’s ≥ 1, use the bottom graph (to the right of 1)

x=1 is the breaking

point of the graph.

To the left is the top

equation.

To the right is the

bottom equation.

Bonuses
• Foot Locker pays its employees a combination of salary and commission. An employee with sales less than \$100,000 earns a \$15,000 salary plus 3% commission. An employee with sales of \$100,000 to \$200,000 earns an \$18,000 salary plus 4% commission.. An employee who earns more than \$200,000 in sales earns a \$20,000 salary plus 5% commission. Write a piecewise model that gives the pay of an employee with xin annual sales.
Step Functions
• Step Functions are piecewise functions whose graphs resemble a set of stairs. It is a special type of function whose graph is a series of line segments.
• A function that increases or decreases abruptly from one constant value to another.
Writing a Piecewise Function
• When writing the equations of a piecewise function
• Find two points, find the slope, and find the equation of the line using the point slope form
• Determine the domain
DO NOW

Graph the Piecewise Functions

f(x) =

f(x) =

Is the statement 0 ≤ -5 true or false?

Recall
• Page 118 #36 and 39
Absolute Value is defined by:

Graph the Piecewise Function

The graph of this piecewise function consists of 2 rays, is V-shaped and opens up.

To the left of

x=0 the line is

y = -x

To the right of

x = 0 the line is

y = x

Notice that the graph is symmetricto the y-axis because

every point (x,y) on the graph, the point (-x,y) is also on it.

y = a |x - h| + k
• General Rules of Absolute Value Functions
• Vertexis at (h, k) & is symmetrical in the line x = h
• V-shaped
• If a< 0 the graph opens down (a is negative)
• If a>0 the graph opens up (a is positive)
• The graph is wider if |a| < 1 (fraction < 1)
• The graph is narrower if |a| > 1
• a is the slope to the right ofthe vertex
To graph y = a |x - h| + k
• Plot the vertex (h, k)
• Note: a|x – (h)| + k
• When h is negative, it becomes a|x + h| + k
• Use the slope to plot another point to the RIGHT of the vertex.
• Use symmetry to plot a 3rdpoint
• Complete the graph
Graph y = -|x + 2| + 3
• V = (-2, 3)
• Apply the slope a=-1 to that point
• Use the line of symmetry x=-2 to plot the 3rd point.
• Complete the graph
Explore
• Graph the functions
• f(x) = |x|
• f(x) = |x|+ 3
• f(x) = |x – 3|
• f(x) = 3|x|
• f(x) = ⅓|x|
• f(x) = 3|x – 3|+ 3
• What do you notice as you shift h and k?

So the equation is:

y = 2|x| -3

• Determine the coordinate of the vertex
• The vertex is @ (0,-3)
• y= a |x - 0| - 3
• Find a (or the slope)
• Substitute the coordinate of a point (2,1) in and solve OR count the slope from the vertex to another point the right)
• Remember
• a is positive if the graph goes up
• a is negative if the graph goes down
Homework
• Chapter 2.8 #12 – 25 ALL
• 34 – 41 ALL