Unit 4 Semester 1 Review 4 of 4

N.Exploring Linear Functions

N. 1 Identify linear functions from graphs and equations
N. 2 Identify linear functions from tables
N. 3 Complete a table and graph a linear function

Identifying Linear Functions

A function whose graph is a straight line is a linear function. It can be identified from graphs, equations, or tables. The graph of a nonlinear functions is not a straight line. Linear functions have a constant rate of change and can be represented by equations in the form y = mx + b, where m is the slope and b is the y-intercept. Linear functions are widely used in various fields, including mathematics, physics, and economics.

Multiple Choice

These are linear functions

• TRUE

• FALSE

Linear Function:

y = mx + b is the equation form of a linear function. It represents a straight line on a graph, where m is the slope and b is the y-intercept. Linear functions are used to model relationships between two variables that have a constant rate of change.

Multiple Choice

What is the equation form of a linear function?

• y = mx + b

• y = ax^2 + bx + c

• y = e^x

• y = ln(x)

Linear vs Nonlinear Functions

Linear functions have a constant rate of change, while nonlinear functions do not. To determine if a function is linear, look for a constant first difference in the range. If the first difference is constant, the function is linear. If not, it is nonlinear. Use tables to identify linear functions by checking for a constant rate of change in the domain.

Multiple Choice

These are linear functions

• FALSE

• TRUE

Constant Rate of Change

Trivia: One way to determine if a function is linear is by checking for a constant rate of change in the domain. This means that for every unit increase in the independent variable, there is a consistent change in the dependent variable. Nonlinear functions do not exhibit this property.

• Linear functions have a constant slope.
• Nonlinear functions have a varying slope.

Multiple Choice

What is one way to determine if a function is linear?

• Look for a constant first difference in the range

• Check for a constant rate of change in the domain

• Examine the title of the passage

• Use tables to identify nonlinear functions

Multiple Choice

This slope in constant so we have a linear function

• Sometimes

• No

• Yes

• I don't know

Exploring Linear Functions

Linear functions are mathematical functions that can be represented by a straight line. They have a constant rate of change, also known as the slope. To determine if a function is linear, check if the rate of change in the domain is constant. Use the formula (Y₂ - Y₁) / (X₂ - X₁) to calculate the slope between any two points. Complete a table and graph the function to visualize the linear relationship. Remember, the general form of a linear function is f(x) = mx + b, where m is the slope and b is the y-intercept.

Reorder

Reorder the following from top to bottom

• 0

• 3

• 6

1

2

3

• 1
• 2
• 3

Slope Formula:

(Y₂ - Y₁) / (X₂ - X₁) is the formula to calculate the slope of a linear function. It represents the change in the y-coordinates divided by the change in the x-coordinates. The slope determines the steepness of the line and can be positive, negative, or zero. Remember, the numerator represents the vertical change and the denominator represents the horizontal change.

Multiple Choice

What is the formula to calculate the slope of a linear function?

• (Y₂ - Y₁) / (X₂ - X₁)

• (X₂ - X₁) / (Y₂ - Y₁)

• (Y₁ - Y₂) / (X₂ - X₁)

• (X₁ - X₂) / (Y₁ - Y₂)

Poll

From left to right, what are the correct solutions?

• The solution should be 9, 11, -7

• The solution should be -7, 11, 9

• The solution should be 11, -7, 9

• The solution should be -7, 9, 11

Open Ended

REVIEW:

What is the slope?

What is the y-intercept?

What does the slope mean? Is it positive or negative?

What does the y-intercept mean?

What is f(9) equal to?

Multiple Select

Which concepts did we explore in today's notes?

• Graphing linear functions

• Completing tables

• Identifying linear functions

• Understanding slope and y-intercept

​What is a Relation?

​A "relation" is a pairing of input values with output values.

​The set of input values is called the domain and the set of output values is called the range.

​Find the domain and range of this relation

​Practice!

​Functions

For every function, there is only one y-value per x-value (or, one output per input). So,

if you can plug in an

input and get two

different answers,

then the relation is

not a function.​

​Functions

We can identify functions one of two ways: checking if any x-value has two or more

​y-values, or using the

​vertical line test on

​relation graphs.

​

​In the vertical line test,

​if you draw a vertical

​line through the graph

​and hit two points on the graph, then it is not a function.

​Is the above relation a function?

​Practice!

Multiple Choice

Identify the domain of the relation.

• -9, -5, 1, 7

• 1, 4, 5, 8

• -9, -5, 1, 1, 7

• 4, 8

​Relations in Two Variables

​An equation in two variables can relate two values like in the equation: y = 2x + 3. The input variable, x, is the independent variable and the output value, y, is the dependent variable (since its value depends on the value of the input).

​Graphing an Equation

​To graph an equation in two variables, we select values for x, plug them in to the equation, and then solve for y. This gives us a coordinate pair (x, y) that we can plot on a Cartesian plane.

​

​The relation to the right is a function and is linear.

​

​NOTE: When working with functions, we sometimes use function notation which is written f(x) = mx + b, and f(x) stands in place of y.

NOTE:​

Functions which contain variables with exponents are not linear.

Examples: y = 4x³ - 8, and y = 2x² + 4x - 16 are not linear since one x-value in each has an exponent attached to it.​

Multiple Choice

Is the following relation linear?

$$y=2x-8$$  

• Yes

• No

Linear Functions

Introduction

       Graph

​

Definition of a Linear Function

• A linear function is any function that graphs to a straight line. 

• A function has either one or two variables with no exponents or powers

• If the function has more variables, the variables must be constants or known variables for the function to remain a linear function.

Identifying Linear Functions

• It must have either one or two real variables.

•  If another variable is present, it must be a known variable or constant. For example, the function C = 2 * pi * r is a linear function because only the C and r are real variables, with the pi being a constant.

• none of the variables can have an exponent or power to them.  All variables must be in the numerator.

•  function must graph to a straight line. Any kind of a curve disqualifies the function.

Multiple Choice

Is the equation y=x^2 linear? Why, or why not?

• No, because the x variable has an exponent.

• No, because the value of y has not been defined.

• Yes, because it's an equation with two variables.

• Yes, because the y variable has no exponent.

Multiple Choice

What is the minimum number of points needed to graph a linear function?

• Three points

• Two points

• One Points

• Zero points

Multiple Choice

For the linear function h=2t-1, what is h when t=4?

• -7

• 7

• 8

• 10

Multiple Choice

A linear function must graph to a _____ line.

• stepped

• curved

• straight

• parabolic

Multiple Choice

Is y=2x+D-10 linear if given that D=4?

• No

• Yes

• Sometimes

What is a Rate of Change?

Rate that describes how one quantity changes in relation to another quantity

If x is the independent variable and y is the dependent variable, then. rate of change=change in y change in x

What is a SLOPE?

slope of real-world situations is often referred to as rate of change.

Rate of change” means the same as “slope.”

 If you are asked to find the rate of change, use the slope formula or make a slope triangle

Types of Slopes of a Line

Positive Slope

A positive slope means the line is increasing when viewed from left to right.


As you can see, Mr. Piggy is having a hard time going up since it costs him an extra effort for an uphill climb.

Negative Slope

A negative slope means the line is decreasing when viewed from left to right.

Thanks to gravity, Mr. Piggy is definitely enjoying the slide because it takes him less effort to go down.

Zero Slope

A zero slope means the line is neither increasing nor decreasing when viewed from left to right, or vice versa. 

Simply put, the slope of a horizontal line is zero,

Undefined Slope

An undefined slope or infinite slope, means the line is neither moving to the left nor to the right such as the case of a vertical line. The slope of a vertical line is either + \,\infty+∞ or - \,\infty−∞.

Types of Slopes

Identify if the Slope is positive, Negative, Zero or Undefined

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Multiple Choice

What is the slope of this line?

• positive

• negative

• zero

• undefined

Multiple Choice

What is the slope of this line?

• positive

• negative

• zero

• undefined

Multiple Choice

What is the slope of this line?

• positive

• negative

• zero

• undefined

Multiple Choice

What is the slope of this line?

• positive

• negative

• zero

• undefined

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Multiple Choice

Calculate the rise and run to find the slope of this line.

• 
  

   $-\frac{5}{3}$  

•  $\frac{5}{3}$  

•  $\frac{3}{5}$  

•  $-\frac{3}{5}$  

Multiple Choice

Calculate the rise and run to find the slope of this line.

• 4

• 3

• 2

• 1

Multiple Choice

Calculate the rise and run to find the slope of this line.

• $\frac{1}{3}$

• $\frac{1}{4}$

• $-\frac{1}{4}$

• $-\frac{1}{3}$

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Dependent and INdependent VARIABLES

By Jennifer Mackie

In any situation, it is the variable affected or the result of changes by another factor.

It is found on the y axis​.

​

​What do you depend on?

​Dependent

Variable​

​

This is the cause of the action. It is found on the x axis .

​

​INdependent

Variable​

​

Multiple Choice

Sara is in NMotion Dance Company. She has to buy a costume every time she performs. Which one is the INdependent variable? (Hint: It's the cause)

• How many dances she performs

• How many costumes she buys

Multiple Choice

Sara is in NMotion Dance Company. She has to buy a costume every time she performs. Which one is the Dependent variable? (Hint: Its the effect)

• How many dances she performs

• How many costumes she buys

Multiple Choice

On a graph, which axis shows the independent variable?

• x-axis
• y-axis
• neither
• it changes

Multiple Choice

The y- axis is the ___________________ number line in the coordinate plane.

• Vertical

• Diagonal

• Horizontal

• origin

Multiple Choice

Name the INDEPENDENT variable from the table below.

• words
• minutes

Multiple Choice

Sophie earned $5 for every A she got on her report card.

m = money

A = A on report card

The dependent variable is:

• m = money

• A = A on report card

Multiple Choice

Which of the following is the label for the independent variable?

• Money Collected

• Raising Money

• Boxes Sold

• None of the above

Multiple Choice

Which label best describes the independent quantities shown in the table?

• Number of tickets (t)
• Cost (c)

Multiple Choice

Which label best describes the dependent quantities shown in the table?

• Number of tickets (t)
• Cost (c)

Multiple Choice

On a graph, which axis shows the dependent variable?

• x-axis
• y-axis
• neither
• it changes

Multiple Choice

Ty earns $12 per hour. Which of the following equations best represents how much money, m, he earns on a shift of h hours?

• m = 12h

• m = 12 + h

• h = 12m

• h = 12 + m

Multiple Choice

Ms. Shi is making a scrapbook. As she adds more photos, the scrapbook grows thicker.

p = the number of photos added

t = the thickness of the scrapbook

• t is the independent variable and p is the dependent variable

• p is the independent variable and t is the dependent variable

Multiple Choice

Is the picture showing an additive or a multiplicative relationship?

• Additive
• Multiplicative

Multiple Choice

Is the picture showing an additive or a multiplicative relationship?

• Additive
• Multiplicative

Multiple Choice

Each pizza at Pizza Hut, p, costs $5.50. Which of the following equations best represents the cost, c, of buying p, pizzas?

• m = 5.50 + c

• c = 5.50 + m

• p = 5.50m

• c = 5.50p

Multiple Choice

On a graph, which axis shows the independent variable?

• x-axis
• y-axis
• neither
• it changes

Multiple Choice

A(n) _____________ variable depends on or is changed by another variable.

• dependent
• independent

Multiple Choice

The y- axis is the ___________________ number line in the coordinate plane.

• Vertical

• Diagonal

• Horizontal

• origin

Multiple Choice

Name the INDEPENDENT variable from the table below.

• words
• minutes

Multiple Choice

In the table, Bubbles is the ______________.

• independent variable

• dependent variable

Multiple Choice

Sophie earned $5 for every A she got on her report card.

m = money

A = A on report card

The dependent variable is:

• m = money

• A = A on report card

Multiple Choice

Which of the following is the label for the independent variable.

• Money Collected

• Raising Money

• Boxes Sold

• None of the above

Multiple Choice

Independent means

• an amount that can stand on its own
• an amount that depends on another amount

Multiple Choice

Which label best describes the dependent quantities shown in the table?

• Number of tickets (t)
• Cost (c)

Multiple Choice

Which label best describes the independent quantities shown in the table?

• Number of tickets (t)
• Cost (c)

Multiple Choice

• the independent variable is distance
• the independent variable is d=25 t
• the dependent variable is time
• the dependent variable is distance

Multiple Choice

On a graph, which axis shows the dependent variable?

• x-axis
• y-axis
• neither
• it changes

Multiple Choice

Jimmy earned $8 for every pizza he delivers.

m = money (dollars)

p = number of pizzas delivered

The dependent variable is:

• m = money

• p = number of pizzas delivered

Multiple Choice

Emma gets one sticker for every paper she completes.

s = stickers earned

p = papers completed

The independent variable is:

• s = stickers earned

• p = papers completed

Multiple Choice

Ty earns $12 per hour. Which of the following equations best represents how much money, m, he earns on a shift of h hours?

• m = 12h

• m = 12 + h

• h = 12m

• h = 12 + m

Multiple Choice

The graph shows the distance traveled by Ms. Luce's family on a road trip. They will travel 25 miles each hour. Which statement is true about the situation?

• the independent variable is distance

• the independent variable is d=25 t

• the dependent variable is time

• the dependent variable is distance

Multiple Choice

Ms. Shi is making a scrapbook. As she adds more photos, the scrapbook grows thicker.

p = the number of photos added

t = the thickness of the scrapbook

• t is the independent variable and p is the dependent variable

• p is the independent variable and t is the dependent variable

Multiple Choice

Is the picture showing an additive or a multiplicative relationship?

• Additive
• Multiplicative

Multiple Choice

Is the picture showing an additive or a multiplicative relationship?

• Additive
• Multiplicative

Multiple Choice

Each pizza at Pizza Hut, p, costs $5.50. Which of the following equations best represents the cost, c, of buying p, pizzas?

• m = 5.50 + c

• c = 5.50 + m

• p = 5.50m

• c = 5.50p

Multiple Select

The range is also called the _________. Choose all that apply.

• input

• output

• independent variable

• dependent variable

Multiple Choice

The rate of change is also called the ___________

• slope

• ratio

• intercept

• point

Multiple Choice

The origin is represented by the ordered pair _______

• (0, 1)

• (2, 0)

• (0, 0)

• 0

Multiple Choice

The point where the graph crosses the x-axis is called the __________

• x-intercept

• y-intercept

Multiple Choice

The y-intercept is a point where the graph crosses the __________

• x-axis

• origin

• boundary

• y-axis

Multiple Choice

In the form y = mx + b, the m represents the __________

• x-intercept

• y-intercept

• origin

• slope

Multiple Choice

In the form y = mx + b, the b represents __________

• slope

• boundary

• y-intercept

• x-intercept

Multiple Choice

The form y - y₁ = m(x - x₁) is called __________ form.

• point-slope

• slope-intercept

• function

• standard

Multiple Choice

The form Ax + By = C is called standard form.

• true

• false

Multiple Choice

The graph of a linear function is a __________

• circle

• line

• parabola

• ellipse

Multiple Select

Choose all that apply.

A relation can be represented by which of the following:

• ordered pairs

• table

• graph

• equation

• mapping

Multiple Choice

A non-linear function can be represented by all of the following EXCEPT:

• parabola

• circle

• hyperbola

• line

Multiple Choice

A relationship between two variables is called a _________

• function

• correlation

• relation

• table

Multiple Choice

A shipping company charges a cost per pound plus a fixed fee to ship a package. The total cost, f(x), in dollars, of shipping x pounds is modeled by

f(x) = 4.99x + 3.29

• 3.29

• 4.99

• 4.99x

• x

Multiple Select

Select ALL the equations that can be represented by a straight line when graphed on the coordinate plane.

• x = 15y - 8

• 2x² + y = -3

• y = -3(x + 7)

• x = -3y³ + y

• y = x(3x + 2) -1

Multiple Select

Select ALL of the correlation coefficients that represent a linear model with a weak correlation.

• -0.987

• 0.333

• 0.204

• -0.001

• 0.675

Multiple Choice

A linear model shows that the relationship between the number of grocery items purchased and the total cost of the grocery bill has a correlation coefficient of 0.89.

• There is a strong relationship between the number of items purchased and the total cost of the grocery bill.

• If a grocery bill has a higher cost, then more items must have been purchased.

• Purchasing more items causes a higher cost of the grocery bill.

• There is no relationship between the number of items purchased and the total cost of the grocery bill.

Multiple Choice

The __________ is the name for the four regions of the coordinate plane.

• quadrilateral

• quartile

• quarter

• quadrant

Multiple Choice

The signs for a point in quadrant II are __________

• (+, +)

• (-, +)

• (-, -)

• (+, -)

Multiple Choice

The Domain of a function is the ...

• x-values

• inputs

• independent variables

• all of the above

Multiple Choice

The Range of a function is all of the possible x-values

• True

• False

Multiple Choice

A rule that assigns exactly one output for each input.

• Function

• Equation

• Expression

• Inequality

Multiple Choice

All of the values that go into a function.

• dependent variables

• inputs

• outputs

• y-values

Multiple Choice

All of the values that come out of a function.

• dependent variables

• x-values

• domain

• outputs

Multiple Choice

If you can draw any vertical line that intersects more than one point on the relationship, then it is not a function.

• True

• False

Multiple Choice

Which is an example of a linear function?

• xy = b

• y = mx + b

• $$f\left(x\right)\ =mx\ ^2+b$$
  

Multiple Choice

Which is an example of a non-linear function

• parabola

• horizontal line

• parallel lines

Multiple Choice

A Relation is a set of input and output values

• True

• False

Multiple Choice

The point where a line or curve crosses the y-axis of a graph is also known as

• x-intercept

• y-intercept

• m-slope

• origin

Multiple Choice

A linear equation written in the form of y=mx+b is called

• Standard Form

• Slope-point form

• Slope-intercept form

• Slope-equation form

Multiple Choice

Slope

• A. The measure of the steepness of a line

• B. The change in y over the change in x

• Neither A nor B

• Both A and B

Multiple Choice

What type of slope does the following graph have?

• Positive
• Negative
• Zero Slope
• Undefined

Multiple Choice

Find the slope:
y = 2x + 3

• 0
• 3
• 1.5
• 2

Multiple Choice

What is the y-intercept?

• -4
• 3
• 4
• 4/3

Multiple Choice

What is the slope in the picture?

• Positive
• Negative
• Zero
• Undefined

Multiple Choice

What is the slope in the picture?

• Positive
• Negative
• Zero
• Undefined

Multiple Choice

What is the slope of the line?

• Undefined
• Zero
• -1
• 2

Multiple Choice

y= $$\frac{2}{3}x$$  +5
In the equation above what is the slope (m)?

• 2/3x

• 5

• 2

• 2/3

Multiple Choice

In the equation below what is the y-intercept?

y=5x-3

• y

• -3

• x-3

• 5x

• 3

Match

Match each slope with the correct equation.

• $$-1$$

• $$1$$

• $$-\frac{1}{2}$$

• $$-2$$

• $$2$$

• $$y=-x-2$$

• $$y=x+2$$

• $$y=-\frac{1}{2}x+1$$

• $$y=-2x-1$$

• $$y=2x-\frac{1}{2}$$

Intercepts from Standard Form

Finding the x and y intercepts from linear functions that are in standard form.

What is an "x - intercept"?

• Where a linear function crosses through the x-axis.

• The x - axis is the horizontal axis.

What is a "y - intercept"?

• The y - intercept is where a linear function crosses through the y - axis on a coordinate plane.

• The y - axis is the vertical axis.

Finding the x-intercept

• Split up the linear function that is written in standard form.

• Set the 'x term' equal to the 'C' integer.

• Example: 2x + 5y = 4

• Set the 'x term' equal to the 'C' integer.

• 2x = 4

• NOW SOLVE FOR THE X INTERCEPT

• x = 4/2 therefore x = 2

Finding the y-intercept

• Split up the linear function that is written in standard form.

• Set the 'y term' equal to the 'C' integer.

• Example: 2x + 5y = 4

• Set the 'y term' equal to the 'C' integer.

• 5y = 4

• NOW SOLVE FOR THE Y INTERCEPT

• y = 4/5

Multiple Choice

Give the y-interecept

3x + y = 5

• y = 0

• y = 5

• y = -3

• y = -5

Multiple Choice

Give the x-intercept

3x + 8y = 24

• x = 8

• x = 0

• x = 3

• x = 24

Multiple Choice

What is the x and y intercept for the following equation ?
4x - 10y = 20

• x - int = 4    
  y-int = -10
• x-int = 5
  y-int = 2
• x - int = 5
  y int = -2
• x -int = 4
  y -int = -10

Multiple Choice

^{What is the x- intercept of the line?}

• x = 1

• x = 2

• x = -1

• x = -2

Poll

Are you comfortable with finding the x and y intercepts from the standard form of a linear function that is provided?

• YES

• NO

Multiple Select

1.Find the y intercept for

4x + 5 = y

• (5, 0)

• (4, 0)

• (0, 5)

• (0, 1)

Multiple Select

2. Find the x-intercept for

x - y = 1

• (1, 0)

• (0, 1)

Multiple Choice

3. Find the y-interecept

3x + y = 5

• (-3, 0)

• (0,5)

• (5, 0)

• (0, -3)

Multiple Choice

4. Find the x-intercept

3x + 8y = 24

• (8, 0)

• (0, 8)

• (3, 0)

• (0, 3)

Multiple Choice

5. Find the y-intercept

y = 3x + 8

• (8, 0)

• (0, 8)

• (3, 0)

• (0, 3)

Multiple Choice

6. True or false:

The x-intercept of a function ALWAYS occurs where y is equal to zero.

• True

• False

Multiple Choice

7. True or false:

The y-intercept of a function ALWAYS occurs where y is equal to zero.

• True

• False

Multiple Choice

8. Find the y-intercept

5x + y = 5

• (1,0)

• (0,5)

• (5,0)

• (0,1)

Multiple Choice

9. Find the y-intercept

x + 2y = 4

• 0

• 2

• 4

• KC Royals

Multiple Choice

10. Which ordered pair could represent a y-intercept on a graph?

• (6, 0)

• (0, 4)

• (-1, 3)

• (2, 0)