1.4 and 1.2 Linear Equations

​Title: 1.4 Linear Functions
EQ: What is slope intercept form? What does each letter stand for?


Warm Up: Talk to a partner
1. what do they remember about linear functions? 2. what is their favorite ice cream flavor?

Be ready to share your partners name and their answer to #2

Slope Intercept Form

What is slope-intercept form of an equation for a line?

​Linear Functions

Slope - write in your notes what you don't remember

• The slope is a number that tells how steep the line will be.

• Slope is the m in y=mx+b. Slope is a fraction... if the number you have is not a fraction, put a 1 under it (3 = 3/1)

• After putting the y-intercept on the graph, the slope tells you where to go next. The top number is the rise (it tells you how far to go up if its positive, or down if its negative), and the bottom number is the run (it tells you how far to go to the right)

• If slope is positive, it will go up to the right ( / ), and if slope is negative, it will go down the the right ( \ )

Multiple Choice

Slope is know as rise/

• run

• fall

• none

• rise

Multiple Choice

What is the slope in the equation: y=4x+3

• 3

• 4x

• 4

• x

Find the slope of the graph.

Remember: Count Rise over Run

Multiple Choice

What type of slope does the following graph have?

• Postive

• Negative

Poll

What is the slope of the graph?

• $$\frac{1}{2}$$

• $$-\frac{1}{2}$$

• $$\frac{2}{1}$$

Multiple Choice

What is the slope in the equation: y=4x+3

• 3

• 4x

• 4

• x

Multiple Choice

What is the rate of change in the equation?
y=8x+6

• 8
• 6

Multiple Choice

Find the rate of change

• 1

• 2

• 11

• Not a constant slope/Non-Linear

Y-Intercept - Write in your notes what you don't remember

• The y-intercept is the number in place of b in y=mx+b.

• This is the starting point when graphing a line!

• The y-intercept is where the line will cross through the y-axis on the coordinate plane.

• Pay attention to the symbol in front of b in the equation! This tells you if it is positive or negative (y=3x-4 means b= -4)

• The y-intercept written as an ordered pair is (0,b)... if b=-4, the ordered pair will be (0,-4)

Linear Graph - Answer as a class, you dont need to write this

• Identify the Slope: ______

• Identify the y-intercept: _______

• Write the slope-intercept form of the equation. ____________________

Identify the slope and y-intercept from the linear equation: y = 4x - 2

• Slope (m) =

• y-intercept (b) =

Identify the slope and y-intercept from the linear equation: y = 5 - x

• slope (m) =

• y-intercept (b) =

Horizontal Graph - answer as a class, you dont need to write this

• Identify the slope (m): _______

• Identify the y-intercept (b): ____

• Identify the slope-intercept form of the equation: ____________

Write the equation of the linear function

Multiple Choice

How would you write the equation with a slope of 2/3 and a y-intercept of -3

• y=3/2x+3

• y=2/3-3

• y=2/3x-3

• y=2/3x+3

Multiple Choice

What is 'b' in this linear equation? y= 5x - 6

• 6

• 5

• y

• -6

Multiple Choice

What is the initial value in this equation?
y=8x+6

• 8
• 6

Multiple Choice

What is the slope in the equation y=2x + 3?

• 2

• 3

• 5

• Cannot be Determined

Multiple Choice

What is the y-intercept in the equation y= -2x + 5

• -2

• 3

• 5

• Cannot be Determined

Multiple Choice

Determine the slope and y-intercept from the following equation: y = 3/2x + 3

• slope: 3

  y-intercept: (0,3)

• slope: 3/2

  y-intercept: (0,3)

• slope: -2

  y-intercept: (0,3/2)

• slope: -3/2

  y-intercept: (3,0)

Multiple Choice

Which letter represents the y-intercept in y=mx+b?

• y

• m

• x

• b

Multiple Choice

What is the y-intercept in the following equation: y=-4x-5

• -4

• -4x

• 5

• -5

Multiple Choice

What does "m" represent in y=mx+b?

• y-intercept

• slope

• nothing

Multiple Choice

Write the slope-intercept form of the linear equation that represents the line with the given slope and y-intercept.

Slope (m) = 2/3

Y-intercept (b) = -6

• y = -6x + 2/3

• y = 2/3x - 6

Multiple Choice

Write the slope-intercept form of the linear equation that represents the line with the given slope and y-intercept.

Slope (m) = -3

Y-intercept (b) = 12

• y = -3x + 12

• y = 12x - 3

Multiple Choice

Identify the slope and y-intercept from the equation.

y = 2x + 3

• slope = 2

  y-intercept = 3

• slope = 3

  y-intercept = 2

Multiple Choice

Identify the slope and y-intercept from the equation.

$$y\ =\ -7\ +\ \frac{1}{3}x$$  

• Slope (m) = -7
  
  y-Intercept (b) =  $$\frac{1}{3}$$  

• Slope (m) =  $$\frac{1}{3}$$  
  
  y-Intercept (b) = -7

Multiple Choice

I can identify the slope and y-intercept to graph a linear function.

Which equation represents the graph shown?

• $$y=\frac{3}{2}x-2$$  

• $$y=\frac{2}{3}x-2$$  

• $$y=-\frac{2}{3}x+2$$  

• $$y=-\frac{3}{2}x-2$$  

​Graphing Linear Equations

1. Plot the y - intercept

2. ​Plug an x-value into the equation to get a y-value

3. ​Plot your (x, y) pair

4. ​Draw a line to connect the (x , y) pair with your y - intercept

​

Let's Practice! Graph y = 2x -3

Multiple Choice

Write an equation that describes the linear function

• $$6x+4y=10$$

• $$y=\frac{1}{2}x-4$$

• $$y=-\frac{2}{3}x+4$$

Draw

Graph the equation y = -2x + 4

Multiple Choice

Compare $$y=x$$  and $$y=5x$$

• $$y=5x$$  is LESS STEEP than $$y=x$$  

• $$y=5x$$  is STEEPER than $$y=x$$  

• $$y=x$$ has the SAME slope as $$y=5x$$  

• $$y=x$$  is STEEPER than $$y=5x$$  

Multiple Choice

Compare $$y=x$$  and $$y=\frac{3}{2}x$$  

• $$y=\frac{3}{2}x$$  is STEEPER than $$y=x$$  

• $$y=x$$  is STEEPER than $$y=\frac{3}{2}x$$  

• $$y=\frac{3}{2}x$$  is LESS STEEP than $$y=x$$  

• $$y=x$$  has the SAME slope as $$y=\frac{3}{2}x$$  

Multiple Choice

Compare $$y=x$$  and $$y=\frac{1}{2}x$$  

• $$y=\frac{1}{2}x$$  is STEEPER than $$y=x$$  

• $$y=\frac{1}{2}x$$  is LESS STEEP than $$y=x$$  

• $$y=x$$  is STEEPER than $$y=\frac{1}{2}x$$  s

• $$y=x$$  has the SAME slope as $$y=\frac{1}{2}x$$  

Multiple Choice

Compare $$y=x$$  and $$y=x-1$$  

• $$y=x-1$$  is STEEPER than $$y=x$$  

• $$y=x-1$$  SHIFTED DOWN 1 unit from $$y=x$$  

• $$y=x-1$$  is LESS STEEP than $$y=x$$  

• $$y=x-1$$  SHIFTED UP 1 unit from $$y=x$$  

Multiple Choice

Compare $$y=x$$  and $$y=x+6$$  

• $$y=x+6$$  SHIFTED DOWN 6 units from $$y=x$$   

• $$y=x+6$$  is STEEPER than $$y=x$$  

• $$y=x+6$$  SHIFTED UP 6 units from $$y=x$$  

• $$y=x+6$$  is LESS STEEP than $$y=x$$  

Multiple Choice

The graphs of linear functions f(x) and g(x) are shown on the grid. Which function is best represented by the graph of g(x) ?

• $$g\left(x\right)=f\left(x\right)+3$$  

• $$g\left(x\right)=f\left(x\right)-3$$  

• $$g\left(x\right)=3f\left(x\right)$$  

• $$g\left(x\right)=-3f\left(x\right)$$  

Multiple Choice

The graphs of linear functions f(x) and g(x) are shown on the grid. Which function is best represented by the graph of g(x) ?

• $$g\left(x\right)=f\left(x\right)+4$$  

• $$g\left(x\right)=f\left(x\right)+5$$  

• $$g\left(x\right)=3f\left(x\right)$$  

• $$g\left(x\right)=\frac{1}{3}f\left(x\right)$$  

Multiple Choice

$$f\left(x\right)=x$$  and  $$g\left(x\right)=2x-4$$  

Which statement about the relationship of these two graphs is true?

• The graph of $$g\left(x\right)$$   is steeper than the graph of $$f\left(x\right)$$  , and the y-intercept has been translated up.

• The graph of $$g\left(x\right)$$ is less steep than the graph of $$f\left(x\right)$$  , and the y-intercept has been translated up

• The graph of $$g\left(x\right)$$   is less steep than the graph of $$f\left(x\right)$$  , and the y-intercept has been translated down.

• The graph of $$g\left(x\right)$$   is steeper than the graph of $$f\left(x\right)$$  , and the y-intercept has been translated down.

Multiple Choice

$$f\left(x\right)=x$$  and  $$g\left(x\right)=-\frac{2}{3}x$$  

Which statement about the relationship between these two graphs is true?

• The graph of g is less steep than the graph of f.

• The graph of g is less steep than the graph of f, and is reflected.

• The graph of g is reflected, and the y-intercept is translated down.

• The graph of g is steeper than the graph of f, and is reflected.

Word Cloud

Type any 1-2 words that you link to a "linear function".

Open Ended

1. How can you tell by looking at a graph that a relationship is linear?

Drag and Drop

Write down in your notes!

x-intercept is the point ​

Placeholder

y-intercept is the point ​

Placeholder

Drag these tiles and drop them in the correct blank above

(2, 0)

(0, 4)

2

4

(0, 2)

(4, 0)

(-2, 0)

(0, -4)

Drag and Drop

Write down in your notes!

x-intercept is the point ​

Placeholder

y-intercept is the point ​

Placeholder

Drag these tiles and drop them in the correct blank above

(6, 0)

(0, -4)

6

-4

(0, 6)

(-4, 0)

(0, 4)

(-6, 0)

Multiple Choice

What is the x and y intercepts for the linear function shown on the graph?

• x intercept: $$\left(0,\ -2\right)$$  

  y intercept: $$\left(-3,\ 0\right)$$  

• x intercept: $$\left(2,\ 0\right)$$  

  y intercept: $$\left(0,\ 3\right)$$  

• x intercept: $$\left(-2,\ 0\right)$$  

  y intercept:  $$\left(0,\ -3\right)$$  

• x intercept: $$\left(0,\ 2\right)$$  

  y intercept: $$\left(3,\ 0\right)$$  

Multiple Choice

In an x intercept, which value is always 0?

• the x value

• the y value

Linear Functions...

are straight lines on a graph

Linear Functions...

have a constant rate of change

Linear Functions...

are represented by equations that have:

* no exponents

* no absolute value signs

* fractions with variables only in the numerator

Non-Linear Functions...

have curves or

two or more different slopes

Non-Linear Functions...

do not have a constant rate of change

Nonlinear Functions...

are represented by equations that have:

* exponents

* absolute value signs

* fractions with variables in the denominator

Nonlinear examples

Multiple Choice

Linear or Nonlinear?

• Linear

• Non-Linear

Multiple Choice

Linear or Nonlinear?

• Linear

• Non-Linear

Multiple Choice

Does the first table represent a linear or nonlinearn function?

• linear

• nonlinear

Multiple Choice

Linear or Nonlinear?

• Linear

• Nonlinear

Multiple Choice

$$\left|x\ +\ 5\right|\ =\ 43$$  

• Linear

• Nonlinear

Multiple Choice

Linear or Nonlinear?

• Linear

• Non-Linear

Multiple Choice

$$y=x^4$$  

• Linear

• Nonlinear

Multiple Choice

$$y\ =\ 6x\ +\ 5$$  

• Linear

• Nonlinear

Multiple Choice

$$y=\frac{6}{x^2}\$$  

• Linear

• Nonlinear

Multiple Choice

The function f(x) = x

has been transformed to

g(x) = x - 5.

Which of the following describes the change?

• vertical shift up 5 units

• vertical shift down 5 units

• horizontal shift right 5 units

• horizontal shift left 5 units

Multiple Choice

If the graph of the parent function f(x) = x is shifted 7 units up and is vertically compressed by a factor of 1/3, what is the equation for the function?

• g(x) = 7x + 1/3

• g(x) = 1/3x - 7

• g(x) = 1/3x + 7

• g(x) = 7x - 1/3

Multiple Select

What are the transformations shown here? Check all that apply

$$-x+3$$  

• reflection

• shifted down 3

• shifted up 3

• compressed by 3

Multiple Select

How did the black function transform from the red parent function? Check all that apply

• Refleted

• Stretched by $$\frac{2}{5}$$

• Compressed by

  $$\frac{2}{5}$$

• shifted up 2

• shifted right 5

Multiple Choice

The cost of Nate’s service is described by y = 4x + 6. The table shows the cost of Beth’s service. Which service is cheaper for 8 movies in one month?

• Beth's service ($38 vs. $40) is cheaper
• Nate’s service ($42 vs. $44) is cheaper
• Nate’s service ($38 vs. $40) is cheaper
• Beth's service ($42 vs. $44) is cheaper

Multiple Choice

Is the equation linear or nonlinear?

• Linear
• Nonlinear

Multiple Choice

Is this function linear or nonlinear?

• Linear
• Nonlinear

Multiple Choice

Which function has the GREATEST rate of change?

Multiple Choice

$$y=x^{2\ }+x$$  Is this function linear?

• Yes

• No

Multiple Choice

$$y-5=4x$$  Is this linear?

• Yes

• No

Multiple Select

Which equations represents y as a linear function of x? Select all that apply

• y=x³

• y=9x

• y=x

• $$y=\frac{1}{x}$$

Poll

Here is the equation y = 6x - 10. Where does this line cross this y-axis?

• 0

• 6

• 10

• -10

Multiple Choice

To prepare for a recent road trip, Jill filled up her 19-gallon tank. She estimates that her SUV will use about three gallons per hour. Write an equation to model the amount of gasoline, G, remaining in her tank after t hours.

• G=19 – 3t

• G=19 + 3t

• G=3 - 19t

• G=3 + 19t

Multiple Choice

The temperature outside is currently 68°F and falling at a steady rate of 4 degrees every hour. Write a linear equation that relates the Fahrenheit temperature, F, to the time in hours, t, it has been falling.

• F = -4x + 68

• F(t) = -4t + 68

• F(t) = 4t + 68

• F(t) = 4x + 64

Multiple Choice

Andy has $310 in his account. Each week, w, he withdraws $30 for his expenses. Write a linear equation to represent the scenario.

• f(x) = -310w + 30

• y = -30w + 310

• y = 30w + 310

• f(x) = -30x + 310

Multiple Choice

The graph of a linear equation is always a straight line.

• true

• false