Semester 1 Review

Unit 1: Right Triangle Trig Review

In this lesson you will learn about Right Triangle Trig.

Some people remember these ratio definitions with the acronym: SOHCAHTOA!

Let's Try an Example

• Using the triangle to the right, find all 3 trig ratios for angle Z.

  • Start by labeling each side (draw an arrow out from Z to find the opposite, identify the longest side as the hypotenuse, and the last side, or the side next to Z, is the adjacent side

• Once you think you have the correct ratios go to the next slide.

How Did You Do?

Multiple Choice

Find the ratio for the given triangle.

• $$\frac{12}{5}$$  

• $$\frac{12}{13}$$  

• $$\frac{5}{12}$$  

• $$\frac{13}{12}$$  

Multiple Choice

Find the ratio for the given triangle. 

• $$\frac{12}{5}$$  

• $$\frac{5}{12}$$  

• $$\frac{13}{12}$$  

• $$\frac{12}{13}$$  

Using Trig Ratios to Solve for Missing Side Lengths

Trig ratios can be used to find missing side lengths.

Let's Try One..

• Use the diagram to solve for the missing side. Solve to the nearest tenth.

• Once you think you solved the problem correctly, go to the next page.

Check Your Answer

Let's Try One..

• Use the diagram to solve for the missing side. Solve to the nearest tenth.

• Once you think you solved the problem correctly, go to the next page.

Check Your Answer

Multiple Choice

Solve for x.

• 31.2

• 5.6

• 14.6

• 6.2

Multiple Choice

Solve for x.

• 17.2

• 27.1

• 8.3

• 30.9

Using Trig Ratios to Solve for Missing Angles
(finding angles starts at 4 minutes)

Let's Try One..

• Use the diagram to the right to solve for the missing angle measure. Round your answer to the nearest tenth.

• Once you think you solved the problem correctly, go to the next page.

Check Your Answer

Let's Try One...

• Use the diagram to the right to solve for the missing angle measure. Round your answer to the nearest tenth.

• Once you think you solved the problem correctly, go to the next page.

Check Your Answer

Multiple Choice

$$\cos\left(\theta\right)=?$$  

• $$\frac{3}{5}$$  

• $$\frac{4}{5}$$  

• $$\frac{3}{4}$$  

• $$\frac{4}{3}$$  

Multiple Choice

Find the measure of the indicated angle (?) round to the nearest tenth.

• 9.4
• 55.2
• 20.15
• 19.4

If you need more practice with this unit, work on the following Delta Math assignments from HW Week 1-4
Multistep Pythagorean Theorem
Simplifying Radicals
Trig Identity Opposite/Adjacent/Hypotenuse
Identifying trig ratios
Using trig ratios to find a side
Using trig to find angles
Trig Word Problems
Finding trig ratios (with radicals)

Unit 2 Functions: definition, composite, inverse, and transformation

Vocab

• Function: a relation in which each element of the domain is mapped to (paired with) exactly one element of the range

• Domain (x): the set of input values for which a function is defined

• Range (y): the set of output values corresponding to the domain values

• Relation: a set of ordered pairs

• Independent Variable: the variable in a function that represents the input values; the first coordinate in an ordered pair - the x value

• Dependent Variable: the variable in a function that represents the output values; the second coordinate in the ordered pair - the y value

Relation or Function?

• All functions are relations, but not all relations are functions.

• A function must have a unique input for every output.

• The domain (x-value) may only be used once.

• The range (y-values) may be used more than once if they have different domains (x-values)

Vertical Line Test

• Visual representation to see if a relation is a function.

• If a vertical line passes through no more than one point, then the graph represents a function.

Multiple Choice

What is the definition of function?

• Has inputs and outputs

• Every input has only ONE output

• Inputs have different outputs every time

• x-values and y-values

Multiple Choice

All of the x values or inputs are called what?

• Domain
• Range
• Relation
• Function

Multiple Choice

How would you write y = 2x - 3 in function notation?

• y = 2x-3

• f = 2x - 3

• f(x) = 2x-3

Multiple Choice

Which graph does NOT pass the vertical line test?

• Graph 1
• Graph 2
• Graph 3
• Graph 4

Multiple Choice

Is this graph a function or not a function? 

• Function 
• Not a Function

Multiple Choice

Is this graph a function or not a function? 

• Function
• Not a Function

Multiple Choice

Does this relation represent a function?
(1,5), (1,6), (1,7), (1,8), (1,9)

• Yes
• No

Multiple Choice

What is the range of the following relation:
(9, -2) (4, 3)  ( 8, 10) ( -4, 8)

• -4, 4, 8, 9
• -2, 3, 8, 10
• (9, -2) ( 4, 3)
• (8, 10) (-4, 8)

Multiple Choice

Is this graph a function or not a function? 

• Function
• Not a Function 

Multiple Choice

Is this mapping a function or not a function? 

• Function
• Not a Function

Multiple Choice

Is the relation a function? Why.

• Yes, because the x-value 11 has two y-values pair with it.
• Yes, because each x-value has only one y-value paired with it.
• No, because the x-value 11 has two y-values pair with it.
• No, because each x-value has only one y-value paired with it.

Multiple Choice

Which set of values is a function?

• (9,5) (10,5) (9,-5) (10,-5)
• (3,4) (4,-3) (7,4) (3, 8)
• (6,-5) (7, -3) (8, -1) (9, 1)
• (2, -2) (5, 9) (5, -7) (1, 4)

Multiple Choice

What is x if
f(x) = -2?

• 4
• 0
• 3
• 8

Multiple Select

Use the graph to identify the definition.

• Function
• Not a Function
• Not a relation

Multiple Choice

Given f(x) =
2x² + 5x - 17, find f(-1). 

• f(-1) = -20
• f(-1) = -24

Multiple Choice

Given f(x) = -4x - 10 and f(x) = 10. Find x

• x = -5
• x = 5
• x = -50
• x = 30

Multiple Choice

The function h(t) represents the height in inches of a plant after t weeks. What does the notation h(4) = 5 mean?

• After 5 week the plant is 4 inches tall
• The plant grows at a rate of 5/4 inch per week
• After 4 weeks the plant is 5 inches tall
• The plant grows at a rate of 1 inch per week

Multiple Choice

What is f(8)? 

• 8
• 0
• 3
• -2

Multiple Choice

Let

$$h\left(x\right)=13-5x$$   What is $$h\left(5\right)=$$  ?

• $$h\left(5\right)=12$$  

• $$h\left(5\right)=3$$  

• $$h\left(5\right)=-2$$  

• $$h\left(5\right)=-12$$  

Poll

What is your level of understanding of functions right now?

• This is easy, I've got this

• I get it, but may have some questions as I work through the assignments.

• I am confused and need help. I will email you to set up a time.

• What class is this?

Standard Mastery

Multiple Choice

Find f(h(5)) based on the graphs of f(x) and g(x)

• -6
• -3
• 6
• 2

Multiple Choice

If the point (-7,5) lies on the graph of f(x) , which of the following points must lie on the graph of its inverse?

• (5,-7)

• (-1/7,1/5)

• (7,-5)

• (1/7,-1/5)

Multiple Choice

The function y=f(x) has an inverse function  $$y=f^{-1}\left(x\right)$$  . If f(a)=-b then which of the following must be true?

• $$f^{-1}\left(-b\right)=-a$$  

• $$f^{-1}\left(\frac{1}{a}\right)=-\frac{1}{b}$$  

• $$f^{-1}\left(-b\right)=a$$  

• $$f^{-1}\left(b\right)=-a$$  

Multiple Choice

What is f(1)?

• 1
• 5
• 0
• 4

Multiple Choice

The graph of the function y=g(x) is shown below. The value of

$$g^{-1}\left(2\right)$$  is

• 2.5

• -4

• 0.4

• -1

Multiple Choice

Find the inverse of

$$y=\frac{1}{4}x-7$$  

• $$f^{-1}\left(x\right)=4x\ +\ 7$$  

• $$f^{-1}\left(x\right)=-4x\ +\ 28$$  

• $$f^{-1}\left(x\right)=4x\ -\ 7$$  

• $$f^{-1}\left(x\right)=4x\ +\ 28$$  

Multiple Choice

What does it mean to find the inverse of a function?

• the x's and y's are switched

• the x's and y's are divided by 2

• the x's and y's are made negative

• the x's and y's are the same

Multiple Choice

If the equation of f(x) goes through (1, 4) and (4, 6), what points does f^-1(x) go through?

• (1, 4) and (4, 6)

• (-4, -1) and (-6, -4)

• (-1, -4) and (-4, -6)

• (4, 1) and (6, 4)

Exponential Functions: The Basics

• the a is the initial value

• the b is the constant ratio(multiplier)

• b> 1 = growth

• b< 1 = decay

• what are the domain and range?

Open Ended

$$y=\ 60\left(\frac{1}{3}\right)^x$$  

Evaluate the function and fill in the table:

Growth and Decay

Multiple Select

Put these functions in order from least to greatest based on their Y-INTERCEPT.

• B C A

• B A C

• C A B

• C B A

Open Ended

Separate these equations into linear and exponential functions:

$$a:\ y=\ 4-2x$$   $$b:\ y=\ \frac{3}{4}x+\ 12$$   $$c:\ y=\ 3\left(2\right)^x$$   $$d:\ y=\ -5\ +\ 3^x$$  

Multiple Select

Find the future value of $500 invested at 12% for 10 years compounded annually.

• $1,052.92

• $1,552.92

• $10,529.20

• $15,529.20

Open Ended

A population of 6 gerbils doubles every 4 months. Write a function to show this growth.

Open Ended

$$y=\ 6\left(2\right)^x$$  

How many gerbils would there be in 5 years (60 months)? Use the function given. 

Open Ended

How can you tell from an equation if an exponential function is showing growth or decay?

Multiple Select

What kind of transformation is represented?

• up 1 unit
• left 1 unit
• right 1 unit 
• down 1 unit

Multiple Select

What is the equation that represents the exponential function in the image below?

• y=3(½)^x
• y=3(2)^x
• y=(2)^x
• y=2(3)^x

Open Ended

Write an exponential function for the table

Multiple Select

What is the asymptote of y = 2^x ?

• y = 0

• x = 0

Open Ended

$$f\left(x\right)\ =\ 20,000\left(0.75\right)^x$$  This function shows a town that had an original population of ________ and is decreasing by _____ percent each year. 

Open Ended

A family buys a house for $165,000 and it increases in value by 4% each year. What will the value for the house be in 5 years?

Multiple Select

If g(x) is a transformation of f(x), describe the transformation.

$$f\left(x\right)=2^x$$      $$g\left(x\right)=2^x+4$$   

• Up 4 units

• Down 4 units

• Left 4 units

• Right 4 units

Multiple Select

Which statement below is TRUE regarding the two functions?

• Both functions are linear

• Both functions are increasing

• Both functions have a y-intercept of 1

• Both functions are exponential growth

Multiple Select

What is the total interest earned on an account with an initial deposit of $500 invested at 12% for 10 years?

• $552.92

• $1,052.92

• $1,552.92

COMPOUND INTEREST

Compound interest is when your principle and any earned interest both earn interest.

Multiple Select

What is the maximum/minimum of a parabola called?

• vertex

• point

• top

• bottom

Multiple Select

• A

• B

• C

• D

Multiple Select

What is the formula above primarily used for?

• Finding the axis of symmetry from standard form
• Finding the vertex from standard form
• Finding the vertex from vertex form
• Finding the vertex from standard form

Multiple Select

What is the name of this form:
y = ax² + bx + c

• vertex form
• standard form
• factored form
• zero form

Multiple Choice

What is the transformation?

f(x) = ( x - 3)²

• right 3

• left 3

• up 3

• down 3

Multiple Select

Does the equation open up or down? 
Y = -3x² +7x - 2

• up
• down
• Neither; it opens to the right.
• Neither; it opens to the left.

Multiple Select

Which is the vertex of y = x²+ 16x + 71 ?

• V(-8, 7)

• V(-8, 3)

• V(8, 10)

• V(16, -2)

Multiple Select

Find the vertex of the quadratic:

y = -2x² + 4x + 3

• (0, 3)

• (-1, 5)

• (1, 5)

• (1, -5)

Multiple Select

Which is the vertex of y = x²+ 16x + 71 ?

• V(-8, 7)

• V(-8, 3)

• V(8, 10)

• V(16, -2)

Multiple Select

Where is the axis of symmetry for this parabola?

• x = −1

• x = 0

• y = 3

• x = −3

Multiple Select

What are the zeros of the parabola? 

• 1, 5
• 3, -5 
• 3, 0
• -6, 1

Multiple Select

What is the maximum/minimum of a parabola called?

• vertex

• point

• top

• bottom

Multiple Choice

What is the y-intercept of this parabola?

• (−1, 3)
• (−3, 11)
• (−2, 5)
• (0, 5)

Multiple Select

What is the vertex of this parabola?

• (−1, 3)

• (−3, 11)

• (−2, 5)

• (0, 5)

Multiple Choice

Does the equation open up or down? 
Y = -3x² +7x - 2

• up
• down
• Neither; it opens to the right.
• Neither; it opens to the left.

Multiple Select

Find the y-intercept of f(x) = 2x² - 2x + 1

• (1,0)

• (-2,0)

• (0,1)

• (0, -1)

Multiple Select

Find the y-intercept y = x² - 6x + 5

• (0, 8)

• (0, - 6)

• (5, 0)

• (0, 5)

Multiple Select

Which is the vertex of y = x²+ 16x + 71 ?

• V(-8, 7)

• V(-8, 3)

• V(8, 10)

• V(16, -2)

Multiple Choice

What is the vertex of this function: (x+4)²-5

• (5,4)

• (4,5)

• (4, -5)

• (-4, -5)

Multiple Choice

What is the axis of symmetry for this function: (x+4)²-5

• x= 4

• x=-4

• x= 5

• x= -5

Multiple Choice

What is the axis of symmetry for this function: 2x²+8x+2

• x= 8

• x= 2

• x= -2

• x= 4

Multiple Choice

The axis of symmetry for the function -x²+10x-17 is 5. What is the vertex?

• (-5, 8)

• (5,8)

• (8,5)

• (-8,5)

Multiple Choice

What is the axis of symmetry for this function: -x²+10x-17

• x= 10

• x= -10

• x= -5

• x= 5

Multiple Choice

What is the parent function of a quadratic equation?

• f(x)=x

• f(x)=x²

• f(x)=ax²+bx+c

• f(x)=a(x-h)²+k

Multiple Select

Describe the transformation that occurred.

h(x) = f(x) + 2

• up 2 units

• left 2 units

• Vertical Strech

• right 2 units

Multiple Select

Describe the transformation that occurred

p(x) = f(x + 3)

• Right 3 units

• Vertical Stretch

• Up 3 units

• Left 3 units

Multiple Choice

f(x) = ( x - 9)² + 5

• left 9

  up 5

• right 9

  up 5

• left 9

  down 5

• left 9

  up 5

Multiple Choice

g(k) = -(k+ 4)³

• reflected over x-axis

  right 4

• reflected over x-axis

  left 4

• reflected over y-axis

  right 4

• reflected over y-axis

  left 4

Multiple Choice

If I know that f(2) = 4, what is another way I can write my answer?

• 2,4

• (2, 4)

• That is the only way to write it.

Multiple Choice

• 9 - √17
• 4
• 2
• √8

​List all transformations you see. Check your answers with Desmos or your graphing calculator.

​

f(x) = -2( x - 7)²​ - 8

​

Multiple Select

Match the correct graph to the equation.

• A

• B

• C

• D

Multiple Choice

Using f(x) , g(x) , or h(x) instead of y is called using ......

• Slope Notation

• Variable Notation

• Intercept Notation

• Function Notation

Multiple Choice

Is this table a function or not a function? 

• Function
• Not a Function

Multiple Choice

f(x) = 3x + 10

g(x) = x - 2

Find g(f(-10))

• -42

• -26

• -18

• -22

Multiple Choice

$$Given:\ f\left(x\right)=x^{2\ }and\ g\left(x\right)=5x+3$$  

If you are asked to find f(g(2)), where do you plug the 2 in first?

• I plug the 2 into the f function

• I plug the 2 into the g function

Multiple Choice

Given h(x) = 5 - 9x, find h(8). 

• h(8) = -360
• h(8) = -67

Multiple Choice

Given f(x) = 3x + 10
and g(x) = x - 2
Find f(g(5))

• 19
• 23
• -10
• None of these.

Multiple Choice

.

• Function
• Not a Function

Multiple Choice

$$Given\ f\left(x\right)=x^2and\ g\left(x\right)=5x+3$$  

Find f(g(2)).  

These are the same functions as the problem above.  After you plug the 2 into the g function,the output value is 13.  For the next step of this problem, do you plug the 2 into f(x) or the 13 into f(x)?

• I plug the 13 into f(x)

• I plug the 2 into f(x)

Multiple Choice

Evaluate for f(1)

• 0
• -3
• 3
• 2

Multiple Choice

If a question asks you to find f(2), what does that mean?

• Let x = 2

• Let y = 2

Multiple Choice

If a question asks you to find f(2), what does that mean?

• Let x = 2

• Let y = 2

Multiple Select

Which graph does NOT pass the vertical line test?

• Graph 1
• Graph 2
• Graph 3
• Graph 4

Multiple Choice

If f(x) = 2x + 1, what would be your first step to find f(1)?

• Plug a 1 in for all the x's

• Set the function equal to 1

• Only plug a 1 into the left side

• Only plug a 1 into the right side

Multiple Choice

f(x) is the same thing as ______.

• y
• x
• input
• an equation

Graphing and finding the Zeroes Quadratic Functions Lesson

Multiple Select

What do we call the graph of a quadratic function?

• parabola
• linear
• cubic
• sinusoidal

Multiple Select

What is the equation for axis of symmetry?

• y = 3

• x = 3

• x = 5

• x = 1

Multiple Select

Where is the axis of symmetry for this parabola?

• x = −1

• x = 0

• y = 3

• x = −3

Poll

Are you ready?