Algebra 2 | Fall Final Review

Algebra 2 | Fall Final Review

Let's revisit some quick topics together!

You MAY use a calculator on any question.

​Reviewing Parent Functions

​Parent functions are the most BASIC form of a function.

​

Your calculator can help you identify key features if you forget!

Multiple Select

What are the key features of the quadratic parent function?

$$f\left(x\right)=x^2$$  

Choose ALL that apply.

• Domain is $$\left(-\infty,\ \infty\right)$$  

• Range is $$\left[0,\infty\right)$$  

• This function has a minimum value of 0.

• There is no symmetry.

Multiple Select

Which funcion(s) below have BOTH a domain and range of ALL REAL NUMBERS?

Select all that apply.

• $$y=x$$  

• $$y=\sqrt{x}$$  

• $$y=x^3$$  

• $$y=\sqrt[3]{x}$$  

Multiple Select

Identify the transformations.

$$3f\left(x+2\right)-5$$

• Shifted right 2

• Shifted down 5

• Vertical stretch by 3

• Shift left 2

• Up by 3

​How can we find inverse functions?

Remember: Inverse functions will ALWAYS undo one another.

​

​We start by switching x and y.

​

​So if (5,3) is on f(x), then (3,5) is on the inverse - x and y are swapped!

Multiple Choice

You are given a table that has the points (-7,-5), (-1,-1), and (8,2).

Which point would be on the INVERSE of this function?

• $$\left(1,1\right)$$  

• $$\left(2,8\right)$$  

• $$\left(5,7\right)$$  

• None would be on the inverse.

Multiple Choice

A linear function is below. Which equation is the INVERSE?

$$f\left(x\right)=\frac{3}{4}x-12$$  

• $$f^{-1}\left(x\right)=\frac{4\left(x+12\right)}{3}$$  

• $$f^{-1}\left(x\right)=-\frac{3}{4}x+12$$  

• $$f^{-1}\left(x\right)=\frac{4}{3}x+12$$  

• $$f^{-1}\left(x\right)=\frac{1}{3}x+48$$  

​Solving Systems

​When we solve systems of equations, we are looking where they CROSS!

​

​We can solve using:

• ​Graphing (2 lines)

• ​Substitution

• ​Elimination

• ​With a matrix

Multiple Select

Corrie was solving the system of equations using her calculator.

$$3x+2y+5z=7$$  

$$2x+3y+z=9$$  

$$4x+y+2z=-2$$  

Select ALL the rows you would use in a matrix.

• $$\left[3\ \ \ \ 2\ \ \ \ 5\ \ \ \ 7\right]$$  

• $$\left[2\ \ \ \ 3\ \ \ \ \ 1\ \ \ \ \ 9\right]$$  

• $$\left[4\ \ \ \ 1\ \ \ \ \ 2\ \ \ \ \ -2\right]$$  

• $$\left[1\ \ \ \ \ 2\ \ \ \ \ 3\ \ \ \ -2\right]$$  

Multiple Choice

Identify the solutions.

$$y=3x-7$$

$$y=-x^2+3x+9$$

• (0,-7)

  (1,-4)

• (4,5)

  (-4,-19)

• 4 and -4

• No solutions.

​Solving Absolute Value

​Since absolute value is the distance from 0, we could have 2 possibilities.

​

​Both 3 and -3 are three away from 0.

​

1. Absolute value by itself?

2. ​Split into TWO equations.

3. ​Solve BOTH.

4. ​Check BOTH solutions.

Multiple Choice

Which statement is TRUE?

$$f\left(x\right)=3\left|x+2\right|-4$$

• There is a minimum value at -4.

• There is a minimum value at -2.

• There is a maximum value at 3.

• There is no minimum or maximum value.

Multiple Select

Solve the absolute value linear equation.

$$\frac{3}{4}\left|4x+8\right|-3=12$$  

Select ALL answers.

• $$x=3$$  

• $$x=-5$$  

• $$x=-7$$  

• $$x=0$$  

​Synthetic Division

​We can use synthetic division to help us divide polynomials FASTER!

Multiple Select

Perform the operation. Identify TWO correct answers.

$$\left(m^3-9\right)\div\ \left(m+1\right)$$

• $$m^2-m+1-\frac{10}{m+1}$$

• $$m^2-m+1+\frac{-10}{m+1}$$

• $$m^2-10$$

• $$m^2+m+1-\frac{8}{m+1}$$

• $$m^2+m+1-\frac{8}{m+1}$$

Multiple Choice

Which of the following is a linear factor of...

$$x^4+8x^3-15x^2-94x+120$$  

• $$\left(x-5\right)$$  

• $$\left(x+3\right)$$  

• $$\left(x+4\right)$$  

• $$\left(x+2\right)$$  

​Factoring Shortcuts

Multiple Choice

What is the factored form of

$$64m^3+27t^3$$

• $$\left(4m+3t\right)^3$$

• $$\left(4m-3t\right)\left(4m+3t\right)$$

• $$\left(4m+3t\right)\left(16m^2-12mt+9t^2\right)$$

• $$\left(4m+3t\right)\left(4m^2+12mt+3t^2\right)$$

Multiple Choice

Which of the following would be the fully factored form?

$$4x^3-8x^2-12x$$

• $$4x\left(x^2-2x-3\right)$$  

• $$4x\left(x+1\right)\left(x-3\right)$$  

• $$\left(4x^2-3\right)\left(x+1\right)$$  

• $$x\left(4x-3\right)\left(x+1\right)$$