Semester Exam Review

Quadratic Inequalities

Graph the quadratic in the calculator or on Desmos.

Choose the solution or set of solutions that fall in the appropriate area:

Multiple Choice

Which correctly shows the inequality $$y\ge-x^2+2x+1$$  ?

Multiple Choice

Which correctly shows the inequality $$y\ge-\left(x+2\right)^2-4$$  ?

Multiple Choice

What is the solution to the inequality $$y\le-x^2+4x-1$$  ? (Hint: use Desmos!)

Systems of Equations

Graph the two equations in the calculator or on Desmos.

The solution(s) may be found at the intersection(s) of the two functions.

Linear Systems can have the following types of solutions

Multiple Choice

Find the solution:
y = x² - 4x + 6
y = x + 2

• (3, 1) and (6, 4)
• (-1, 1) and (4, 2)
• (1, 3) and (4, 6)
• (1, 4) and (3, 6)

Multiple Choice

• A
• B
• C
• D

Multiple Choice

What is the solution(s) to the system shown?

$$y=-2x^2-4x-1$$

$$y=2x+4$$

• (-9,0)

• (3,-1) & (5,8)

• No Solution

• (-2,2) & (0,6)

Cubic Functions

The inverse of a cubic function is a cube root function.

Inverses are reflections of each other about the diagonal line y = x

To find the inverse, you can compare the graphs OR switch the x-and y-values.

Multiple Choice

Which functions are inverses of each other?

Multiple Choice

Find the inverse of $$f\left(x\right)=\sqrt{x+2}+9$$  

• $$f\left(x\right)=\left(x-9\right)^2-2$$  

• $$f\left(x\right)=\left(x+9\right)^2+2$$  

• $$f\left(x\right)=\sqrt{x+9}+2$$  

• $$f\left(x\right)=\sqrt{x-9}-2$$  

Multiple Choice

Find the inverse function of $$y=-2+\sqrt{x+7}$$  

• $$y=\left(x+2\right)^2-7$$  

• $$y=\left(x-2\right)^2+7$$  

• $$y=\left(x+7\right)^2-2$$  

• $$y=\left(x-7\right)^2+2$$  

Operations on Polynomials

Adding = Combining Like Terms

Operations on Polynomials

Multiplying = Use the Box Method

Multiple Choice

(4x³-5x²+3x)+(-2x³-x²+6x)

• 2x³-6x²-9x
• 2x³+6x²+9x
• -2x³-6x²+9x
• 2x³-6x²+9x

Multiple Choice

Subtract these polynomials:

(x⁵ + x³) - (6x - x³ + 6x⁵)

• -5x⁵ + 2x³ - 6x

• 7x⁵ - 6x

• -5x³ + 2x² - 6x

• -5x⁵ - 6x

Multiple Choice

$$\left(m+5\right)\left(m^2+4m-8\right)$$  

• $$m^3+4m^2-8m+5m^2+20m-40$$  

• $$m^2+40m+m^3$$  

• $$m^3+9m^2+12m-40$$  

Multiple Choice

(4a + 2)(6a² − a + 2)

• 24a³ + 16a² + 6a + 4
• 24a³ + 8a² + 10a + 4
• 24a³ + 8a² + 6a + 4
• 24a³ - 8a² - 6a + 4

What is synthetic division?

• Shortcut way to divide polynomials

• Uses the roots (not factor form) as the divisor

• Uses the coefficients of the terms, and not the variables, as the dividend

Multiple Choice

x³-3x²-10x+24 ÷ x+3

• x²-7x+5
• x-8
• x²-6x+8
• x+9

Multiple Choice

Use Synthetic Division $$\left(x^3-5x+6\right)\div\left(x-5\right)$$  

• $$x^3+6$$  

• $$x^2+5x-30+\frac{156}{x-5}$$  

• $$x^2+5x+20+\frac{106}{x-5}$$  

• $$x^2+6$$  

Multiple Choice

• 2x³ + 3x + (7/x-4)
• 2x² + 3x + 8 + (7/x-4)
• 2x² + 3x + 8
• 2x² - x + 10 + (6/x-4)

Roots of Polynomials

Names for roots

Polynomial functions have roots, which are also known as the x-intercepts, zeros, or solutions to the function. These are the "answers" when you factor and solve.

Multiple Choice

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

• 2
• -2
• 1
• -1

Multiple Choice

What is another name for the x-intercepts of a quadratic?

• y-intercepts
• zeros
• x-axis
• none of these

Multiple Choice

What are the zeros of the function?

• 4,1
• 3,1,-2,-5
• -3, -1, 2, 5
• -2,4