1A Review - Algebra II

1A Review - Algebra II

What We've Done So Far

Topics We've Done So Far

• Domain and Range

• Points of Intersection

• Rational Functions/Asymptotes

• Solving Systems of Equations (Algebraically + Graphically)

• Solving Nonlinear Systems

Multiple Choice

How would you describe the domain of a function?

• The possible x-values, or inputs

• The possible y-values, or outputs

Multiple Choice

How would you describe the range of a function?

• The possible x-values, or inputs

• The possible y-values, or outputs

Domain + Range

• To find the domain, we use the x-axis

• To find the range, we use the y-axis

Multiple Choice

What is the domain?

• $-5\le x\le\infty$

• $-5\ \le\ y\ \le\ \infty$

• $-\infty\le y\le\infty$

• $-\infty<x<\infty$

Multiple Choice

What is the range?

• $-5\le x\le\infty$

•  $-5\ \le\ y\ <\ \infty$ 

•  $-\infty<y<\infty$ 

• $-\infty<x<\infty$

Multiple Choice

What is the domain?

•  $0\le x<\infty$ 

•  $0\ <\ y\ <\ \infty$ 

•  $-\infty<y<\infty$ 

• $-\infty<x<\infty$

Multiple Choice

What is the range?

•  $0\le x<\infty$ 

•  $0\ <\ y\ <\ \infty$ 

•  $-\infty<y<\infty$ 

• $-\infty<x<\infty$

Interval Notation vs. Inequality Notation

We talked about two different ways to denote the domain and range of a graph.

Inequality Notation uses

 $<\ and\ \le$  
Interval Notation uses
( ) and [ ]

Inequality Notation

 $<$  is used when a number is not included in the domain/range 
 $\le$  is used when a number is included in the domain/range

Ex. In the figure, the domain is  $4\le x<\infty$  where 4 is included (solid dot) and infinity is not because it is not a discrete number

Inequality Notation

 $\left(\ \right)$  is used when a number is not included in the domain/range 
 $\left[\ \right]$  is used when a number is included in the domain/range

Ex. In the figure, range is  $\left(-5,\ 5\right)$  where -5 nor 5 are included (open circles)

Multiple Choice

What is the range in interval notation?

• (-3, 5)

• (-5, 3)

• [-3, 5)

• [-3, 5]

Multiple Choice

What is the domain in interval notation?

• (-3, 5)

• (-5, 3)

• [-3, 5)

• [-5, 3]

Points of Intersection

Then, we began to look at solutions to systems. One way was looking at where they intersect either using graphs or algebra

Algebraically

• Suppose we have y=x+5 and y=-2x+2

• Points of intersection are solutions to systems, or in other words, where they are equal to each other

   $x+5=-2x+2$  
  

• (-1, 4)

Solution

(0.5, 7.5)

Rational Functions

Rational Functions are functions that take the form of fractions

Multiple Choice

The biggest thing to note about rational functions is that because they are fractions, we have to watch out for a 0 in the denominator. When a value of x causes us to have a zero in the denominator, what happens to the graph?

• It stops abruptly

• There is an asymptote

• It spirals

Open Ended

How would you describe an asymptote?

Vertical and Horizontal Asympotes

An asymptote is a line that the graph approaches but never meets

Vertical Asymptote

The vertical asymptote can be found by looking at the denominator - what value of x will cause a 0 to be in the denominator?

 $y=\frac{1}{x-8}$  


x = 8

Horizontal Asymptote

The horizontal asymptote can be found by looking at the constant at the end of the function. Let's edit our function a bit:

 $y=\frac{1}{x-8}+3$  

y = 3

Multiple Choice

 $f\left(x\right)=\frac{1}{x-2}+5$  

What is the vertical asymptote?

• x = 2

• x = 5

•  y = 5

• y = 1

Multiple Choice

 $f\left(x\right)=\frac{1}{x+10}+1$  

What is the vertical asymptote?

• x = 1

• x = -10

•  y = 10

• y = 1

Multiple Choice

 $f\left(x\right)=\frac{1}{x+10}-6$  

What is the horizontal asymptote?

• x = -10

• x = 6

•  y = 10

• y = -6

Extraneous Solutions

What makes a solution extraneous?

Extraneous Solutions

A solution is extraneous if it does not make the original equation true. When plugged back in, the end result will be false.

Multiple Select

Check all that are true given the equation and that x = 3 and

x = 7.

• x = 3 is a real solution

• x = 7 is a real solution

• x = 3 is an extraneous solution

• x = 7 is an extraneous solution

Multiple Select

Check all that are true given the equation and that x = -3 and

x = -5.

• x = -3 is a real solution

• x = -5 is a real solution

• x = -3 is an extraneous solution

• x = -5 is an extraneous solution

Systems of Equations

Algebraically, Graphically + Nonlinear

Linear Combination

Use either method: substitution or elimination

Systems of Equations

Using either substitution or elimination, describe how you would solve this system.

 $3x-y=8$  

 $x+2y=5$  

Nonlinear Combination

Circle and a line

Nonlinear Systems

Remember that we have three cases of solutions