5.1 CFA Remediation

TRANSFORMATION RULES

​Now lets put those two concepts togethers

Multiple Choice

How is this function transformed from the parent function ?

$$y=\frac{1}{x}$$  

• Translated Right 5 and Up 1

• Translated Right 5 and Down 1

• Translated Left 5 and Up 1

• Translated Left 5 and Down 1

Multiple Choice

Describe the transformations from the parent function

$$y=\frac{1}{x}$$  

• Right 5  and  Up 7

• Left 7  and  Up 5

• Reflect over the x-axis  and  Up 7

• Left 5  and  Up 7

Horizontal vs Vertical

• Horizontal asymptotes are horizontal lines that stop the graph from approaching some y value. Example: y = 1

• Vertical asymptotes are vertical lines that stop the graph from approaching some x value. Example : x = 3

Multiple Choice

What are the asymptotes?

• $$x=-3,\ y=1$$

• $$x=-3,\ y=-1$$

• $$x=1,\ y=-3$$

• $$x=1,\ y=3$$

Multiple Choice

What are the asymptotes?

• x=1, x= 2, y =1, y= 2

• x= 2, x=-2, y = 1

• x=2 y =-1

• x=1 y =2, y =-2

Multiple Choice

What are the asymptotes?

• $$x=-3,\ y=1$$

• $$x=-3,\ y=-1$$

• $$x=1,\ y=-3$$

• $$x=1,\ y=3$$

Multiple Choice

What are the asymptotes?

• x=-2 and y=-1

• x=-2 and y=1

• x=2 and y=1

• x=2 and y = -1

Multiple Choice

Identify the transformations & Asymptotes for $$f\left(x\right)=-\frac{1}{x+3}$$  

• Move Right 3 and Up 0

  VA: x = 3

  HA: y = 0

• Move Left 3 and Up 0

  VA: x = -3

  HA: y = 0

• Move Right 3 and Down 1

  VA: x = 3

  HA: y = -1

• Move Left 3 and Down 1

  VA: x = -3

  HA: y = -1

Multiple Choice

Which function matches this graph?

• $$f\left(x\right)=\frac{1}{x-1}+2$$

• $$f\left(x\right)=\frac{1}{x+2}-1$$

• $$f\left(x\right)=\frac{1}{x+1}+2$$

• $$f\left(x\right)=\frac{1}{x-2}+1$$

Multiple Choice

What is the equation of the rational function graphed?

Finding Vertical Asymptotes

To find the vertical asymptotes, factor the denominator to find the possible asymptotes

Use the zero product property to identify possible x-values for vertical asymptotes

Graph the function to find the vertical asymptotes

Finding Vertical Asymptotes

Multiple Choice

Factor the rational function:

$$f\left(x\right)\ =\ \frac{x+3}{x^2-8x+12}$$

• $$\frac{x+3}{\left(x-3\right)\left(x-4\right)}$$

• $$\frac{x+3}{\left(x-6\right)\left(x-2\right)}$$

• $$\frac{x+3}{\left(x+3\right)\left(x+4\right)}$$

• $$\frac{x+3}{\left(x+6\right)\left(x+2\right)}$$

Match

Match the following

• $$f\left(x\right)=\frac{6}{x-1}$$

• $$f\left(x\right)=\frac{x+4}{x-5}$$

• $$f\left(x\right)=\frac{-3x-1}{3x+2}$$

• $$f\left(x\right)=\frac{6x+1}{3x-5}$$

• $$f\left(x\right)=\frac{-8x+3}{4x}$$

VA at y = 1

VA at y = 5

VA at y = -2/3

VA at x = 5/3

VA at y = 0

• VA at y = 1

• VA at y = 5

• VA at y = -2/3

• VA at x = 5/3

• VA at y = 0

Multiple Choice

Find the Vertical Asymptotes for the following function

$$g\left(x\right)\ =\frac{2x^2+x-9}{x^2-2x-8}$$  

• $$x=-4,\ x=2$$  

• $$x=-2,\ x=4$$  

• $$x=-6,\ x\ =\ -2$$  

• $$x=\ 2,\ x=6$$  

Finding Horizontal Asymptotes

Finding Horizontal Asymptotes

How to find Domain/Range:

• Domain- Read left to right and 'skip' over the Vertical Asymptote

• Range - Read bottom to top and 'skip' over the Horizontal Asymptote

Finding Horizontal Asymptotes

Multiple Choice

What is the horizontal asymptote of the following function?

$$f\left(x\right)=\frac{3x^2}{4x^2-1}$$  

• $$y=\frac{3}{4}$$  

• $$y=\frac{4}{3}$$  

• $$y=0$$  

• No asymptote

Multiple Choice

The rational function, $$f\left(x\right)=3+\frac{2}{x-4}$$   has a horizontal asymptote at...

• y = -4

• y = 2

• y = 3

• y= 4

Multiple Choice

What's the horizontal asymptote of the function?

$$f\left(x\right)=\frac{3x+2}{x^2+1}$$  

• y = 3

• No horizontal asymptote

• y = 0

• y = 2

Multiple Choice

What is the horizontal asymptote for the following function?

$$f\left(x\right)=\frac{4x+3}{x^2-4}$$  

• $$y=4$$  

• $$y=\frac{1}{4}$$  

• $$y=0$$  

• No Asymptote

Multiple Choice

What are the asymptotes for the following rational function?

$$f\left(x\right)\ =\ \frac{6x}{2x+1}$$  ?

• $$y\ =3,\ x\ =\ \frac{1}{2}$$  

• $$y=-\frac{1}{2},\ x\ =\ 3$$  

• $$y\ =3,\ x\ =-\frac{1}{2}$$  

• $$y=\frac{1}{2},\ x\ =\ 3$$  y=1/2

Domain/Range of this graph?

Multiple Choice

The vertical asymptote here x= -3, so what is the Domain?

• $$\left(-\infty,\ \infty\right)$$ All Reals

• $$\left(-3,\infty\right)$$ x-values bigger than -3

• $$\left(-\infty,3\right)U\left(3,\infty\right)$$ All reals but x $$\ne$$  3

• $$\left(-\infty,-3\right)U\left(-3,\infty\right)$$ All reals but x $$\ne$$   -3 
  

Multiple Choice

The horizontal asymptote here y=0, so what is the Range?

• $$\left(-\infty,\ \infty\right)$$ All Reals

• $$\left(0,\infty\right)$$ y-values bigger than 0

• $$\left(-\infty,0\right)U\left(0,\infty\right)$$ All Reals but y $$\ne$$   0

• $$\left(-\infty,-3\right)U\left(-3,\infty\right)$$ All reals but y $$\ne$$  -3 
  

Multiple Choice

What is the domain and range?

• $$D:\ \left(-\infty,2\right)\cup\left(2,\infty\right)$$
  

  $$R:\ \left(-\infty,1\right)\cup\left(1,\infty\right)$$
  

• $$D:\ \left(-\infty,1\right)\cup\left(1,\infty\right)$$
  $$R:\ \left(-\infty,2\right)\cup\left(2,\infty\right)$$

• $$null$$
  $$R:\ \left(-\infty,\frac{1}{2}\right)\cup\left(\frac{1}{2},\infty\right)$$

• $$D:\ \left(-\infty,-2\right)\cup\left(-2,\infty\right)$$
  $$R:\ \left(-\infty,-1\right)\cup\left(-1,\infty\right)$$

Multiple Choice

What is the domain and range?

• D : $$\left(-\infty,-3\right)\cup\left(-3,\infty\right)$$
  R: $$\left(-\infty,1\right)\cup\left(1,\infty\right)$$

• D : $$\left(-\infty,1\right)\cup\left(1,\infty\right)$$
  R: $$\left(-\infty,-3\right)\cup\left(-3,\infty\right)$$

• D : $$\left(-\infty,3\right)\cup\left(3,\infty\right)$$
  R: undefined

• D: $$\left(-\infty,-1\right)\cup\left(-1,\infty\right)$$
  R: $$\left(-\infty,3\right)\cup\left(3,\infty\right)$$

Finding the hole

Multiple Choice

Find the vertical asymptotes and holes of the following function:

$$f\left(x\right)\ =\ \frac{x^2-3x}{3x^2+6x}$$

Remember to factor first!

• No Hole.

  VA at x = -2

• Hole at x = 0,

  VA at x = -2

• Hole at x = 3,

  VA at x = 2

• Hole at x = -2,

  VA at x = 3

Multiple Choice

What are the coordinates of the hole, if one exists.

• (-1, -2)

• (-1, ½)

• (-1, -½)

• there isn't one