Section 2.5 worked examples

[Eigenvalue]

Precalculus
Michael Sullivan
Edition 4
Chapter 2, Section 2.5

Here are some worked examples for section 2.5 (transformations of functions).

[Eigenvalue]

Questions 19-26; write the function whose graph is the graph of y=x³ but is:

19. Shifted to the right 4 units
y=(x-4)³

20. Shifted to the left 4 units
y=(x+4)³

21. Shifted up 4 units
y=x³+4

22. Shifted down 4 units
y=x³-4

23. Reflected about the x-axis
(x,y)[tex]\rightarrow[/tex] (x,-y)
-y=x³
y=-x³

24. Reflected about the y-axis
(x,y)[tex]\rightarrow[/tex](-x,y)
y=(-x)³
y=-x³

25. Vertically stretched by a factor of 4
f(x)[tex]\rightarrow[/tex]af(x)
(x,y)[tex]\rightarrow[/tex] (x, ay)
y=4x³

26. Horizontally stretched by a factor of 4
f(x)[tex]\rightarrow[/tex]f(ax)
(x,y)[tex]\rightarrow[/tex]([tex]\frac{x}{a}[/tex],y)
y=([tex]\frac{1}{4}[/tex]x)³=[tex]\frac{1}{64}[/tex]x³

[Eigenvalue]

Questions 39-45

39. f(x)=x²-1
Parent function: y=x²
Shift down 1 unit
(x,y)[tex]\rightarrow[/tex](x,y-1)
Domain: all real numbers
Range: [-1, infinity)

40. f(x)=x²+4
Parent function: y=x²
Shift up 4 units
(x,y)[tex]\rightarrow[/tex](x, y+4)
Domain: all real numbers
Range: [4, infinity)

41. g(x)=x³+1
Parent function: y=x³
Shift up 1 unit
(x,y)[tex]\rightarrow[/tex](x, y+1)
Domain: all real numbers
Range: all real numbers

42. g(x)=x³-1
Parent function: y=x³
Shift down 1 unit
(x,y)[tex]\rightarrow[/tex](x, y-1)
Domain: all real numbers
Range: all real numbers

43. h(x)=[tex]\sqrt{x-2}[/tex]
Parent function: y=[tex]\sqrt{x}[/tex]
Shift right 2 units
(x,y)[tex]\rightarrow[/tex](x+2, y)
Domain: [2, infinity), as x-2 must be greater than or equal to 0
Range: [0, infinity) as a radicand must be greater than or equal to 0 to produce a real number result

44. h(x)=[tex]\sqrt{x+1}[/tex]
Parent function: y=[tex]\sqrt{x}[/tex]
Shift left 1 unit
(x,y)[tex]\rightarrow[/tex](x-1, y)
Domain: [-1, infinity), as x+1 must be greater than or equal to 0
Range: [0, infinity) as a radicand must be greater than or equal to 0 to produce a real number result

45. f(x)=(x-1)³+2
Parent function: y=x³
Shift right 1 unit
Shift up 2 units
(x,y)[tex]\rightarrow[/tex](x+1,y+2)
Domain: all real numbers
Range: all real numbers

[nycmath]

Precalculus
Michael Sullivan
Edition 4
Chapter 2, Section 2.5

Here are some worked examples for section 2.5 (transformations of functions).

Again, it is your choice to do all this work. Boy, you must have a lot of time on your hands! THE CHAPTER 1 AND 2 TESTS IS ENOUGH REVIEW OF SECTION 1.1 TO 2.6. If you find pleasure in doing all this work, fabulous!!