starting with the graph of a basic function, graph the following function using the techniques of shifting…
starting with the graph of a basic function, graph the following function using the techniques of shifting, compressing, stretching, and/or reflecting. be sure to identify at least three key points. find the domain and range of the function.\ng(x)=(x + 4)^3+1\ncomplete the table of coordinates that lie on the graph of y = x^3 and the corresponding points that lie on the graph of g(x)=(x + 4)^3+1.\n(type ordered pairs. simplify your answers.)\npoints that lie on the graph of y = x^3 corresponding points that lie on the graph of g(x)=(x + 4)^3+1\n(-1,)\n(0,)\n(1,)
Answer
Explanation:
Step1: Find ( y ) values for ( y = x^{3} )
For ( x=-1 ), ( y=(-1)^{3}=-1 ). So the point is ((-1,-1)). For ( x = 0 ), ( y=(0)^{3}=0 ). So the point is ((0,0)). For ( x = 1 ), ( y=(1)^{3}=1 ). So the point is ((1,1)).
Step2: Transform the points for ( g(x)=(x + 4)^{3}+1 )
The transformation ( y=f(x)\to y=f(x + h)+k ) (here ( h = 4,k = 1 )) shifts the graph of ( y = f(x)) left by ( |h| ) units (since ( h>0 )) and up by ( |k| ) units. For the point ((x,y)) on ( y=x^{3} ), the corresponding point on ( g(x)=(x + 4)^{3}+1 ) is ((x-4,y + 1)).
- For ((-1,-1)) on ( y=x^{3} ): ( x=-1-4=-5,y=-1 + 1=0 ). The point is ((-5,0)).
- For ((0,0)) on ( y=x^{3} ): ( x=0-4=-4,y=0 + 1=1 ). The point is ((-4,1)).
- For ((1,1)) on ( y=x^{3} ): ( x=1-4=-3,y=1 + 1=2 ). The point is ((-3,2)).
Answer:
| Points that lie on the graph of ( y = x^{3} ) | Corresponding points that lie on the graph of ( g(x)=(x + 4)^{3}+1 ) |
|---|---|
| ((-1,-1)) | ((-5,0)) |
| ((0,0)) | ((-4,1)) |
| ((1,1)) | ((-3,2)) |