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,-1) 28\n(0,0) 65\n(1,1) 126

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,-1) 28\n(0,0) 65\n(1,1) 126

Answer

Explanation:

Step1: Analyze the transformation

The function (g(x)=(x + 4)^{3}+1) is a transformation of (y = x^{3}). The transformation is a horizontal shift (4) units to the left (because of (x+4)) and a vertical shift (1) unit up (because of (+1)). For a point ((x,y)) on (y=x^{3}), the corresponding point on (g(x)) is ((x-4,y + 1))

Step2: Find the corresponding points

  • For the point ((-1,-1)) on (y=x^{3}): Substitute (x=-1) into (g(x)): (g(-1)=(-1 + 4)^{3}+1=3^{3}+1=27 + 1=28). The corresponding point is ((-5,0)) (since (x=-1-4=-5) and (y=-1 + 1=0))
  • For the point ((0,0)) on (y=x^{3}): Substitute (x = 0) into (g(x)): (g(0)=(0 + 4)^{3}+1=64+1=65). The corresponding point is ((-4,1)) (since (x=0-4=-4) and (y=0 + 1=1))
  • For the point ((1,1)) on (y=x^{3}): Substitute (x = 1) into (g(x)): (g(1)=(1 + 4)^{3}+1=125+1=126). The corresponding point is ((-3,2)) (since (x=1-4=-3) and (y=1 + 1=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))

The domain of (g(x)) is ((-\infty,\infty)) (since for any real - number (x), the expression ((x + 4)^{3}+1) is defined). The range of (g(x)) is ((-\infty,\infty)) (because the cube function (y = u^{3}) where (u=x + 4) has a range of ((-\infty,\infty)) and adding (1) just shifts the graph vertically, not changing the range).