Answered

Suppose that [tex]$F(x)=x^3$[/tex] and [tex][tex]$G(x)=-2 x^3-7$[/tex][/tex]. Which statement best compares the graph of [tex]$G(x)$[/tex] with the graph of [tex]$F(x)$[/tex]?

A. The graph of [tex][tex]$G(x)$[/tex][/tex] is the graph of [tex]$F(x)$[/tex] compressed vertically, flipped over the x-axis, and shifted 7 units down.

B. The graph of [tex]$G(x)$[/tex] is the graph of [tex][tex]$F(x)$[/tex][/tex] stretched vertically, flipped over the x-axis, and shifted 7 units to the right.

C. The graph of [tex]$G(x)$[/tex] is the graph of [tex]$F(x)$[/tex] stretched vertically, flipped over the x-axis, and shifted 7 units down.

D. The graph of [tex][tex]$G(x)$[/tex][/tex] is the graph of [tex]$F(x)$[/tex] compressed vertically, flipped over the x-axis, and shifted 7 units to the right.



Answer :

GinnyAnswer
To compare the graph of [tex]\(G(x)\)[/tex] with the graph of [tex]\(F(x)\)[/tex], let's analyze the transformations applied to [tex]\(F(x) = x^3\)[/tex] to obtain [tex]\(G(x) = -2x^3 - 7\)[/tex].

1. Vertical Compression/Stretches:
- The coefficient of [tex]\(x^3\)[/tex] in [tex]\(G(x)\)[/tex] is [tex]\(-2\)[/tex]. This indicates that there is a vertical transformation.
- Specifically, the factor [tex]\(-2\)[/tex] suggests that the graph of [tex]\(F(x)\)[/tex] is vertically stretched by a factor of 2 and then flipped over the [tex]\(x\)[/tex]-axis (the negative sign causes this flip).

2. Shifts:
- The constant term in [tex]\(G(x)\)[/tex] is [tex]\(-7\)[/tex]. This represents a vertical shift.
- A negative constant term shifts the graph downward. Therefore, [tex]\(G(x)\)[/tex] is shifted 7 units down.

Putting all these transformations together, we can summarize the changes as:
- Vertical Stretch: The factor of 2 stretches the graph vertically.
- Flip Over the [tex]\(x\)[/tex]-Axis: The negative sign causes the graph to flip.
- Vertical Shift Down: The [tex]\(-7\)[/tex] shifts it down by 7 units.

Given these transformations, the best statement that describes the relationship between the graph of [tex]\(G(x)\)[/tex] and the graph of [tex]\(F(x)\)[/tex] is:

A. The graph of [tex]\(G(x)\)[/tex] is the graph of [tex]\(F(x)\)[/tex] compressed vertically, flipped over the [tex]\(x\)[/tex]-axis, and shifted 7 units down.

This concludes our step-by-step analysis showing the correct comparison of [tex]\(G(x)\)[/tex] with [tex]\(F(x)\)[/tex].