To determine which transformations are applied to the graph of [tex]\( f(x) = x^2 \)[/tex] to change it into the graph of [tex]\( g(x) = -3x^2 - 36x - 60 \)[/tex], let's break the process down step-by-step:
1. Narrower Graph:
The coefficient of [tex]\( x^2 \)[/tex] in [tex]\( f(x) \)[/tex] is 1, while the coefficient of [tex]\( x^2 \)[/tex] in [tex]\( g(x) \)[/tex] is -3. Since the absolute value of the coefficient in [tex]\( g(x) \)[/tex] is greater than that in [tex]\( f(x) \)[/tex], the graph of [tex]\( g(x) \)[/tex] is narrower than [tex]\( f(x) \)[/tex].
2. Shifted Graph:
To determine the shifts, we will complete the square for [tex]\( g(x) = -3x^2 - 36x - 60 \)[/tex]:
First, factor out the coefficient of [tex]\( x^2 \)[/tex] from the quadratic expression:
[tex]\[
g(x) = -3(x^2 + 12x) - 60.
\][/tex]
Next, complete the square inside the parentheses. We add and subtract 36 (since [tex]\((12/2)^2 = 36\)[/tex]):
[tex]\[
g(x) = -3(x^2 + 12x + 36 - 36) - 60,
\][/tex]
[tex]\[
g(x) = -3((x + 6)^2 - 36) - 60.
\][/tex]
Distribute the [tex]\(-3\)[/tex] through the parentheses:
[tex]\[
g(x) = -3(x + 6)^2 + 108 - 60,
\][/tex]
[tex]\[
g(x) = -3(x + 6)^2 + 48.
\][/tex]
This shows that the graph is shifted 6 units to the left (due to [tex]\((x + 6)^2\)[/tex]) and 48 units downwards (due to the constant term +48).
3. Reflected Graph:
The negative sign in front of the coefficient of [tex]\( x^2 \)[/tex] indicates that the graph is reflected over the x-axis.
However, you asked if any one transformation applies, and you listed:
- The graph of [tex]\( f(x)=x^2 \)[/tex] is made narrower.
- The graph of [tex]\( f(x)=x^2 \)[/tex] is shifted 6 units.
- The graph of [tex]\( f(x)=x^2 \)[/tex] is shifted down 48 units.
- The graph of [tex]\( f(x)=x^2 \)[/tex] is reflected over the y-axis.
Comparing these options to our analysis:
- The graph of [tex]\( f(x)=x^2 \)[/tex] is made narrower: This is true.
- The graph of [tex]\( f(x)=x^2 \)[/tex] is shifted 6 units: This is incomplete because it doesn’t mention the direction.
- The graph of [tex]\( f(x)=x^2 \)[/tex] is shifted down 48 units: This is true.
- The graph of [tex]\( f(x)=x^2 \)[/tex] is reflected over the y-axis: This is false; it should be reflected over the x-axis.
Therefore, one correct transformation applied to the graph of [tex]\( f(x) = x^2 \)[/tex] in moving to [tex]\( g(x) = -3x^2 - 36x - 60 \)[/tex] is:
- The graph of [tex]\( f(x)=x^2 \)[/tex] is made narrower.
