Let's analyze the transformation of the function [tex]\( f(x) = x^2 \)[/tex] into [tex]\( g(x) = -3x^2 - 36x - 60 \)[/tex] step-by-step.
1. Identify the quadratic functions:
- The original function is [tex]\( f(x) = x^2 \)[/tex].
- The transformed function is [tex]\( g(x) = -3x^2 - 36x - 60 \)[/tex].
2. Write the transformed function in vertex form:
The equation [tex]\( g(x) = -3x^2 - 36x - 60 \)[/tex] can be transformed into vertex form by completing the square.
[tex]\[
g(x) = -3x^2 - 36x - 60
\][/tex]
First, factor out the coefficient [tex]\(-3\)[/tex] from the quadratic and linear terms:
[tex]\[
g(x) = -3(x^2 + 12x) - 60
\][/tex]
Next, complete the square inside the parentheses. To do this, add and subtract [tex]\((\frac{12}{2})^2 = 36\)[/tex] inside the parentheses:
[tex]\[
g(x) = -3(x^2 + 12x + 36 - 36) - 60
\][/tex]
Simplify inside the parentheses:
[tex]\[
g(x) = -3((x + 6)^2 - 36) - 60
\][/tex]
Distribute the [tex]\(-3\)[/tex]:
[tex]\[
g(x) = -3(x + 6)^2 + 108 - 60
\][/tex]
Combine constants:
[tex]\[
g(x) = -3(x + 6)^2 + 48
\][/tex]
So, the vertex form of [tex]\( g(x) \)[/tex] is:
[tex]\[
g(x) = -3(x + 6)^2 + 48
\][/tex]
3. Determine the transformations:
- The coefficient of [tex]\((x + 6)^2\)[/tex] is [tex]\(-3\)[/tex]. This indicates the parabola is narrower because the absolute value of the coefficient (3) is greater than 1, and it opens downwards since the sign is negative.
- The expression [tex]\((x + 6)^2\)[/tex] indicates a horizontal shift 6 units to the left (since it is [tex]\(x + 6\)[/tex]).
- The constant [tex]\(+ 48\)[/tex] indicates a vertical shift 48 units up.
4. Analyze the options:
- The graph of [tex]\( f(x) = x^2 \)[/tex] is made narrower. (True, the coefficient 3 makes it narrower.)
- The graph of [tex]\( f(x)=x^2 \)[/tex] is shifted right 6 units. (False, it is shifted left by 6 units.)
- The graph of [tex]\( f(x)=x^2 \)[/tex] is shifted down 48 units. (False, it is shifted up by 48 units.)
- The graph of [tex]\( f(x)=x^2 \)[/tex] is reflected over the y-axis. (False, it is reflected over the x-axis, not the y-axis.)
Thus, the correct transformation applied is:
[tex]\[
\boxed{\text{The graph of } f(x)= x^2 \text{ is made narrower.}}
\][/tex]
