To compare the graphs of [tex]\( f(x) = x^2 \)[/tex] and [tex]\( g(x) = -\frac{1}{5} x^2 \)[/tex], let’s break down the components of each function and understand their transformations.
1. Original Function [tex]\( f(x) = x^2 \)[/tex]:
- This is a basic quadratic function, which has a parabola shape opening upwards with the vertex at the origin (0,0).
2. Transformed Function [tex]\( g(x) = -\frac{1}{5} x^2 \)[/tex]:
- The negative sign in front of the [tex]\(\frac{1}{5}\)[/tex] means the parabola will be flipped over the x-axis. Essentially, this transformation reflects the graph of [tex]\( f(x) \)[/tex], making it open downwards.
- The factor of [tex]\(\frac{1}{5}\)[/tex] compresses the graph vertically. Instead of stretching it, it will shrink by a factor of 5. This means that for any given x-value, the y-value of [tex]\( g(x) \)[/tex] is [tex]\(\frac{1}{5}\)[/tex] of the corresponding y-value of [tex]\( f(x) \)[/tex].
Now that we understand the transformations:
- Point by point: For a value [tex]\( x \)[/tex]:
- If [tex]\( f(x) = x^2 \)[/tex] gives a y-value,
- Then [tex]\( g(x) = -\frac{1}{5} x^2 \)[/tex] gives a y-value that is [tex]\(\frac{1}{5}\)[/tex] of [tex]\( f(x) \)[/tex] but in the opposite direction (since it's negative).
Putting it all together, we see that:
- The graph of [tex]\( g(x) \)[/tex] is flipped over the x-axis.
- It is vertically compressed by a factor of 5.
Thus, the statement that best compares the graph of [tex]\( g(x) \)[/tex] with the graph of [tex]\( f(x) \)[/tex] is:
C. The graph of [tex]\( g(x) \)[/tex] is vertically compressed by a factor of 5 and flipped over the x-axis.
