Suppose that [tex]$f(x)=x^2$[/tex] and [tex]$g(x)=\frac{2}{3} x^2$[/tex]. Which statement best compares the graph of [tex][tex]$g(x)$[/tex][/tex] with the graph of [tex]$f(x)$[/tex]?

A. The graph of [tex]$g(x)$[/tex] is the graph of [tex][tex]$f(x)$[/tex][/tex] stretched vertically.
B. The graph of [tex]$g(x)$[/tex] is the graph of [tex]$f(x)$[/tex] stretched vertically and flipped over the [tex][tex]$x$[/tex]-axis[/tex].
C. The graph of [tex]$g(x)$[/tex] is the graph of [tex]$f(x)$[/tex] compressed vertically and flipped over the [tex][tex]$x$[/tex]-axis[/tex].
D. The graph of [tex]$g(x)$[/tex] is the graph of [tex]$f(x)$[/tex] compressed vertically.



Answer :

Let's examine the given functions [tex]\( f(x) = x^2 \)[/tex] and [tex]\( g(x) = \frac{2}{3} x^2 \)[/tex]. We need to determine how the graph of [tex]\( g(x) \)[/tex] compares with the graph of [tex]\( f(x) \)[/tex].

1. Function Forms:
- [tex]\( f(x) = x^2 \)[/tex] is a standard quadratic function with a vertex at the origin [tex]\((0, 0)\)[/tex] and a parabola opening upwards.
- [tex]\( g(x) = \frac{2}{3} x^2 \)[/tex] is also a quadratic function, but with a coefficient in front of the [tex]\( x^2 \)[/tex] term that affects the shape of the parabola.

2. Comparison of Coefficients:
- The coefficient of [tex]\( x^2 \)[/tex] in [tex]\( f(x) \)[/tex] is [tex]\( 1 \)[/tex].
- The coefficient of [tex]\( x^2 \)[/tex] in [tex]\( g(x) \)[/tex] is [tex]\( \frac{2}{3} \)[/tex].
- Since [tex]\( \frac{2}{3} \)[/tex] is less than [tex]\( 1 \)[/tex], this means that the graph of [tex]\( g(x) \)[/tex] compared to [tex]\( f(x) \)[/tex] will be affected in the vertical direction.

3. Vertical Compression and Stretch:
- If the coefficient is greater than [tex]\( 1 \)[/tex], the graph stretches vertically.
- If the coefficient is between [tex]\( 0 \)[/tex] and [tex]\( 1 \)[/tex], the graph compresses vertically.
- If the coefficient were negative, the graph would flip over the [tex]\( x \)[/tex]-axis, but the absence of a negative sign in [tex]\( \frac{2}{3} \)[/tex] indicates no flipping occurs here.

4. Conclusion:
- By comparing the coefficients, [tex]\( g(x) = \frac{2}{3} x^2 \)[/tex] compresses the graph of [tex]\( f(x) = x^2 \)[/tex] vertically because the coefficient [tex]\( \frac{2}{3} \)[/tex] is less than [tex]\( 1 \)[/tex].
- There is no vertical flipping as the coefficient is positive.

So, the correct statement that best compares the graph of [tex]\( g(x) \)[/tex] with the graph of [tex]\( f(x) \)[/tex] is:

D. The graph of [tex]\( g(x) \)[/tex] is the graph of [tex]\( f(x) \)[/tex] compressed vertically.