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.
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.
