To understand the graph of [tex]\( g(x) = \frac{1}{2} f(x) \)[/tex] where [tex]\( f(x) = x^2 \)[/tex], let's derive [tex]\( g(x) \)[/tex] and plot its key points.
1. Define the functions:
- [tex]\( f(x) = x^2 \)[/tex]
- [tex]\( g(x) = \frac{1}{2} f(x) = \frac{1}{2} x^2 \)[/tex]
2. Determine points on the graph of [tex]\( g(x) \)[/tex]:
- Calculate [tex]\( g(x) \)[/tex] at specific values of [tex]\( x \)[/tex]:
- For [tex]\( x = -2 \)[/tex]:
[tex]\[
g(-2) = \frac{1}{2} (-2)^2 = \frac{1}{2} \times 4 = 2.0
\][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[
g(-1) = \frac{1}{2} (-1)^2 = \frac{1}{2} \times 1 = 0.5
\][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
g(0) = \frac{1}{2} (0)^2 = \frac{1}{2} \times 0 = 0.0
\][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
g(1) = \frac{1}{2} (1)^2 = \frac{1}{2} \times 1 = 0.5
\][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
g(2) = \frac{1}{2} (2)^2 = \frac{1}{2} \times 4 = 2.0
\][/tex]
3. Compile the points:
- The points on the graph of [tex]\( g(x) \)[/tex] are:
[tex]\[
(-2, 2.0), (-1, 0.5), (0, 0.0), (1, 0.5), (2, 2.0)
\][/tex]
4. Sketching the graph:
- The function [tex]\( g(x) \)[/tex] is similar to the graph of [tex]\( f(x) = x^2 \)[/tex], a parabola, but vertically scaled by a factor of [tex]\( \frac{1}{2} \)[/tex].
- This means that the graph will open upward, and it’ll be less steep compared to [tex]\( f(x) = x^2 \)[/tex].
Based on the points calculated, the graph of [tex]\( g(x) = \frac{1}{2} x^2 \)[/tex] can be plotted, showing a parabolic shape that passes through the points mentioned above.
