To transform the graph of [tex]\( f(x) = x^2 \)[/tex] to the graph of [tex]\( g(x) = -5x^2 + 100x - 450 \)[/tex], let's break down the transformations step-by-step:
1. Reflection over the [tex]\( x \)[/tex]-axis:
The graph of [tex]\( f(x) = x^2 \)[/tex] is reflected over the [tex]\( x \)[/tex]-axis to produce [tex]\( -x^2 \)[/tex]. This changes the parabolic shape from opening upwards to opening downwards.
2. Vertical Stretch by a factor of 5:
The coefficient of [tex]\( x^2 \)[/tex] is -5 in [tex]\( g(x) \)[/tex]. This indicates a vertical stretch by a factor of 5. The parabola becomes narrower as it is stretched vertically.
3. Horizontal Shift to the right by 10 units:
To determine the horizontal shift, we complete the square on the quadratic function:
[tex]\[
g(x) = -5x^2 + 100x - 450 \\
= -5(x^2 - 20x) - 450 \\
= -5((x - 10)^2 - 100) - 450 \\
= -5(x - 10)^2 + 500 - 450 \\
= -5(x - 10)^2 + 50
\][/tex]
This shows a horizontal shift to the right by 10 units.
4. Vertical Shift up by 50 units:
From the completed square form [tex]\(-5(x - 10)^2 + 50\)[/tex], we can see that there is a vertical shift of the graph upwards by 50 units.
Based on the above breakdown, the transformations applied are:
- The graph of [tex]\( f(x) = x^2 \)[/tex] is shifted up 50 units.
- The graph of [tex]\( f(x) = x^2 \)[/tex] is shifted right 10 units.
- The graph of [tex]\( f(x) = x^2 \)[/tex] is reflected over the [tex]\( x \)[/tex]-axis.
Thus, the selected transformations are:
- The graph of [tex]\( f(x) = x^2 \)[/tex] is shifted up 50 units.
- The graph of [tex]\( f(x) = x^2 \)[/tex] is shifted right 10 units.
- The graph of [tex]\( f(x) = x^2 \)[/tex] is reflected over the [tex]\( x \)[/tex]-axis.
