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To determine the transformations applied to the graph of [tex]\( f(x) = x^2 \)[/tex] to produce the graph of [tex]\( g(x) = -5x^2 + 100x - 450 \)[/tex], let's analyze the given information and the transformations available.
1. Reflection Over the [tex]\( x \)[/tex]-Axis:
The original function [tex]\( f(x) = x^2 \)[/tex] is a parabola opening upwards. The presence of the negative coefficient [tex]\( -5 \)[/tex] in front of the [tex]\( x^2 \)[/tex] term in [tex]\( g(x) \)[/tex] indicates that the parabola opens downwards. This means the graph has been reflected over the [tex]\( x \)[/tex]-axis.
2. Vertical Stretch and Shrink:
The coefficient of [tex]\(-5\)[/tex] also indicates a vertical stretch by a factor of 5. However, this transformation isn't listed among the options provided, so we won't focus on this one further.
3. Shifting:
To see how the graph has shifted, consider converting [tex]\( g(x) = -5x^2 + 100x - 450 \)[/tex] into vertex form. By completing the square or analyzing vertices, the vertex form helps us identify the shifts.
We find the vertex [tex]\((h, k)\)[/tex] of the quadratic equation by:
[tex]\[ h = \frac{-b}{2a} \quad \text{and} \quad k = c - a(h)^2 \][/tex]
Given [tex]\( a = -5 \)[/tex], [tex]\( b = 100 \)[/tex], and [tex]\( c = -450 \)[/tex]:
[tex]\[ h = \frac{-100}{2(-5)} = \frac{-100}{-10} = 10 \][/tex]
[tex]\[ k = -450 - (-5)(10)^2 = -450 - (-5)(100) = -450 + 500 = 50 \][/tex]
This tells us the vertex of [tex]\( g(x) \)[/tex] is [tex]\((10, 50)\)[/tex].
Comparing this with the original vertex of [tex]\( f(x) = x^2 \)[/tex], which is at the origin [tex]\((0, 0)\)[/tex], we see:
- The graph moves right by 10 units (shift to (10, y)).
- The graph moves up by 50 units (shift to (x, 50)).
Combining these observations, we have the following transformations:
- The graph of [tex]\( f(x) = x^2 \)[/tex] is reflected over the [tex]\( x \)[/tex]-axis.
- The graph of [tex]\( f(x) = x^2 \)[/tex] is shifted right 10 units.
- The graph of [tex]\( f(x) = x^2 \)[/tex] is shifted up 50 units.
Therefore, the correct options are:
1. The graph of [tex]\( f(x) = x^2 \)[/tex] is shifted up 50 units.
2. The graph of [tex]\( f(x) = x^2 \)[/tex] is shifted right 10 units.
3. The graph of [tex]\( f(x) = x^2 \)[/tex] is reflected over the [tex]\( x \)[/tex]-axis.
1. Reflection Over the [tex]\( x \)[/tex]-Axis:
The original function [tex]\( f(x) = x^2 \)[/tex] is a parabola opening upwards. The presence of the negative coefficient [tex]\( -5 \)[/tex] in front of the [tex]\( x^2 \)[/tex] term in [tex]\( g(x) \)[/tex] indicates that the parabola opens downwards. This means the graph has been reflected over the [tex]\( x \)[/tex]-axis.
2. Vertical Stretch and Shrink:
The coefficient of [tex]\(-5\)[/tex] also indicates a vertical stretch by a factor of 5. However, this transformation isn't listed among the options provided, so we won't focus on this one further.
3. Shifting:
To see how the graph has shifted, consider converting [tex]\( g(x) = -5x^2 + 100x - 450 \)[/tex] into vertex form. By completing the square or analyzing vertices, the vertex form helps us identify the shifts.
We find the vertex [tex]\((h, k)\)[/tex] of the quadratic equation by:
[tex]\[ h = \frac{-b}{2a} \quad \text{and} \quad k = c - a(h)^2 \][/tex]
Given [tex]\( a = -5 \)[/tex], [tex]\( b = 100 \)[/tex], and [tex]\( c = -450 \)[/tex]:
[tex]\[ h = \frac{-100}{2(-5)} = \frac{-100}{-10} = 10 \][/tex]
[tex]\[ k = -450 - (-5)(10)^2 = -450 - (-5)(100) = -450 + 500 = 50 \][/tex]
This tells us the vertex of [tex]\( g(x) \)[/tex] is [tex]\((10, 50)\)[/tex].
Comparing this with the original vertex of [tex]\( f(x) = x^2 \)[/tex], which is at the origin [tex]\((0, 0)\)[/tex], we see:
- The graph moves right by 10 units (shift to (10, y)).
- The graph moves up by 50 units (shift to (x, 50)).
Combining these observations, we have the following transformations:
- The graph of [tex]\( f(x) = x^2 \)[/tex] is reflected over the [tex]\( x \)[/tex]-axis.
- The graph of [tex]\( f(x) = x^2 \)[/tex] is shifted right 10 units.
- The graph of [tex]\( f(x) = x^2 \)[/tex] is shifted up 50 units.
Therefore, the correct options are:
1. The graph of [tex]\( f(x) = x^2 \)[/tex] is shifted up 50 units.
2. The graph of [tex]\( f(x) = x^2 \)[/tex] is shifted right 10 units.
3. The graph of [tex]\( f(x) = x^2 \)[/tex] is reflected over the [tex]\( x \)[/tex]-axis.
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