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Sagot :
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], we need to rewrite [tex]\( g(x) \)[/tex] in a form that clearly shows these transformations. The form we aim for is [tex]\( g(x) = a(x-h)^2 + k \)[/tex], as this form reveals vertical and horizontal shifts, stretches, and reflections.
Step-by-step solution:
1. Rewrite [tex]\( g(x) \)[/tex] in a more usable form:
Start with the given function:
[tex]\[ g(x) = -5x^2 + 100x - 450 \][/tex]
2. Factor out the coefficient of [tex]\( x^2 \)[/tex]:
Factor out [tex]\(-5\)[/tex] from the terms involving [tex]\( x \)[/tex]:
[tex]\[ g(x) = -5(x^2 - 20x) - 450 \][/tex]
3. Complete the square:
To complete the square inside the parentheses, we first need to find the term that completes the square.
Given [tex]\( x^2 - 20x \)[/tex], the completing term will be [tex]\((\frac{-20}{2})^2\)[/tex], which is [tex]\( (-10)^2 = 100 \)[/tex].
To balance the equation, we add and subtract 100 inside the parentheses:
[tex]\[ g(x) = -5(x^2 - 20x + 100 - 100) - 450 \][/tex]
[tex]\[ g(x) = -5\left[(x - 10)^2 - 100\right] - 450 \][/tex]
4. Simplify further:
Distribute the [tex]\(-5\)[/tex] and combine like terms:
[tex]\[ g(x) = -5(x - 10)^2 + 500 - 450 \][/tex]
[tex]\[ g(x) = -5(x - 10)^2 + 50 \][/tex]
Now, [tex]\( g(x) \)[/tex] is in the form [tex]\( g(x) = a(x-h)^2 + k \)[/tex], where [tex]\( a = -5 \)[/tex], [tex]\( h = 10 \)[/tex], and [tex]\( k = 50 \)[/tex].
From this form, we can identify the transformations:
1. Reflected over the x-axis:
The negative coefficient [tex]\(-5\)[/tex] indicates a reflection over the x-axis.
2. Vertical stretch:
The coefficient [tex]\(-5\)[/tex] (ignoring the sign) indicates a vertical stretch by a factor of 5.
3. Shifted right:
The term [tex]\((x - 10)\)[/tex] indicates a horizontal shift to the right by 10 units.
4. Shifted up:
The term [tex]\(+50\)[/tex] indicates a vertical shift upwards by 50 units.
Hence, the transformations applied are:
1. The graph of [tex]\( f(x) = x^2 \)[/tex] is reflected over the x-axis.
2. The graph of [tex]\( f(x) = x^2 \)[/tex] is stretched vertically by a factor of 5.
3. The graph of [tex]\( f(x) = x^2 \)[/tex] is shifted right by 10 units.
4. The graph of [tex]\( f(x) = x^2 \)[/tex] is shifted up by 50 units.
Since the question asks for three options and specifies "shifted down 50 units" (which is incorrect), the three correct transformations out of the four are:
- Reflected over the x-axis (Option 1)
- Stretched vertically by a factor of 5 (Option 2)
- Shifted right by 10 units (Option 3)
The incorrect option (shifted down 50 units) should not be selected.
Step-by-step solution:
1. Rewrite [tex]\( g(x) \)[/tex] in a more usable form:
Start with the given function:
[tex]\[ g(x) = -5x^2 + 100x - 450 \][/tex]
2. Factor out the coefficient of [tex]\( x^2 \)[/tex]:
Factor out [tex]\(-5\)[/tex] from the terms involving [tex]\( x \)[/tex]:
[tex]\[ g(x) = -5(x^2 - 20x) - 450 \][/tex]
3. Complete the square:
To complete the square inside the parentheses, we first need to find the term that completes the square.
Given [tex]\( x^2 - 20x \)[/tex], the completing term will be [tex]\((\frac{-20}{2})^2\)[/tex], which is [tex]\( (-10)^2 = 100 \)[/tex].
To balance the equation, we add and subtract 100 inside the parentheses:
[tex]\[ g(x) = -5(x^2 - 20x + 100 - 100) - 450 \][/tex]
[tex]\[ g(x) = -5\left[(x - 10)^2 - 100\right] - 450 \][/tex]
4. Simplify further:
Distribute the [tex]\(-5\)[/tex] and combine like terms:
[tex]\[ g(x) = -5(x - 10)^2 + 500 - 450 \][/tex]
[tex]\[ g(x) = -5(x - 10)^2 + 50 \][/tex]
Now, [tex]\( g(x) \)[/tex] is in the form [tex]\( g(x) = a(x-h)^2 + k \)[/tex], where [tex]\( a = -5 \)[/tex], [tex]\( h = 10 \)[/tex], and [tex]\( k = 50 \)[/tex].
From this form, we can identify the transformations:
1. Reflected over the x-axis:
The negative coefficient [tex]\(-5\)[/tex] indicates a reflection over the x-axis.
2. Vertical stretch:
The coefficient [tex]\(-5\)[/tex] (ignoring the sign) indicates a vertical stretch by a factor of 5.
3. Shifted right:
The term [tex]\((x - 10)\)[/tex] indicates a horizontal shift to the right by 10 units.
4. Shifted up:
The term [tex]\(+50\)[/tex] indicates a vertical shift upwards by 50 units.
Hence, the transformations applied are:
1. The graph of [tex]\( f(x) = x^2 \)[/tex] is reflected over the x-axis.
2. The graph of [tex]\( f(x) = x^2 \)[/tex] is stretched vertically by a factor of 5.
3. The graph of [tex]\( f(x) = x^2 \)[/tex] is shifted right by 10 units.
4. The graph of [tex]\( f(x) = x^2 \)[/tex] is shifted up by 50 units.
Since the question asks for three options and specifies "shifted down 50 units" (which is incorrect), the three correct transformations out of the four are:
- Reflected over the x-axis (Option 1)
- Stretched vertically by a factor of 5 (Option 2)
- Shifted right by 10 units (Option 3)
The incorrect option (shifted down 50 units) should not be selected.
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