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Sagot :
To determine the transformations applied to the parent function [tex]\( f(x) = x^2 \)[/tex] to get the new function [tex]\( g(x) = -x^2 + 6x - 5 \)[/tex], we analyze the different aspects of the quadratic function:
1. Axis of Symmetry:
The standard form of a quadratic equation is [tex]\( ax^2 + bx + c \)[/tex]. For [tex]\( g(x) = -x^2 + 6x - 5 \)[/tex], the coefficients are [tex]\( a = -1 \)[/tex], [tex]\( b = 6 \)[/tex], and [tex]\( c = -5 \)[/tex]. The axis of symmetry for a quadratic function is given by the formula [tex]\( x = -\frac{b}{2a} \)[/tex].
[tex]\[ x = -\frac{6}{2(-1)} = 3 \][/tex]
Therefore, [tex]\( g(x) \)[/tex] has an axis of symmetry at [tex]\( x = 3 \)[/tex]. This statement is True.
2. Vertical Shift:
To determine the vertical shift, we compare the constant term of [tex]\( g(x) \)[/tex] with the parent function [tex]\( f(x) \)[/tex]. Since [tex]\( f(x) = x^2 \)[/tex] and there is no constant term, when we look at [tex]\( g(x) \)[/tex], the constant term is [tex]\(-5\)[/tex]. This indicates that [tex]\( g(x) \)[/tex] is shifted downwards by 5 units from [tex]\( f(x) \)[/tex]. This statement is True.
3. Horizontal Shift:
To identify the horizontal shift, we rewrite [tex]\( g(x) \)[/tex] in vertex form [tex]\( a(x-h)^2 + k \)[/tex], where [tex]\( (h, k) \)[/tex] is the vertex of the parabola. Completing the square:
[tex]\[ g(x) = -x^2 + 6x - 5 \\ g(x) = - (x^2 - 6x) - 5 \\ g(x) = - (x^2 - 6x + 9 - 9) - 5 \\ g(x) = - (x^2 - 6x + 9) + 9 - 5 \\ g(x) = - (x - 3)^2 + 4 \][/tex]
The vertex form shows [tex]\( (h, k) = (3, 4) \)[/tex]. The horizontal shift [tex]\( h = 3 \)[/tex] means [tex]\( g(x) \)[/tex] is shifted right by 3 units. This statement is True.
4. Vertical Shift (upwards):
The vertex form [tex]\( - (x - 3)^2 + 4 \)[/tex] also shows that the vertex [tex]\( k = 4 \)[/tex] indicates the graph is moved 4 units up from the transformed graph [tex]\( y = - (x - 3)^2 \)[/tex]. This matches our form [tex]\( - (x-3)^2 + 4 \)[/tex]. Hence, [tex]\( g(x) \)[/tex] is shifted up by 4 units. This statement is True.
5. Width of the Parabola:
The width of a parabola determined by the coefficient [tex]\( a \)[/tex]. For [tex]\( f(x) = x^2 \)[/tex], [tex]\( a = 1 \)[/tex] and for [tex]\( g(x) = -x^2 + 6x - 5 \)[/tex], [tex]\( a = -1 \)[/tex]. Since the absolute value of [tex]\( a \)[/tex] is the same for both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] (both are 1), [tex]\( g(x) \)[/tex] is not narrower than [tex]\( f(x) \)[/tex]. This statement is False.
Thus, the transformations can be checked as follows:
- [tex]\( g(x) \)[/tex] has an axis of symmetry at [tex]\( x = 3 \)[/tex]. True
- [tex]\( g(x) \)[/tex] is shifted down 5 units from the graph of [tex]\( f(x) \)[/tex]. True
- [tex]\( g(x) \)[/tex] is shifted right 3 units from the graph of [tex]\( f(x) \)[/tex]. True
- [tex]\( g(x) \)[/tex] is shifted up 4 units from the graph of [tex]\( f(x) \)[/tex]. True
- [tex]\( g(x) \)[/tex] is narrower than [tex]\( f(x) \)[/tex]. False
1. Axis of Symmetry:
The standard form of a quadratic equation is [tex]\( ax^2 + bx + c \)[/tex]. For [tex]\( g(x) = -x^2 + 6x - 5 \)[/tex], the coefficients are [tex]\( a = -1 \)[/tex], [tex]\( b = 6 \)[/tex], and [tex]\( c = -5 \)[/tex]. The axis of symmetry for a quadratic function is given by the formula [tex]\( x = -\frac{b}{2a} \)[/tex].
[tex]\[ x = -\frac{6}{2(-1)} = 3 \][/tex]
Therefore, [tex]\( g(x) \)[/tex] has an axis of symmetry at [tex]\( x = 3 \)[/tex]. This statement is True.
2. Vertical Shift:
To determine the vertical shift, we compare the constant term of [tex]\( g(x) \)[/tex] with the parent function [tex]\( f(x) \)[/tex]. Since [tex]\( f(x) = x^2 \)[/tex] and there is no constant term, when we look at [tex]\( g(x) \)[/tex], the constant term is [tex]\(-5\)[/tex]. This indicates that [tex]\( g(x) \)[/tex] is shifted downwards by 5 units from [tex]\( f(x) \)[/tex]. This statement is True.
3. Horizontal Shift:
To identify the horizontal shift, we rewrite [tex]\( g(x) \)[/tex] in vertex form [tex]\( a(x-h)^2 + k \)[/tex], where [tex]\( (h, k) \)[/tex] is the vertex of the parabola. Completing the square:
[tex]\[ g(x) = -x^2 + 6x - 5 \\ g(x) = - (x^2 - 6x) - 5 \\ g(x) = - (x^2 - 6x + 9 - 9) - 5 \\ g(x) = - (x^2 - 6x + 9) + 9 - 5 \\ g(x) = - (x - 3)^2 + 4 \][/tex]
The vertex form shows [tex]\( (h, k) = (3, 4) \)[/tex]. The horizontal shift [tex]\( h = 3 \)[/tex] means [tex]\( g(x) \)[/tex] is shifted right by 3 units. This statement is True.
4. Vertical Shift (upwards):
The vertex form [tex]\( - (x - 3)^2 + 4 \)[/tex] also shows that the vertex [tex]\( k = 4 \)[/tex] indicates the graph is moved 4 units up from the transformed graph [tex]\( y = - (x - 3)^2 \)[/tex]. This matches our form [tex]\( - (x-3)^2 + 4 \)[/tex]. Hence, [tex]\( g(x) \)[/tex] is shifted up by 4 units. This statement is True.
5. Width of the Parabola:
The width of a parabola determined by the coefficient [tex]\( a \)[/tex]. For [tex]\( f(x) = x^2 \)[/tex], [tex]\( a = 1 \)[/tex] and for [tex]\( g(x) = -x^2 + 6x - 5 \)[/tex], [tex]\( a = -1 \)[/tex]. Since the absolute value of [tex]\( a \)[/tex] is the same for both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] (both are 1), [tex]\( g(x) \)[/tex] is not narrower than [tex]\( f(x) \)[/tex]. This statement is False.
Thus, the transformations can be checked as follows:
- [tex]\( g(x) \)[/tex] has an axis of symmetry at [tex]\( x = 3 \)[/tex]. True
- [tex]\( g(x) \)[/tex] is shifted down 5 units from the graph of [tex]\( f(x) \)[/tex]. True
- [tex]\( g(x) \)[/tex] is shifted right 3 units from the graph of [tex]\( f(x) \)[/tex]. True
- [tex]\( g(x) \)[/tex] is shifted up 4 units from the graph of [tex]\( f(x) \)[/tex]. True
- [tex]\( g(x) \)[/tex] is narrower than [tex]\( f(x) \)[/tex]. False
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