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Ali graphs the function [tex]f(x)=-(x+2)^2-1[/tex] as shown.

Which best describes the error in the graph?

A. The axis of symmetry should be [tex]x=-1[/tex].
B. The axis of symmetry should be [tex]x=2[/tex].
C. The vertex should be a maximum.
D. The vertex should be [tex](-2,1)[/tex].

Sagot :

GinnyAnsweridn
Let's analyze the given function step by step.

The function provided is [tex]\( f(x) = -(x + 2)^2 - 1 \)[/tex].

1. Rewrite the function in vertex form:
The given function is already in vertex form:
[tex]\[ y = a(x - h)^2 + k \][/tex]
Here, [tex]\( a = -1 \)[/tex], [tex]\( h = -2 \)[/tex], and [tex]\( k = -1 \)[/tex]. Therefore, the function is:
[tex]\[ y = -(x + 2)^2 - 1 \][/tex]

2. Determine the vertex:
The vertex of a parabola in the form [tex]\( y = a(x - h)^2 + k \)[/tex] is given by the point [tex]\( (h, k) \)[/tex].
Here, [tex]\( h = -2 \)[/tex] and [tex]\( k = -1 \)[/tex]. Therefore, the vertex of the function is:
[tex]\[ (-2, -1) \][/tex]

3. Find the axis of symmetry:
The axis of symmetry for a parabola in the form [tex]\( y = a(x - h)^2 + k \)[/tex] is the vertical line given by [tex]\( x = h \)[/tex].
Here, [tex]\( h = -2 \)[/tex]. Therefore, the equation for the axis of symmetry is:
[tex]\[ x = -2 \][/tex]

4. Determine if the vertex is a maximum or minimum:
The coefficient [tex]\( a \)[/tex] determines whether the parabola opens upwards or downwards.
- If [tex]\( a > 0 \)[/tex], the parabola opens upwards, and the vertex is a minimum.
- If [tex]\( a < 0 \)[/tex], the parabola opens downwards, and the vertex is a maximum.

Here, [tex]\( a = -1 \)[/tex], which is less than 0, indicating that the parabola opens downwards. Therefore, the vertex at [tex]\( (-2, -1) \)[/tex] is a maximum.

Next, let us analyze the provided options considering our conclusions about the function:

1. The axis of symmetry should be [tex]\( x = -1 \)[/tex]: This is incorrect. The axis of symmetry is [tex]\( x = -2 \)[/tex].

2. The axis of symmetry should be [tex]\( x = 2 \)[/tex]: This is also incorrect. As stated earlier, the axis of symmetry is [tex]\( x = -2 \)[/tex].

3. The vertex should be a maximum: This is correct. Since the coefficient [tex]\( a \)[/tex] is negative, the parabola opens downward, making the vertex a maximum.

4. The vertex should be [tex]\((-2, 1)\)[/tex]: This is incorrect. The correct vertex is [tex]\((-2, -1)\)[/tex].

Therefore, the best description of the error in Ali's graph is:

The vertex should be a maximum.

Thus, the correct option is:

The vertex should be a maximum.
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