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To determine which graph represents the function [tex]\( f(x) = (x+2)^2 - 3 \)[/tex], we'll follow a series of steps to analyze its characteristics:
1. Identify the General Form:
The given function is a quadratic function of the form [tex]\( f(x) = a(x - h)^2 + k \)[/tex]. For our function:
[tex]\[ f(x) = (x + 2)^2 - 3 \][/tex]
We can rewrite it to better match the standard form:
[tex]\[ f(x) = (x - (-2))^2 - 3 \][/tex]
Hence, this function is in the standard form [tex]\( f(x) = a(x - h)^2 + k \)[/tex] with [tex]\( a = 1 \)[/tex], [tex]\( h = -2 \)[/tex], and [tex]\( k = -3 \)[/tex].
2. Vertex of the Parabola:
The vertex [tex]\((h, k)\)[/tex] of the parabola is [tex]\((-2, -3)\)[/tex].
3. Direction of the Parabola:
Since the coefficient [tex]\( a \)[/tex] (which is 1 in this case) is positive, the parabola opens upwards.
4. Axis of Symmetry:
The axis of symmetry for the parabola is the vertical line that passes through the vertex, given by [tex]\( x = -2 \)[/tex].
5. Y-Intercept:
To find the y-intercept, set [tex]\( x = 0 \)[/tex] and solve for [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = (0 + 2)^2 - 3 = 4 - 3 = 1 \][/tex]
So, the y-intercept is [tex]\( (0, 1) \)[/tex].
6. Additional Points for Accuracy:
To get a better sense of the graph, calculate the values of the function at a couple of additional points:
[tex]\[ f(-4) = (-4 + 2)^2 - 3 = (-2)^2 - 3 = 4 - 3 = 1 \][/tex]
[tex]\[ f(-1) = (-1 + 2)^2 - 3 = 1 - 3 = -2 \][/tex]
Therefore, some additional points are [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -2)\)[/tex].
7. Sketch the Function:
Based on the above information, we can sketch the graph:
- The vertex is at [tex]\((-2, -3)\)[/tex].
- The parabola opens upwards.
- The axis of symmetry is the line [tex]\( x = -2 \)[/tex].
- The y-intercept is at [tex]\( (0, 1) \)[/tex].
- Other points include [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -2)\)[/tex].
8. Graphical Representation:
The graph has a parabola with its minimum point (vertex) at [tex]\((-2, -3)\)[/tex]. From there, it opens upwards, and symmetry about the vertical line [tex]\( x = -2 \)[/tex] ensures that points on either side of this line are mirrored.
9. Conclusion:
The graph representing [tex]\( f(x) = (x+2)^2 - 3 \)[/tex] should exhibit these main features:
- An upward-opening parabola
- Vertex at [tex]\((-2, -3)\)[/tex]
- Axis of symmetry along [tex]\( x = -2 \)[/tex]
- Passes through the points [tex]\( (0, 1) \)[/tex], [tex]\((-4, 1)\)[/tex], and [tex]\((-1, -2)\)[/tex]
Identify or draw a plot that satisfies these conditions to represent the function [tex]\( f(x) = (x+2)^2 - 3 \)[/tex].
1. Identify the General Form:
The given function is a quadratic function of the form [tex]\( f(x) = a(x - h)^2 + k \)[/tex]. For our function:
[tex]\[ f(x) = (x + 2)^2 - 3 \][/tex]
We can rewrite it to better match the standard form:
[tex]\[ f(x) = (x - (-2))^2 - 3 \][/tex]
Hence, this function is in the standard form [tex]\( f(x) = a(x - h)^2 + k \)[/tex] with [tex]\( a = 1 \)[/tex], [tex]\( h = -2 \)[/tex], and [tex]\( k = -3 \)[/tex].
2. Vertex of the Parabola:
The vertex [tex]\((h, k)\)[/tex] of the parabola is [tex]\((-2, -3)\)[/tex].
3. Direction of the Parabola:
Since the coefficient [tex]\( a \)[/tex] (which is 1 in this case) is positive, the parabola opens upwards.
4. Axis of Symmetry:
The axis of symmetry for the parabola is the vertical line that passes through the vertex, given by [tex]\( x = -2 \)[/tex].
5. Y-Intercept:
To find the y-intercept, set [tex]\( x = 0 \)[/tex] and solve for [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = (0 + 2)^2 - 3 = 4 - 3 = 1 \][/tex]
So, the y-intercept is [tex]\( (0, 1) \)[/tex].
6. Additional Points for Accuracy:
To get a better sense of the graph, calculate the values of the function at a couple of additional points:
[tex]\[ f(-4) = (-4 + 2)^2 - 3 = (-2)^2 - 3 = 4 - 3 = 1 \][/tex]
[tex]\[ f(-1) = (-1 + 2)^2 - 3 = 1 - 3 = -2 \][/tex]
Therefore, some additional points are [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -2)\)[/tex].
7. Sketch the Function:
Based on the above information, we can sketch the graph:
- The vertex is at [tex]\((-2, -3)\)[/tex].
- The parabola opens upwards.
- The axis of symmetry is the line [tex]\( x = -2 \)[/tex].
- The y-intercept is at [tex]\( (0, 1) \)[/tex].
- Other points include [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -2)\)[/tex].
8. Graphical Representation:
The graph has a parabola with its minimum point (vertex) at [tex]\((-2, -3)\)[/tex]. From there, it opens upwards, and symmetry about the vertical line [tex]\( x = -2 \)[/tex] ensures that points on either side of this line are mirrored.
9. Conclusion:
The graph representing [tex]\( f(x) = (x+2)^2 - 3 \)[/tex] should exhibit these main features:
- An upward-opening parabola
- Vertex at [tex]\((-2, -3)\)[/tex]
- Axis of symmetry along [tex]\( x = -2 \)[/tex]
- Passes through the points [tex]\( (0, 1) \)[/tex], [tex]\((-4, 1)\)[/tex], and [tex]\((-1, -2)\)[/tex]
Identify or draw a plot that satisfies these conditions to represent the function [tex]\( f(x) = (x+2)^2 - 3 \)[/tex].
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