Answer :
To graph the piecewise function [tex]\( h(x) \)[/tex], we need to analyze each segment of the function and plot them separately. Here is a step-by-step breakdown of the function and its corresponding graph:
1. Function Analysis:
[tex]\[ h(x) = \begin{cases} -x^2 + 4 & \text{if } x \leq -2 \\ -0.5x & \text{if } -2 < x < 2 \\ x^2 - 5 & \text{if } x \geq 2 \end{cases} \][/tex]
2. Segment for [tex]\( x \leq -2 \)[/tex]:
- This segment is described by the function [tex]\( h(x) = -x^2 + 4 \)[/tex].
- It is a downward-facing parabola.
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ h(-2) = -(-2)^2 + 4 = -4 + 4 = 0 \][/tex]
- As [tex]\( x \)[/tex] decreases, the value of [tex]\( h(x) \)[/tex] increases until it reaches a maximum at [tex]\( x = 0 \)[/tex] (outside the domain of this segment).
3. Segment for [tex]\( -2 < x < 2 \)[/tex]:
- This segment is described by the function [tex]\( h(x) = -0.5x \)[/tex].
- It is a straight line with a negative slope.
- For [tex]\( x =-2 \)[/tex] (from left transition point):
[tex]\[ \text{From } h(x) = -0.5x, \, h(-2) = -0.5*(-2) = 1 \][/tex]
- For [tex]\( x = 2 \)[/tex] (from right transition point):
[tex]\[ \text{From } h(x) = -0.5x, \, h(2) = -0.5*2 = -1 \][/tex]
4. Segment for [tex]\( x \geq 2 \)[/tex]:
- This segment is described by the function [tex]\( h(x) = x^2 - 5 \)[/tex].
- It is an upward-facing parabola.
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ h(2) = 2^2 - 5 = 4 - 5 = -1 \][/tex]
- As [tex]\( x \)[/tex] increases, the value of [tex]\( h(x) \)[/tex] increases quadratically.
5. Combining the Segments:
- For [tex]\( x \leq -2 \)[/tex]: The points in the segment will follow the curve of the downward parabola until [tex]\((x, h(x)) = (-2, 0)\)[/tex].
- For [tex]\( -2 < x < 2 \)[/tex]: The points will follow the straight line segment from just over [tex]\( -2 \)[/tex], [tex]\( (x, h(x)) \approx (-2, 1) \)[/tex] to just under [tex]\( 2 \)[/tex], [tex]\( (x, h(x)) \approx (2, -1) \)[/tex].
- For [tex]\( x \geq 2 \)[/tex]: The points will follow the parabola, starting at [tex]\((2, -1)\)[/tex], going upwards.
6. Key Transitions:
- At [tex]\( x = -2 \)[/tex], there is a transition discontinuity from [tex]\( (x, h(x)) = (-2, 0) \)[/tex] with the parabolic pattern downward to [tex]\( (x, h(x)) = (-2, 1) \)[/tex] with the linear pattern as [tex]\( x \)[/tex] moves just above [tex]\( -2 \)[/tex].
- At [tex]\( x = 2 \)[/tex], [tex]\( (x, h(x)) = (2, -1) \)[/tex] aligns with both linear and parabolic.
Graph Sketch:
- x < -2: Parabola [tex]\((-x^2 + 4)\)[/tex] starting from below [tex]\((-2, 0)\)[/tex].
- -2 < x < 2: Line segment [tex]\((-0.5x)\)[/tex] from [tex]\((-2, 1)\)[/tex] to [tex]\((2, -1)\)[/tex].
- x \geq 2: Parabola [tex]\((x^2 - 5)\)[/tex] starting from below [tex]\((2, -1)\)[/tex].
By combining these segments with precise transition points explained, the resulting graph of [tex]\( h(x) \)[/tex] should follow the patterns of the piecewise segments described above.
1. Function Analysis:
[tex]\[ h(x) = \begin{cases} -x^2 + 4 & \text{if } x \leq -2 \\ -0.5x & \text{if } -2 < x < 2 \\ x^2 - 5 & \text{if } x \geq 2 \end{cases} \][/tex]
2. Segment for [tex]\( x \leq -2 \)[/tex]:
- This segment is described by the function [tex]\( h(x) = -x^2 + 4 \)[/tex].
- It is a downward-facing parabola.
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ h(-2) = -(-2)^2 + 4 = -4 + 4 = 0 \][/tex]
- As [tex]\( x \)[/tex] decreases, the value of [tex]\( h(x) \)[/tex] increases until it reaches a maximum at [tex]\( x = 0 \)[/tex] (outside the domain of this segment).
3. Segment for [tex]\( -2 < x < 2 \)[/tex]:
- This segment is described by the function [tex]\( h(x) = -0.5x \)[/tex].
- It is a straight line with a negative slope.
- For [tex]\( x =-2 \)[/tex] (from left transition point):
[tex]\[ \text{From } h(x) = -0.5x, \, h(-2) = -0.5*(-2) = 1 \][/tex]
- For [tex]\( x = 2 \)[/tex] (from right transition point):
[tex]\[ \text{From } h(x) = -0.5x, \, h(2) = -0.5*2 = -1 \][/tex]
4. Segment for [tex]\( x \geq 2 \)[/tex]:
- This segment is described by the function [tex]\( h(x) = x^2 - 5 \)[/tex].
- It is an upward-facing parabola.
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ h(2) = 2^2 - 5 = 4 - 5 = -1 \][/tex]
- As [tex]\( x \)[/tex] increases, the value of [tex]\( h(x) \)[/tex] increases quadratically.
5. Combining the Segments:
- For [tex]\( x \leq -2 \)[/tex]: The points in the segment will follow the curve of the downward parabola until [tex]\((x, h(x)) = (-2, 0)\)[/tex].
- For [tex]\( -2 < x < 2 \)[/tex]: The points will follow the straight line segment from just over [tex]\( -2 \)[/tex], [tex]\( (x, h(x)) \approx (-2, 1) \)[/tex] to just under [tex]\( 2 \)[/tex], [tex]\( (x, h(x)) \approx (2, -1) \)[/tex].
- For [tex]\( x \geq 2 \)[/tex]: The points will follow the parabola, starting at [tex]\((2, -1)\)[/tex], going upwards.
6. Key Transitions:
- At [tex]\( x = -2 \)[/tex], there is a transition discontinuity from [tex]\( (x, h(x)) = (-2, 0) \)[/tex] with the parabolic pattern downward to [tex]\( (x, h(x)) = (-2, 1) \)[/tex] with the linear pattern as [tex]\( x \)[/tex] moves just above [tex]\( -2 \)[/tex].
- At [tex]\( x = 2 \)[/tex], [tex]\( (x, h(x)) = (2, -1) \)[/tex] aligns with both linear and parabolic.
Graph Sketch:
- x < -2: Parabola [tex]\((-x^2 + 4)\)[/tex] starting from below [tex]\((-2, 0)\)[/tex].
- -2 < x < 2: Line segment [tex]\((-0.5x)\)[/tex] from [tex]\((-2, 1)\)[/tex] to [tex]\((2, -1)\)[/tex].
- x \geq 2: Parabola [tex]\((x^2 - 5)\)[/tex] starting from below [tex]\((2, -1)\)[/tex].
By combining these segments with precise transition points explained, the resulting graph of [tex]\( h(x) \)[/tex] should follow the patterns of the piecewise segments described above.
