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To graph the given piecewise function [tex]\( h(x) \)[/tex], we need to analyze and plot each piece of the function over its respective domain. Let's break this down step by step:
### 1. Understanding the Function Pieces and Their Domains
The function [tex]\( h(x) \)[/tex] is defined differently over three intervals. We need to consider each interval separately:
1. For [tex]\( x \leq -2 \)[/tex]:
[tex]\[ h(x) = -x^2 + 4 \][/tex]
2. For [tex]\( -2 < x < 2 \)[/tex]:
[tex]\[ h(x) = -0.5x \][/tex]
3. For [tex]\( x \geq 2 \)[/tex]:
[tex]\[ h(x) = x^2 - 5 \][/tex]
### 2. Plotting Each Piece
a. For [tex]\( x \leq -2 \)[/tex]: [tex]\( h(x) = -x^2 + 4 \)[/tex]
This part of the function is a downward parabola opening downwards because of the [tex]\(-x^2\)[/tex] term. To plot this:
- Calculate points: [tex]\( x = -2 \rightarrow h(-2) = -(-2)^2 + 4 = -4 + 4 = 0 \)[/tex]
- As [tex]\( x \)[/tex] gets more negative, [tex]\( h(x) \)[/tex] will decrease.
b. For [tex]\( -2 < x < 2 \)[/tex]: [tex]\( h(x) = -0.5x \)[/tex]
This part of the function is a straight line with a slope of [tex]\(-0.5\)[/tex]. To plot this:
- Calculate points: [tex]\( x = -2 \rightarrow h(x) = -0.5(-2) = 1 \)[/tex]
- [tex]\( x = 2 \rightarrow h(2) = -0.5(2) = -1 \)[/tex]
c. For [tex]\( x \geq 2 \)[/tex]: [tex]\( h(x) = x^2 - 5 \)[/tex]
This part of the function is an upward parabola because of the [tex]\( x^2 \)[/tex] term. To plot this:
- Calculate points: [tex]\( x = 2 \rightarrow h(2) = 2^2 - 5 = 4 - 5 = -1 \)[/tex]
- As [tex]\( x \)[/tex] gets larger, [tex]\( h(x) \)[/tex] will increase.
### 3. Combining the Plots
Now, we combine the three pieces, taking into account their respective domains.
- For [tex]\( x \leq -2 \)[/tex], plot the points and curve for [tex]\( h(x) = -x^2 + 4 \)[/tex].
- For [tex]\( -2 < x < 2 \)[/tex], plot the straight line connecting the points [tex]\((2, -1)\)[/tex] and [tex]\((-2, 1)\)[/tex]. Note the domain does not include the endpoints, so these should be open circles.
- For [tex]\( x \geq 2 \)[/tex], plot the points and curve for [tex]\( h(x) = x^2 - 5 \)[/tex].
### 4. Drawing the Graph
- For [tex]\( x \leq -2 \)[/tex], draw the downward parabola passing through [tex]\((-2, 0)\)[/tex].
- For [tex]\(-2 < x < 2 \)[/tex], draw a straight line from [tex]\((-2, 1)\)[/tex] to [tex]\((2, -1)\)[/tex] with open circles at [tex]\((-2, 1)\)[/tex] and [tex]\((2, -1)\)[/tex].
- For [tex]\( x \geq 2 \)[/tex], draw the upward parabola starting from [tex]\((2, -1)\)[/tex].
### 5. Confirming the Graph
Here is a summary of the points:
1. Downward Parabola: Starts at [tex]\((-2, 0)\)[/tex] and goes downward as [tex]\( x \)[/tex] decreases.
2. Straight Line: From [tex]\((-2, 1)\)[/tex] (not included) to [tex]\((2, -1)\)[/tex] (not included).
3. Upward Parabola: Starts from [tex]\((2, -1)\)[/tex] and goes upward as [tex]\( x \)[/tex] increases.
When you plot these, the graph should look like a combination of a downward parabola, a straight line, and an upward parabola, each fitting the designated domains.
This is the complete, detailed graphing procedure for the given piecewise function [tex]\( h(x) \)[/tex].
### 1. Understanding the Function Pieces and Their Domains
The function [tex]\( h(x) \)[/tex] is defined differently over three intervals. We need to consider each interval separately:
1. For [tex]\( x \leq -2 \)[/tex]:
[tex]\[ h(x) = -x^2 + 4 \][/tex]
2. For [tex]\( -2 < x < 2 \)[/tex]:
[tex]\[ h(x) = -0.5x \][/tex]
3. For [tex]\( x \geq 2 \)[/tex]:
[tex]\[ h(x) = x^2 - 5 \][/tex]
### 2. Plotting Each Piece
a. For [tex]\( x \leq -2 \)[/tex]: [tex]\( h(x) = -x^2 + 4 \)[/tex]
This part of the function is a downward parabola opening downwards because of the [tex]\(-x^2\)[/tex] term. To plot this:
- Calculate points: [tex]\( x = -2 \rightarrow h(-2) = -(-2)^2 + 4 = -4 + 4 = 0 \)[/tex]
- As [tex]\( x \)[/tex] gets more negative, [tex]\( h(x) \)[/tex] will decrease.
b. For [tex]\( -2 < x < 2 \)[/tex]: [tex]\( h(x) = -0.5x \)[/tex]
This part of the function is a straight line with a slope of [tex]\(-0.5\)[/tex]. To plot this:
- Calculate points: [tex]\( x = -2 \rightarrow h(x) = -0.5(-2) = 1 \)[/tex]
- [tex]\( x = 2 \rightarrow h(2) = -0.5(2) = -1 \)[/tex]
c. For [tex]\( x \geq 2 \)[/tex]: [tex]\( h(x) = x^2 - 5 \)[/tex]
This part of the function is an upward parabola because of the [tex]\( x^2 \)[/tex] term. To plot this:
- Calculate points: [tex]\( x = 2 \rightarrow h(2) = 2^2 - 5 = 4 - 5 = -1 \)[/tex]
- As [tex]\( x \)[/tex] gets larger, [tex]\( h(x) \)[/tex] will increase.
### 3. Combining the Plots
Now, we combine the three pieces, taking into account their respective domains.
- For [tex]\( x \leq -2 \)[/tex], plot the points and curve for [tex]\( h(x) = -x^2 + 4 \)[/tex].
- For [tex]\( -2 < x < 2 \)[/tex], plot the straight line connecting the points [tex]\((2, -1)\)[/tex] and [tex]\((-2, 1)\)[/tex]. Note the domain does not include the endpoints, so these should be open circles.
- For [tex]\( x \geq 2 \)[/tex], plot the points and curve for [tex]\( h(x) = x^2 - 5 \)[/tex].
### 4. Drawing the Graph
- For [tex]\( x \leq -2 \)[/tex], draw the downward parabola passing through [tex]\((-2, 0)\)[/tex].
- For [tex]\(-2 < x < 2 \)[/tex], draw a straight line from [tex]\((-2, 1)\)[/tex] to [tex]\((2, -1)\)[/tex] with open circles at [tex]\((-2, 1)\)[/tex] and [tex]\((2, -1)\)[/tex].
- For [tex]\( x \geq 2 \)[/tex], draw the upward parabola starting from [tex]\((2, -1)\)[/tex].
### 5. Confirming the Graph
Here is a summary of the points:
1. Downward Parabola: Starts at [tex]\((-2, 0)\)[/tex] and goes downward as [tex]\( x \)[/tex] decreases.
2. Straight Line: From [tex]\((-2, 1)\)[/tex] (not included) to [tex]\((2, -1)\)[/tex] (not included).
3. Upward Parabola: Starts from [tex]\((2, -1)\)[/tex] and goes upward as [tex]\( x \)[/tex] increases.
When you plot these, the graph should look like a combination of a downward parabola, a straight line, and an upward parabola, each fitting the designated domains.
This is the complete, detailed graphing procedure for the given piecewise function [tex]\( h(x) \)[/tex].
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