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Sagot :
Sure, let's graph the equation [tex]\( y = |x| + 4 \)[/tex] by substituting values and plot its points. Then, we will reflect the graph across the line [tex]\( y = x \)[/tex] to obtain the graph of its inverse.
### Step 1: Plotting the Equation [tex]\( y = |x| + 4 \)[/tex]
First, let's choose a set of values for [tex]\( x \)[/tex] and calculate the corresponding [tex]\( y \)[/tex] values for the equation [tex]\( y = |x| + 4 \)[/tex].
[tex]\[ \begin{aligned} &\text{For } x = -4: & y &= |-4| + 4 = 4 + 4 = 8 \\ &\text{For } x = -2: & y &= |-2| + 4 = 2 + 4 = 6 \\ &\text{For } x = -1: & y &= |-1| + 4 = 1 + 4 = 5 \\ &\text{For } x = 0: & y &= |0| + 4 = 0 + 4 = 4 \\ &\text{For } x = 1: & y &= |1| + 4 = 1 + 4 = 5 \\ &\text{For } x = 2: & y &= |2| + 4 = 2 + 4 = 6 \\ &\text{For } x = 4: & y &= |4| + 4 = 4 + 4 = 8 \\ \end{aligned} \][/tex]
These points [tex]\((-4, 8)\)[/tex], [tex]\((-2, 6)\)[/tex], [tex]\((-1, 5)\)[/tex], [tex]\((0, 4)\)[/tex], [tex]\((1, 5)\)[/tex], [tex]\((2, 6)\)[/tex], [tex]\((4, 8)\)[/tex] help us visualize the function.
### Step 2: Drawing the Graph
On a coordinate plane:
1. Plot the points: [tex]\((-4, 8)\)[/tex], [tex]\((-2, 6)\)[/tex], [tex]\((-1, 5)\)[/tex], [tex]\((0, 4)\)[/tex], [tex]\((1, 5)\)[/tex], [tex]\((2, 6)\)[/tex], [tex]\((4, 8)\)[/tex].
2. Connect these points. The graph will be V-shaped with the vertex at [tex]\((0,4)\)[/tex].
### Step 3: Reflecting the Graph Across [tex]\( y = x \)[/tex]
To find the inverse, we reflect each point across the line [tex]\( y = x \)[/tex]. This means we swap the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] coordinates of each point plotted above.
[tex]\[ \begin{aligned} &\text{Original Point} &\quad &\text{Reflected Point} \\ &(-4, 8) &\quad &(8, -4) \\ &(-2, 6) &\quad &(6, -2) \\ &(-1, 5) &\quad &(5, -1) \\ &(0, 4) &\quad &(4, 0) \\ &(1, 5) &\quad &(5, 1) \\ &(2, 6) &\quad &(6, 2) \\ &(4, 8) &\quad &(8, 4) \\ \end{aligned} \][/tex]
### Step 4: Drawing the Reflected Graph
1. Plot the reflected points: [tex]\((8, -4)\)[/tex], [tex]\((6, -2)\)[/tex], [tex]\((5, -1)\)[/tex], [tex]\((4, 0)\)[/tex], [tex]\((5, 1)\)[/tex], [tex]\((6, 2)\)[/tex], [tex]\((8, 4)\)[/tex].
2. Connect these points. This reflected graph is an inverse of the original function.
### Step 5: Summary and Visual Reference
The graph of [tex]\( y = |x| + 4 \)[/tex] is V-shaped with the vertex at [tex]\((0, 4)\)[/tex]. The inverse graph, obtained by reflecting across the line [tex]\( y = x \)[/tex], will not correspond to a single function because it will comprise two branches for each segment of the V-shape.
[tex]\[ \begin{aligned} & \text{Function} \quad y = |x| + 4 \\ & \text{Inverse Relation} \quad x = |y| + 4 \end{aligned} \][/tex]
Note that the inverse relation [tex]\( x = |y| + 4 \)[/tex] is not a function by the vertical line test, but it helps us understand the reflection and behavior of the original function.
By plotting these graphs on a coordinate plane, we observe how the reflection of the original function across the line [tex]\( y = x \)[/tex] provides the inverse relation.
### Step 1: Plotting the Equation [tex]\( y = |x| + 4 \)[/tex]
First, let's choose a set of values for [tex]\( x \)[/tex] and calculate the corresponding [tex]\( y \)[/tex] values for the equation [tex]\( y = |x| + 4 \)[/tex].
[tex]\[ \begin{aligned} &\text{For } x = -4: & y &= |-4| + 4 = 4 + 4 = 8 \\ &\text{For } x = -2: & y &= |-2| + 4 = 2 + 4 = 6 \\ &\text{For } x = -1: & y &= |-1| + 4 = 1 + 4 = 5 \\ &\text{For } x = 0: & y &= |0| + 4 = 0 + 4 = 4 \\ &\text{For } x = 1: & y &= |1| + 4 = 1 + 4 = 5 \\ &\text{For } x = 2: & y &= |2| + 4 = 2 + 4 = 6 \\ &\text{For } x = 4: & y &= |4| + 4 = 4 + 4 = 8 \\ \end{aligned} \][/tex]
These points [tex]\((-4, 8)\)[/tex], [tex]\((-2, 6)\)[/tex], [tex]\((-1, 5)\)[/tex], [tex]\((0, 4)\)[/tex], [tex]\((1, 5)\)[/tex], [tex]\((2, 6)\)[/tex], [tex]\((4, 8)\)[/tex] help us visualize the function.
### Step 2: Drawing the Graph
On a coordinate plane:
1. Plot the points: [tex]\((-4, 8)\)[/tex], [tex]\((-2, 6)\)[/tex], [tex]\((-1, 5)\)[/tex], [tex]\((0, 4)\)[/tex], [tex]\((1, 5)\)[/tex], [tex]\((2, 6)\)[/tex], [tex]\((4, 8)\)[/tex].
2. Connect these points. The graph will be V-shaped with the vertex at [tex]\((0,4)\)[/tex].
### Step 3: Reflecting the Graph Across [tex]\( y = x \)[/tex]
To find the inverse, we reflect each point across the line [tex]\( y = x \)[/tex]. This means we swap the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] coordinates of each point plotted above.
[tex]\[ \begin{aligned} &\text{Original Point} &\quad &\text{Reflected Point} \\ &(-4, 8) &\quad &(8, -4) \\ &(-2, 6) &\quad &(6, -2) \\ &(-1, 5) &\quad &(5, -1) \\ &(0, 4) &\quad &(4, 0) \\ &(1, 5) &\quad &(5, 1) \\ &(2, 6) &\quad &(6, 2) \\ &(4, 8) &\quad &(8, 4) \\ \end{aligned} \][/tex]
### Step 4: Drawing the Reflected Graph
1. Plot the reflected points: [tex]\((8, -4)\)[/tex], [tex]\((6, -2)\)[/tex], [tex]\((5, -1)\)[/tex], [tex]\((4, 0)\)[/tex], [tex]\((5, 1)\)[/tex], [tex]\((6, 2)\)[/tex], [tex]\((8, 4)\)[/tex].
2. Connect these points. This reflected graph is an inverse of the original function.
### Step 5: Summary and Visual Reference
The graph of [tex]\( y = |x| + 4 \)[/tex] is V-shaped with the vertex at [tex]\((0, 4)\)[/tex]. The inverse graph, obtained by reflecting across the line [tex]\( y = x \)[/tex], will not correspond to a single function because it will comprise two branches for each segment of the V-shape.
[tex]\[ \begin{aligned} & \text{Function} \quad y = |x| + 4 \\ & \text{Inverse Relation} \quad x = |y| + 4 \end{aligned} \][/tex]
Note that the inverse relation [tex]\( x = |y| + 4 \)[/tex] is not a function by the vertical line test, but it helps us understand the reflection and behavior of the original function.
By plotting these graphs on a coordinate plane, we observe how the reflection of the original function across the line [tex]\( y = x \)[/tex] provides the inverse relation.
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