Westonci.ca is the ultimate Q&A platform, offering detailed and reliable answers from a knowledgeable community. Join our Q&A platform to connect with experts dedicated to providing precise answers to your questions in different areas. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
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.
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
