Get the answers you need at Westonci.ca, where our expert community is dedicated to providing you with accurate information. Discover comprehensive solutions to your questions from a wide network of experts on our user-friendly platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
Sure, I'd be delighted to walk you through the process of graphing the function:
[tex]\[ f(x) = -\frac{1}{2}(x - 2)^2 - 4 \][/tex]
To graph this function, we will follow these steps:
### Step 1: Understand the function's form
The given function is:
[tex]\[ f(x) = -\frac{1}{2}(x - 2)^2 - 4 \][/tex]
This is a quadratic function in the form of
[tex]\[ y = a(x - h)^2 + k \][/tex]
#### Key characteristics:
- Vertex: [tex]\((h, k)\)[/tex], where [tex]\(h = 2\)[/tex] and [tex]\(k = -4\)[/tex]. Thus, the vertex is [tex]\((2, -4)\)[/tex].
- Opens downward: Since [tex]\(a = -\frac{1}{2}\)[/tex] (negative value), the parabola opens downward.
### Step 2: Determine the vertex
The vertex of the parabola is [tex]\((2, -4)\)[/tex]. This is the highest point on the graph because the parabola opens downward.
### Step 3: Identify the axis of symmetry
The axis of symmetry for the parabola is the vertical line that passes through the vertex. Hence, it is:
[tex]\[ x = 2 \][/tex]
### Step 4: Determine points on either side of the vertex
Choose some x-values to the left and right of the vertex and compute their corresponding [tex]\(y\)[/tex]-values to get a sense of the shape of the parabola.
We can plug in values of [tex]\(x\)[/tex] and calculate [tex]\(f(x)\)[/tex]:
1. At [tex]\(x = 0\)[/tex]:
[tex]\[ f(0) = -\frac{1}{2}(0 - 2)^2 - 4 = -\frac{1}{2}(4) - 4 = -2 - 4 = -6 \][/tex]
Point: [tex]\((0, -6)\)[/tex]
2. At [tex]\(x = 4\)[/tex]:
[tex]\[ f(4) = -\frac{1}{2}(4 - 2)^2 - 4 = -\frac{1}{2}(2)^2 - 4 = -2 - 4 = -6 \][/tex]
Point: [tex]\((4, -6)\)[/tex]
3. At [tex]\(x= -2\)[/tex]:
[tex]\[ f(-2) = -\frac{1}{2}((-2) - 2)^2 - 4 = -\frac{1}{2}(4 + 0)^2 -4 = -\frac{1}{2}(16) - 4= -8 -4 = -12 \][/tex]
Point: [tex]\((-2, -12)\)[/tex]
### Step 5: Plot these points and draw the parabola
Based on the given numerical values, we have many points that lie on the graph of the function. Here are a few important points that we calculated:
- Vertex: [tex]\((2, -4)\)[/tex]
- Symmetric Points: [tex]\((0, -6)\)[/tex] and [tex]\((4, -6)\)[/tex]
- Leftward Point: [tex]\((-2, -12)\)[/tex]
### Step 6: Sketch the function
Using these points, we can sketch the graph. Here’s a general description of how the points and the curve appear:
- Plot the vertex [tex]\((2, -4)\)[/tex].
- Plot the symmetric points [tex]\((0, -6)\)[/tex] and [tex]\((4, -6)\)[/tex].
- Plot [tex]\((-2, -12)\)[/tex] or any other points to ensure the shape.
The graph should look like a downward-facing parabola with a maximum at the vertex [tex]\((2, -4)\)[/tex].
### Visual Representation
Below is a basic sketch of how the parabolic graph should appear on a coordinate plane:
[tex]\[ \begin{array}{c|c} \text{x} & \text{y=f(x)} \\\hline -2 & -12 \\ 0 & -6\\ 2 & -4 \\ 4 & -6 \\ 10 & -41 \end{array} \][/tex]
```
y
12
+---+
10 -
+ :
8 -:
+ :
6 -:
+ : :
4 -: :
+ : :
2 -: :
+ : :
0 - :
+-:-+--*-----2----+-----4----+----5
-2 -: :
+ :
-4 ===================================================
+ :
- 6 :
+ :
-8 - :
+ :
-10- :
-+ :
-12 ==================================================
x
```
So, in this graph, our key points including the vertex are plotted, and the general shape of the parabola from the vertex extending downwards is shown clearly.
[tex]\[ f(x) = -\frac{1}{2}(x - 2)^2 - 4 \][/tex]
To graph this function, we will follow these steps:
### Step 1: Understand the function's form
The given function is:
[tex]\[ f(x) = -\frac{1}{2}(x - 2)^2 - 4 \][/tex]
This is a quadratic function in the form of
[tex]\[ y = a(x - h)^2 + k \][/tex]
#### Key characteristics:
- Vertex: [tex]\((h, k)\)[/tex], where [tex]\(h = 2\)[/tex] and [tex]\(k = -4\)[/tex]. Thus, the vertex is [tex]\((2, -4)\)[/tex].
- Opens downward: Since [tex]\(a = -\frac{1}{2}\)[/tex] (negative value), the parabola opens downward.
### Step 2: Determine the vertex
The vertex of the parabola is [tex]\((2, -4)\)[/tex]. This is the highest point on the graph because the parabola opens downward.
### Step 3: Identify the axis of symmetry
The axis of symmetry for the parabola is the vertical line that passes through the vertex. Hence, it is:
[tex]\[ x = 2 \][/tex]
### Step 4: Determine points on either side of the vertex
Choose some x-values to the left and right of the vertex and compute their corresponding [tex]\(y\)[/tex]-values to get a sense of the shape of the parabola.
We can plug in values of [tex]\(x\)[/tex] and calculate [tex]\(f(x)\)[/tex]:
1. At [tex]\(x = 0\)[/tex]:
[tex]\[ f(0) = -\frac{1}{2}(0 - 2)^2 - 4 = -\frac{1}{2}(4) - 4 = -2 - 4 = -6 \][/tex]
Point: [tex]\((0, -6)\)[/tex]
2. At [tex]\(x = 4\)[/tex]:
[tex]\[ f(4) = -\frac{1}{2}(4 - 2)^2 - 4 = -\frac{1}{2}(2)^2 - 4 = -2 - 4 = -6 \][/tex]
Point: [tex]\((4, -6)\)[/tex]
3. At [tex]\(x= -2\)[/tex]:
[tex]\[ f(-2) = -\frac{1}{2}((-2) - 2)^2 - 4 = -\frac{1}{2}(4 + 0)^2 -4 = -\frac{1}{2}(16) - 4= -8 -4 = -12 \][/tex]
Point: [tex]\((-2, -12)\)[/tex]
### Step 5: Plot these points and draw the parabola
Based on the given numerical values, we have many points that lie on the graph of the function. Here are a few important points that we calculated:
- Vertex: [tex]\((2, -4)\)[/tex]
- Symmetric Points: [tex]\((0, -6)\)[/tex] and [tex]\((4, -6)\)[/tex]
- Leftward Point: [tex]\((-2, -12)\)[/tex]
### Step 6: Sketch the function
Using these points, we can sketch the graph. Here’s a general description of how the points and the curve appear:
- Plot the vertex [tex]\((2, -4)\)[/tex].
- Plot the symmetric points [tex]\((0, -6)\)[/tex] and [tex]\((4, -6)\)[/tex].
- Plot [tex]\((-2, -12)\)[/tex] or any other points to ensure the shape.
The graph should look like a downward-facing parabola with a maximum at the vertex [tex]\((2, -4)\)[/tex].
### Visual Representation
Below is a basic sketch of how the parabolic graph should appear on a coordinate plane:
[tex]\[ \begin{array}{c|c} \text{x} & \text{y=f(x)} \\\hline -2 & -12 \\ 0 & -6\\ 2 & -4 \\ 4 & -6 \\ 10 & -41 \end{array} \][/tex]
```
y
12
+---+
10 -
+ :
8 -:
+ :
6 -:
+ : :
4 -: :
+ : :
2 -: :
+ : :
0 - :
+-:-+--*-----2----+-----4----+----5
-2 -: :
+ :
-4 ===================================================
+ :
- 6 :
+ :
-8 - :
+ :
-10- :
-+ :
-12 ==================================================
x
```
So, in this graph, our key points including the vertex are plotted, and the general shape of the parabola from the vertex extending downwards is shown clearly.
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.