Answer :
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.
