Answer :
To graph the equation [tex]\( y = x^2 - 2x - 3 \)[/tex], we need to find the key points of the parabola, including the roots, the vertex, and other additional points. Let’s go through the steps to identify these points and then plot the graph.
1. Finding the Vertex:
The vertex of a parabola given by [tex]\( y = ax^2 + bx + c \)[/tex] is found using the formula [tex]\( x = -\frac{b}{2a} \)[/tex].
For the equation [tex]\( y = x^2 - 2x - 3 \)[/tex], we have:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -2 \)[/tex]
- [tex]\( c = -3 \)[/tex]
The x-coordinate of the vertex is:
[tex]\[ x = -\frac{-2}{2 \cdot 1} = \frac{2}{2} = 1 \][/tex]
To find the y-coordinate, we substitute [tex]\( x = 1 \)[/tex] into the equation:
[tex]\[ y = (1)^2 - 2(1) - 3 = 1 - 2 - 3 = -4 \][/tex]
Therefore, the vertex of the parabola is [tex]\((1, -4)\)[/tex].
2. Finding the Roots:
The roots of the quadratic equation can be found using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For [tex]\( y = x^2 - 2x - 3 \)[/tex]:
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-3)}}{2 \cdot 1} \][/tex]
Simplifying inside the square root:
[tex]\[ x = \frac{2 \pm \sqrt{4 + 12}}{2} = \frac{2 \pm \sqrt{16}}{2} \][/tex]
Therefore:
[tex]\[ x = \frac{2 \pm 4}{2} \][/tex]
Which gives us two roots:
[tex]\[ x = \frac{2 + 4}{2} = 3 \quad \text{and} \quad x = \frac{2 - 4}{2} = -1 \][/tex]
So, the roots are [tex]\((3, 0)\)[/tex] and [tex]\((-1, 0)\)[/tex].
3. Finding Additional Points:
Let's calculate the values of the function at [tex]\( x = -2 \)[/tex], [tex]\( x = 0 \)[/tex], and [tex]\( x = 2 \)[/tex].
When [tex]\( x = -2 \)[/tex]:
[tex]\[ y = (-2)^2 - 2(-2) - 3 = 4 + 4 - 3 = 5 \][/tex]
So, we have the point [tex]\((-2, 5)\)[/tex].
When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = (0)^2 - 2(0) - 3 = -3 \][/tex]
So, we have the point [tex]\((0, -3)\)[/tex].
When [tex]\( x = 2 \)[/tex]:
[tex]\[ y = (2)^2 - 2(2) - 3 = 4 - 4 - 3 = -3 \][/tex]
So, we have the point [tex]\((2, -3)\)[/tex].
4. Plotting the Points and Drawing the Graph:
Now that we have all the key points, we can plot them on the graph:
- Vertex: [tex]\((1, -4)\)[/tex]
- Roots: [tex]\((3, 0)\)[/tex] and [tex]\((-1, 0)\)[/tex]
- Additional points: [tex]\((-2, 5)\)[/tex], [tex]\((0, -3)\)[/tex], and [tex]\((2, -3)\)[/tex]
Once these points are plotted on the set of axes, you can draw a smooth curve through these points to complete the graph of the parabola [tex]\( y = x^2 - 2x - 3 \)[/tex].
Here is a visual representation of the points:
```
y
^
|
|
|
|____________|_____________|____________|____________|____________|____________|___ x
-3 -2 -1 0 1 2 3
```
- The asterisks ([tex]\(*\)[/tex]) represent the plotted points.
- Draw a smooth curve through these points to form the parabola.
1. Finding the Vertex:
The vertex of a parabola given by [tex]\( y = ax^2 + bx + c \)[/tex] is found using the formula [tex]\( x = -\frac{b}{2a} \)[/tex].
For the equation [tex]\( y = x^2 - 2x - 3 \)[/tex], we have:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -2 \)[/tex]
- [tex]\( c = -3 \)[/tex]
The x-coordinate of the vertex is:
[tex]\[ x = -\frac{-2}{2 \cdot 1} = \frac{2}{2} = 1 \][/tex]
To find the y-coordinate, we substitute [tex]\( x = 1 \)[/tex] into the equation:
[tex]\[ y = (1)^2 - 2(1) - 3 = 1 - 2 - 3 = -4 \][/tex]
Therefore, the vertex of the parabola is [tex]\((1, -4)\)[/tex].
2. Finding the Roots:
The roots of the quadratic equation can be found using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For [tex]\( y = x^2 - 2x - 3 \)[/tex]:
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-3)}}{2 \cdot 1} \][/tex]
Simplifying inside the square root:
[tex]\[ x = \frac{2 \pm \sqrt{4 + 12}}{2} = \frac{2 \pm \sqrt{16}}{2} \][/tex]
Therefore:
[tex]\[ x = \frac{2 \pm 4}{2} \][/tex]
Which gives us two roots:
[tex]\[ x = \frac{2 + 4}{2} = 3 \quad \text{and} \quad x = \frac{2 - 4}{2} = -1 \][/tex]
So, the roots are [tex]\((3, 0)\)[/tex] and [tex]\((-1, 0)\)[/tex].
3. Finding Additional Points:
Let's calculate the values of the function at [tex]\( x = -2 \)[/tex], [tex]\( x = 0 \)[/tex], and [tex]\( x = 2 \)[/tex].
When [tex]\( x = -2 \)[/tex]:
[tex]\[ y = (-2)^2 - 2(-2) - 3 = 4 + 4 - 3 = 5 \][/tex]
So, we have the point [tex]\((-2, 5)\)[/tex].
When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = (0)^2 - 2(0) - 3 = -3 \][/tex]
So, we have the point [tex]\((0, -3)\)[/tex].
When [tex]\( x = 2 \)[/tex]:
[tex]\[ y = (2)^2 - 2(2) - 3 = 4 - 4 - 3 = -3 \][/tex]
So, we have the point [tex]\((2, -3)\)[/tex].
4. Plotting the Points and Drawing the Graph:
Now that we have all the key points, we can plot them on the graph:
- Vertex: [tex]\((1, -4)\)[/tex]
- Roots: [tex]\((3, 0)\)[/tex] and [tex]\((-1, 0)\)[/tex]
- Additional points: [tex]\((-2, 5)\)[/tex], [tex]\((0, -3)\)[/tex], and [tex]\((2, -3)\)[/tex]
Once these points are plotted on the set of axes, you can draw a smooth curve through these points to complete the graph of the parabola [tex]\( y = x^2 - 2x - 3 \)[/tex].
Here is a visual representation of the points:
```
y
^
|
|
|
|____________|_____________|____________|____________|____________|____________|___ x
-3 -2 -1 0 1 2 3
```
- The asterisks ([tex]\(*\)[/tex]) represent the plotted points.
- Draw a smooth curve through these points to form the parabola.
