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Sagot :
To sketch the graph of the parabola [tex]\( y = x^2 - 2x - 3 \)[/tex], we need to determine a few key features: the vertex, axis of symmetry, and [tex]\( y \)[/tex]-intercept.
### Step 1: Find the Vertex
The general form of a quadratic equation is [tex]\( y = ax^2 + bx + c \)[/tex]. In this case, [tex]\( a = 1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = -3 \)[/tex].
The vertex ([tex]\( h, k \)[/tex]) of a parabola can be found using the vertex formula:
[tex]\[ h = -\frac{b}{2a} \][/tex]
Plugging in the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ h = -\frac{-2}{2 \cdot 1} = \frac{2}{2} = 1.0 \][/tex]
Next, to find the [tex]\( k \)[/tex]-coordinate, we substitute [tex]\( x = h \)[/tex] back into the original equation:
[tex]\[ k = (1)^2 - 2(1) - 3 = 1 - 2 - 3 = -4.0 \][/tex]
So, the vertex of the parabola is at [tex]\( (1.0, -4.0) \)[/tex].
### Step 2: Determine the Axis of Symmetry
The axis of symmetry is a vertical line that passes through the vertex. The equation for the axis of symmetry is [tex]\( x = h \)[/tex].
Since [tex]\( h = 1.0 \)[/tex], the axis of symmetry is:
[tex]\[ x = 1.0 \][/tex]
### Step 3: Find the [tex]\( y \)[/tex]-Intercept
The [tex]\( y \)[/tex]-intercept is the point where the parabola crosses the [tex]\( y \)[/tex]-axis. This occurs when [tex]\( x = 0 \)[/tex].
Substituting [tex]\( x = 0 \)[/tex] into the equation [tex]\( y = x^2 - 2x - 3 \)[/tex]:
[tex]\[ y = 0^2 - 2(0) - 3 = -3 \][/tex]
So, the [tex]\( y \)[/tex]-intercept is [tex]\( (0, -3) \)[/tex].
### Step 4: Sketch the Graph
1. Plot the Vertex: Mark the point [tex]\( (1.0, -4.0) \)[/tex] on the coordinate plane.
2. Draw the Axis of Symmetry: Draw a vertical dashed line through [tex]\( x = 1.0 \)[/tex].
3. Plot the [tex]\( y \)[/tex]-Intercept: Mark the point [tex]\( (0, -3) \)[/tex].
4. Draw the Parabola: Using the vertex and [tex]\( y \)[/tex]-intercept as reference points, sketch the shape of the parabola. The parabola will open upwards since [tex]\( a > 0 \)[/tex].
Ensure the parabola is symmetric about the axis of symmetry. The graph should look similar to this:
```
|
5 |
4 |
3 |
2 | (0,-3)
1 | |
| |
-1 | |
-2 | |
-3 |- - - - - - - - - + - - - -
-4 | (1.0, -4.0)
------------------------------
-5 -4 -3 -2 -1 0 1 2 3 4 5
```
### Labels:
- Vertex: [tex]\( (1.0, -4.0) \)[/tex]
- Axis of Symmetry: [tex]\( x = 1.0 \)[/tex]
- [tex]\( y \)[/tex]-Intercept: [tex]\( (0, -3) \)[/tex]
This completes the sketch of the graph for the parabola [tex]\( y = x^2 - 2x - 3 \)[/tex].
### Step 1: Find the Vertex
The general form of a quadratic equation is [tex]\( y = ax^2 + bx + c \)[/tex]. In this case, [tex]\( a = 1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = -3 \)[/tex].
The vertex ([tex]\( h, k \)[/tex]) of a parabola can be found using the vertex formula:
[tex]\[ h = -\frac{b}{2a} \][/tex]
Plugging in the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ h = -\frac{-2}{2 \cdot 1} = \frac{2}{2} = 1.0 \][/tex]
Next, to find the [tex]\( k \)[/tex]-coordinate, we substitute [tex]\( x = h \)[/tex] back into the original equation:
[tex]\[ k = (1)^2 - 2(1) - 3 = 1 - 2 - 3 = -4.0 \][/tex]
So, the vertex of the parabola is at [tex]\( (1.0, -4.0) \)[/tex].
### Step 2: Determine the Axis of Symmetry
The axis of symmetry is a vertical line that passes through the vertex. The equation for the axis of symmetry is [tex]\( x = h \)[/tex].
Since [tex]\( h = 1.0 \)[/tex], the axis of symmetry is:
[tex]\[ x = 1.0 \][/tex]
### Step 3: Find the [tex]\( y \)[/tex]-Intercept
The [tex]\( y \)[/tex]-intercept is the point where the parabola crosses the [tex]\( y \)[/tex]-axis. This occurs when [tex]\( x = 0 \)[/tex].
Substituting [tex]\( x = 0 \)[/tex] into the equation [tex]\( y = x^2 - 2x - 3 \)[/tex]:
[tex]\[ y = 0^2 - 2(0) - 3 = -3 \][/tex]
So, the [tex]\( y \)[/tex]-intercept is [tex]\( (0, -3) \)[/tex].
### Step 4: Sketch the Graph
1. Plot the Vertex: Mark the point [tex]\( (1.0, -4.0) \)[/tex] on the coordinate plane.
2. Draw the Axis of Symmetry: Draw a vertical dashed line through [tex]\( x = 1.0 \)[/tex].
3. Plot the [tex]\( y \)[/tex]-Intercept: Mark the point [tex]\( (0, -3) \)[/tex].
4. Draw the Parabola: Using the vertex and [tex]\( y \)[/tex]-intercept as reference points, sketch the shape of the parabola. The parabola will open upwards since [tex]\( a > 0 \)[/tex].
Ensure the parabola is symmetric about the axis of symmetry. The graph should look similar to this:
```
|
5 |
4 |
3 |
2 | (0,-3)
1 | |
| |
-1 | |
-2 | |
-3 |- - - - - - - - - + - - - -
-4 | (1.0, -4.0)
------------------------------
-5 -4 -3 -2 -1 0 1 2 3 4 5
```
### Labels:
- Vertex: [tex]\( (1.0, -4.0) \)[/tex]
- Axis of Symmetry: [tex]\( x = 1.0 \)[/tex]
- [tex]\( y \)[/tex]-Intercept: [tex]\( (0, -3) \)[/tex]
This completes the sketch of the graph for the parabola [tex]\( y = x^2 - 2x - 3 \)[/tex].
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