Find the best solutions to your questions at Westonci.ca, the premier Q&A platform with a community of knowledgeable experts. Discover a wealth of knowledge from experts across different disciplines on our comprehensive Q&A platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
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].
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
