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To graph the parabola defined by the equation [tex]\(y = -2x^2 - 4x + 2\)[/tex], we need to plot five crucial points. These five points include the vertex of the parabola, two points to the left of the vertex, and two points to the right of the vertex. Here is the step-by-step solution to obtain these points:
### Step 1: Identify the Vertex
The vertex of a parabola given by the equation [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formula for the x-coordinate of the vertex:
[tex]\[ x = -\frac{b}{2a} \][/tex]
For the equation [tex]\( y = -2x^2 - 4x + 2 \)[/tex]:
- [tex]\( a = -2 \)[/tex]
- [tex]\( b = -4 \)[/tex]
- [tex]\( c = 2 \)[/tex]
Plug [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula:
[tex]\[ x = -\frac{-4}{2 \cdot -2} = -\frac{4}{-4} = 1 \][/tex]
So, the x-coordinate of the vertex is [tex]\( x = -1 \)[/tex]. Now, substitute [tex]\( x = -1 \)[/tex] back into the equation to find the y-coordinate:
[tex]\[ y = -2(-1)^2 - 4(-1) + 2 = -2(1) + 4 + 2 = -2 + 4 + 2 = 4 \][/tex]
Therefore, the vertex is at [tex]\( (-1, 4) \)[/tex].
### Step 2: Points to the Left of the Vertex
#### Point 1: One unit to the left of the vertex
The x-coordinate is [tex]\( x = -1 - 1 = -2 \)[/tex]. Now, substitute [tex]\( x = -2 \)[/tex] into the equation to find the y-coordinate:
[tex]\[ y = -2(-2)^2 - 4(-2) + 2 = -2(4) + 8 + 2 = -8 + 8 + 2 = 2 \][/tex]
So, the first point to the left of the vertex is [tex]\( (-2, 2) \)[/tex].
#### Point 2: Two units to the left of the vertex
The x-coordinate is [tex]\( x = -1 - 2 = -3 \)[/tex]. Now, substitute [tex]\( x = -3 \)[/tex] into the equation to find the y-coordinate:
[tex]\[ y = -2(-3)^2 - 4(-3) + 2 = -2(9) + 12 + 2 = -18 + 12 + 2 = -4 \][/tex]
So, the second point to the left of the vertex is [tex]\( (-3, -4) \)[/tex].
### Step 3: Points to the Right of the Vertex
#### Point 1: One unit to the right of the vertex
The x-coordinate is [tex]\( x = -1 + 1 = 0 \)[/tex]. Now, substitute [tex]\( x = 0 \)[/tex] into the equation to find the y-coordinate:
[tex]\[ y = -2(0)^2 - 4(0) + 2 = 2 \][/tex]
So, the first point to the right of the vertex is [tex]\( (0, 2) \)[/tex].
#### Point 2: Two units to the right of the vertex
The x-coordinate is [tex]\( x = -1 + 2 = 1 \)[/tex]. Now, substitute [tex]\( x = 1 \)[/tex] into the equation to find the y-coordinate:
[tex]\[ y = -2(1)^2 - 4(1) + 2 = -2(1) - 4 + 2 = -2 - 4 + 2 = -4 \][/tex]
So, the second point to the right of the vertex is [tex]\( (1, -4) \)[/tex].
### Summary of Points
Here are the five points to plot on the graph of the parabola:
1. Vertex: [tex]\((-1, 4)\)[/tex]
2. Left Point 1: [tex]\((-2, 2)\)[/tex]
3. Left Point 2: [tex]\((-3, -4)\)[/tex]
4. Right Point 1: [tex]\((0, 2)\)[/tex]
5. Right Point 2: [tex]\((1, -4)\)[/tex]
### Graphing
To graph the function, plot these points on the coordinate plane and draw a smooth curve through them to represent the parabola [tex]\( y = -2x^2 - 4x + 2 \)[/tex]. The parabola will open downwards since the coefficient of [tex]\( x^2 \)[/tex] is negative ([tex]\( -2 \)[/tex]). Points are as follows:
- [tex]\((-1, 4)\)[/tex]
- [tex]\((-2, 2)\)[/tex]
- [tex]\((-3, -4)\)[/tex]
- [tex]\((0, 2)\)[/tex]
- [tex]\((1, -4)\)[/tex]
You can use your graphing tool or graph-a-function button to plot these points and visualize the parabola.
### Step 1: Identify the Vertex
The vertex of a parabola given by the equation [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formula for the x-coordinate of the vertex:
[tex]\[ x = -\frac{b}{2a} \][/tex]
For the equation [tex]\( y = -2x^2 - 4x + 2 \)[/tex]:
- [tex]\( a = -2 \)[/tex]
- [tex]\( b = -4 \)[/tex]
- [tex]\( c = 2 \)[/tex]
Plug [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula:
[tex]\[ x = -\frac{-4}{2 \cdot -2} = -\frac{4}{-4} = 1 \][/tex]
So, the x-coordinate of the vertex is [tex]\( x = -1 \)[/tex]. Now, substitute [tex]\( x = -1 \)[/tex] back into the equation to find the y-coordinate:
[tex]\[ y = -2(-1)^2 - 4(-1) + 2 = -2(1) + 4 + 2 = -2 + 4 + 2 = 4 \][/tex]
Therefore, the vertex is at [tex]\( (-1, 4) \)[/tex].
### Step 2: Points to the Left of the Vertex
#### Point 1: One unit to the left of the vertex
The x-coordinate is [tex]\( x = -1 - 1 = -2 \)[/tex]. Now, substitute [tex]\( x = -2 \)[/tex] into the equation to find the y-coordinate:
[tex]\[ y = -2(-2)^2 - 4(-2) + 2 = -2(4) + 8 + 2 = -8 + 8 + 2 = 2 \][/tex]
So, the first point to the left of the vertex is [tex]\( (-2, 2) \)[/tex].
#### Point 2: Two units to the left of the vertex
The x-coordinate is [tex]\( x = -1 - 2 = -3 \)[/tex]. Now, substitute [tex]\( x = -3 \)[/tex] into the equation to find the y-coordinate:
[tex]\[ y = -2(-3)^2 - 4(-3) + 2 = -2(9) + 12 + 2 = -18 + 12 + 2 = -4 \][/tex]
So, the second point to the left of the vertex is [tex]\( (-3, -4) \)[/tex].
### Step 3: Points to the Right of the Vertex
#### Point 1: One unit to the right of the vertex
The x-coordinate is [tex]\( x = -1 + 1 = 0 \)[/tex]. Now, substitute [tex]\( x = 0 \)[/tex] into the equation to find the y-coordinate:
[tex]\[ y = -2(0)^2 - 4(0) + 2 = 2 \][/tex]
So, the first point to the right of the vertex is [tex]\( (0, 2) \)[/tex].
#### Point 2: Two units to the right of the vertex
The x-coordinate is [tex]\( x = -1 + 2 = 1 \)[/tex]. Now, substitute [tex]\( x = 1 \)[/tex] into the equation to find the y-coordinate:
[tex]\[ y = -2(1)^2 - 4(1) + 2 = -2(1) - 4 + 2 = -2 - 4 + 2 = -4 \][/tex]
So, the second point to the right of the vertex is [tex]\( (1, -4) \)[/tex].
### Summary of Points
Here are the five points to plot on the graph of the parabola:
1. Vertex: [tex]\((-1, 4)\)[/tex]
2. Left Point 1: [tex]\((-2, 2)\)[/tex]
3. Left Point 2: [tex]\((-3, -4)\)[/tex]
4. Right Point 1: [tex]\((0, 2)\)[/tex]
5. Right Point 2: [tex]\((1, -4)\)[/tex]
### Graphing
To graph the function, plot these points on the coordinate plane and draw a smooth curve through them to represent the parabola [tex]\( y = -2x^2 - 4x + 2 \)[/tex]. The parabola will open downwards since the coefficient of [tex]\( x^2 \)[/tex] is negative ([tex]\( -2 \)[/tex]). Points are as follows:
- [tex]\((-1, 4)\)[/tex]
- [tex]\((-2, 2)\)[/tex]
- [tex]\((-3, -4)\)[/tex]
- [tex]\((0, 2)\)[/tex]
- [tex]\((1, -4)\)[/tex]
You can use your graphing tool or graph-a-function button to plot these points and visualize the parabola.
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