To graph the quadratic function \( y = x^2 + 2x + 4 \), we can first identify its vertex and axis of symmetry.
Step 1: Identify the Axis of Symmetry
The formula for the axis of symmetry for a quadratic equation of the form \( y = ax^2 + bx + c \) is given by:
\[ x = -\frac{b}{2a} \]
In our equation, \( a = 1 \) and \( b = 2 \).
Plugging in the values:
\[ x = -\frac{2}{2 \cdot 1} = -1 \]
So, the axis of symmetry is \( x = -1 \).
Step 2: Find the Vertex
To find the vertex, we substitute \( x = -1 \) back into the quadratic equation:
\[ y = (-1)^2 + 2(-1) + 4 \] \[ y = 1 - 2 + 4 = 3 \]
Thus, the vertex of the parabola is at the point \( (-1, 3) \).
Step 3: Determine Additional Points for Graphing
To graph the quadratic, we can find some additional points:
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For \( x = -2 \): \[ y = (-2)^2 + 2(-2) + 4 = 4 - 4 + 4 = 4 \quad \text{(point: } (-2, 4)) \]
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For \( x = 0 \): \[ y = (0)^2 + 2(0) + 4 = 4 \quad \text{(point: } (0, 4)) \]
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For \( x = 1 \): \[ y = (1)^2 + 2(1) + 4 = 1 + 2 + 4 = 7 \quad \text{(point: } (1, 7)) \]
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For \( x = -3 \): \[ y = (-3)^2 + 2(-3) + 4 = 9 - 6 + 4 = 7 \quad \text{(point: } (-3, 7)) \]
Step 4: Sketch the Graph
Now, we can sketch the graph using the points we found:
- Vertex: \( (-1, 3) \)
- Points: \( (-2, 4), (0, 4), (1, 7), (-3, 7) \)
Given these points, we can see the general shape of the parabola, which opens upwards since \( a > 0 \).
Summary
- Axis of symmetry: \( x = -1 \)
- Vertex: \( (-1, 3) \)
You can use this information to draw the graph of the function on a coordinate plane. The parabola will be symmetric about the line \( x = -1 \) and will have its vertex point at \( (-1, 3) \).