To graph the quadratic equation \( y = x^2 + 2x + 4 \), we can first determine the vertex and the axis of symmetry.
The general form of a quadratic equation is given by \( y = ax^2 + bx + c \). For this equation, we have:
- \( a = 1 \)
- \( b = 2 \)
- \( c = 4 \)
The formula for the axis of symmetry for a quadratic equation is given by: \[ x = -\frac{b}{2a} \]
Substituting the values of \( a \) and \( b \): \[ x = -\frac{2}{2 \cdot 1} = -\frac{2}{2} = -1 \]
So, the axis of symmetry for the given quadratic equation is \( x = -1 \).
To plot the quadratic:
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Find the vertex by substituting the axis of symmetry \( x = -1 \) back into the quadratic equation: \[ y = (-1)^2 + 2(-1) + 4 = 1 - 2 + 4 = 3 \] The vertex is at the point \((-1, 3)\).
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Identify the y-intercept (where \( x = 0 \)): \[ y = 0^2 + 2(0) + 4 = 4 \] The y-intercept is \( (0, 4) \).
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Find another point on the graph (let's use \( x = -2 \)): \[ y = (-2)^2 + 2(-2) + 4 = 4 - 4 + 4 = 4 \] The point \((-2, 4)\) lies on the graph.
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The graph is a parabola that opens upwards since \( a > 0 \).
Summary of Points:
- Vertex: \((-1, 3)\)
- y-intercept: \((0, 4)\)
- Another Point: \((-2, 4)\)
Finally, you can sketch the graph of the quadratic, plotting these points and ensuring the parabola is symmetrical about the line \( x = -1 \).
Conclusion:
The axis of symmetry for the quadratic \( y = x^2 + 2x + 4 \) is \( x = -1 \).
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