To determine the correct graph of the equation \( y = x^2 + 4x + 3 \), we can analyze the function by completing the square or by finding the vertex.
Step 1: Identify the vertex using the vertex formula
The vertex of a parabola in the form \( y = ax^2 + bx + c \) can be found using: \[ x = -\frac{b}{2a} \] Here, \( a = 1 \) and \( b = 4 \): \[ x = -\frac{4}{2 \cdot 1} = -2 \]
Step 2: Calculate the y-coordinate
Now, substitute \( x = -2 \) back into the equation to find \( y \): \[ y = (-2)^2 + 4(-2) + 3 = 4 - 8 + 3 = -1 \] So, the vertex of the parabola is at \( (-2, -1) \).
Step 3: Finding x-intercepts (roots)
To find the x-intercepts, we can set \( y = 0 \): \[ 0 = x^2 + 4x + 3 \] This can be factored as: \[ 0 = (x + 1)(x + 3) \] Setting each factor to zero gives us: \[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \] \[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \] So, the x-intercepts are at \( (-1, 0) \) and \( (-3, 0) \).
Conclusion
Since the vertex is at \( (-2, -1) \) and the parabola opens upward (as the coefficient of \( x^2 \) is positive), the correct graph must show:
- A vertex at \( (-2, -1) \)
- The parabola passing through the x-intercept at \( (-3, 0) \)
Therefore, the correct response is the first option: A parabola is shown on coordinate grid opening upward with vertex at \( (-2, -1) \), passing through \( (-3, 0) \).