Q. The graph shows the equation \(y = x^2 + 4x + 3\).

Step-by-step solution and full explanation

1. Write down the equation.

   The parabola is given by the quadratic function

   \[ y = x^{2} + 4x + 3 \]

2. Find the x-intercepts (zeros) by setting y equal to 0 and solving the quadratic equation.

   Set

   \[ 0 = x^{2} + 4x + 3. \]

   Factor the quadratic trinomial. We look for two numbers whose product is 3 and whose sum is 4. Those numbers are 1 and 3, so the factorization is

   \[ x^{2} + 4x + 3 = (x + 1)(x + 3). \]

   Set each factor equal to 0 to find the roots:

   \[ x + 1 = 0 \quad \Rightarrow \quad x = -1, \]

   \[ x + 3 = 0 \quad \Rightarrow \quad x = -3. \]

   Thus the x-intercepts are

   \[ (-1,\,0) \quad\text{and}\quad (-3,\,0). \]

3. Find the y-intercept by evaluating y at x = 0.

   Substitute x = 0 into the function:

   \[ y(0) = 0^{2} + 4\cdot 0 + 3 = 3. \]

   The y-intercept is

   \[ (0,\,3). \]

4. Find the vertex by completing the square.

   Start with the quadratic expression:

   \[ y = x^{2} + 4x + 3. \]

   Complete the square for the x-terms. Take half of the coefficient of x (which is 4), square it: \((\tfrac{4}{2})^{2} = 4\). Add and subtract 4 inside the expression:

   \[ y = (x^{2} + 4x + 4) + 3 – 4. \]

   Simplify the perfect square trinomial and constants:

   \[ y = (x + 2)^{2} – 1. \]

   From this vertex form, the vertex is at

   \[ (-2,\, -1). \]

5. Determine the axis of symmetry.

   The axis of symmetry is the vertical line through the x-coordinate of the vertex:

   \[ x = -2. \]

6. Determine the direction the parabola opens and the minimum/maximum value.

   Because the coefficient of \(x^{2}\) is positive (1), the parabola opens upward. Therefore the vertex is a minimum point.

   The minimum value of y is the y-coordinate of the vertex:

   \[ y_{\text{min}} = -1 \quad\text{at}\quad x = -2. \]

7. Summary of key features (useful for sketching the graph).
   • Equation: \[ y = x^{2} + 4x + 3 = (x + 2)^{2} – 1. \]
   • Vertex: \[ (-2,\, -1). \]
   • Axis of symmetry: \[ x = -2. \]
   • X-intercepts (zeros): \[ (-1,\,0) \text{ and } (-3,\,0). \]
   • Y-intercept: \[ (0,\,3). \]
   • Parabola opens upward; minimum value \[ y_{\text{min}} = -1. \]
8. How to sketch the graph using these features (step-by-step instructions).
   1. Plot the vertex at \((-2,-1)\).
   2. Draw the axis of symmetry as the vertical line \(x=-2\).
   3. Plot the x-intercepts at \((-1,0)\) and \((-3,0)\); they are symmetric about the axis of symmetry.
   4. Plot the y-intercept at \((0,3)\); check that its reflection across the axis of symmetry is at \((-4,3)\) (optional point).
   5. Draw a smooth upward-opening parabola through these points, symmetric about \(x=-2\).