d the turning point of $$ y=x^{2}+4 x-3 $$ by completing the square

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To find the turning point of the quadratic function $$ y = x^2 + 4x - 3 $$ by completing the square, we will follow these steps:

1. **Identify the quadratic and linear coefficients**: The quadratic term is $$ x^2 $$ and the linear term is $$ 4x $$.

2. **Complete the square**:
   - Start with the expression $$ x^2 + 4x $$.
   - To complete the square, take half of the coefficient of $$ x $$ (which is $$ 4 $$), square it, and add and subtract that value inside the equation.
   - Half of $$ 4 $$ is $$ 2 $$, and squaring it gives $$ 4 $$.

3. **Rewrite the expression**:
   $$
   y = (x^2 + 4x + 4) - 4 - 3
   $$
   $$
   y = (x + 2)^2 - 7
   $$

4. **Identify the turning point**:
   - The vertex form of a quadratic function is $$ y = a(x - h)^2 + k $$, where $$ (h, k) $$ is the vertex (turning point).
   - From our completed square form $$ y = (x + 2)^2 - 7 $$, we can see that:
     - $$ h = -2 $$
     - $$ k = -7 $$

Thus, the turning point of the function $$ y = x^2 + 4x - 3 $$ is at the point $$ (-2, -7) $$.
The turning point of $$ y = x^2 + 4x - 3 $$ is at $$ (-2, -7) $$.

d the turning point of \( y=x^{2}+4 x-3 \) by completing the square

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