Question Complete the square to re-write the quadratic function in vertex form: $$ y=x^{2}-x-1 $$ Answer Attempt 1 out of 2

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To rewrite the quadratic function $$ y = x^2 - x - 1 $$ in vertex form, we will complete the square. Here are the steps:

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

2. **Factor out the coefficient of $$ x^2 $$**: In this case, the coefficient is 1, so we can proceed without factoring anything out.

3. **Take half of the linear coefficient**: The linear coefficient is $$-1$$. Half of $$-1$$ is $$-\frac{1}{2}$$.

4. **Square this value**: Squaring $$-\frac{1}{2}$$ gives us $$\left(-\frac{1}{2}\right)^2 = \frac{1}{4}$$.

5. **Add and subtract this square inside the equation**: We will add and subtract $$\frac{1}{4}$$ within the equation to maintain equality.

$$
   y = x^2 - x + \frac{1}{4} - \frac{1}{4} - 1
   $$

6. **Rewrite the equation**: Now we can group the perfect square trinomial and simplify the constants.

$$
   y = \left(x - \frac{1}{2}\right)^2 - \frac{1}{4} - 1
   $$

$$
   y = \left(x - \frac{1}{2}\right)^2 - \frac{5}{4}
   $$

Now, the quadratic function in vertex form is:

$$
y = \left(x - \frac{1}{2}\right)^2 - \frac{5}{4}
$$

The vertex of the parabola represented by this function is at the point $$\left(\frac{1}{2}, -\frac{5}{4}\right)$$.
The quadratic function $$ y = x^2 - x - 1 $$ in vertex form is: $$ y = \left(x - \frac{1}{2}\right)^2 - \frac{5}{4} $$ The vertex of the parabola is at $$\left(\frac{1}{2}, -\frac{5}{4}\right)$$.

Question Complete the square to re-write the quadratic function in vertex form: \[ y=x^{2}-x-1 \] Answer Attempt 1 out of 2

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