f(x) =-2x^2-4x+5
vertex form
You asked:
Write the following equation in vertex form: \(f\left( x \right) = -\left( 2 {x}^{2} \right) - 4 x + 5\).
MathBot Answer:
\[f(x) = - 2 \left(x - -1\right)^{2} + 7\]
The quadratic expression \(a (x-x_0)^2 + y_0\) is said to be in vertex form.
The quadratic expression \(a x^2 + b x + c\) can be converted to vertex form by completing the square: \[\begin{aligned}a x^2 + b x + c &= a\left(x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 \right) + c\\ &= a \left(x + \frac{b}{2a}\right)^2 + c - \frac{b^2}{4a}\\&= a(x - x_0)^2 + y_0\end{aligned}\] where \(x_0 = -\frac{b}{2 a}\) and \(y_0 = c - \frac{b^2}{4a}\).
The term vertex form comes from the fact that \((x_0, y_0)\) is the vertex of the parabola \(y = a (x-x_0)^2 + y_0\).
The quadratic expression is \(- 2 x^{2} - 4 x + 5\), so \(a = -2\), \(b = -4 \), and \(c = 5\).
Therefore, \(x_0 = -\frac{-4}{2 \cdot -2} = -1\) and \(y_0 = 5 - \frac{\left(-4\right)^{2}}{4 \cdot -2} = 7\).