Questions: Question Rewrite the following quadratic function in standard (vertex) form. f(x)=-7x^2-4x-6 Enter exact values and use improper fractions, if necessary. Provide your answer below: f(x)=

Solution Steps

To rewrite the quadratic function in standard (vertex) form, we need to complete the square. This involves rearranging the quadratic expression into a perfect square trinomial plus a constant. The standard form of a quadratic function is \( f(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola.

To rewrite the quadratic function in standard (vertex) form, we will complete the square.

The given quadratic function is: \[ f(x) = -7x^2 - 4x - 6 \] Here, the coefficients are \(a = -7\), \(b = -4\), and \(c = -6\).

First, factor out the leading coefficient \(-7\) from the terms involving \(x\): \[ f(x) = -7(x^2 + \frac{4}{7}x) - 6 \]

To complete the square, take the coefficient of \(x\) inside the parentheses, \(\frac{4}{7}\), divide it by 2, and square it: \[ \left(\frac{4}{7} \times \frac{1}{2}\right)^2 = \left(\frac{2}{7}\right)^2 = \frac{4}{49} \]

Add and subtract this square inside the parentheses: \[ f(x) = -7\left(x^2 + \frac{4}{7}x + \frac{4}{49} - \frac{4}{49}\right) - 6 \]

Rewrite the expression by completing the square: \[ f(x) = -7\left((x + \frac{2}{7})^2 - \frac{4}{49}\right) - 6 \]

Distribute the \(-7\) and simplify: \[ f(x) = -7(x + \frac{2}{7})^2 + \frac{28}{49} - 6 \]

Convert \(\frac{28}{49}\) to a simpler form: \[ \frac{28}{49} = \frac{4}{7} \]

Thus, the function becomes: \[ f(x) = -7(x + \frac{2}{7})^2 + \frac{4}{7} - 6 \]

Combine the constants: \[ f(x) = -7(x + \frac{2}{7})^2 - \frac{38}{7} \]

Final Answer