N. 11 Write a quadratic function from its opening up or down has vertex \( (-7,-3) \) and passes through \( \left(7,-\frac{61}{4}\right) \), Write its equation in vertex form. Simplify any fractions.

To write the quadratic function in vertex form, we start with the vertex form of a quadratic function, which is given by: \[ f(x) = a(x - h)^2 + k \] where \((h, k)\) is the vertex of the parabola. In this case, the vertex is \((-7, -3)\), so we can substitute \(h = -7\) and \(k = -3\): \[ f(x) = a(x + 7)^2 - 3 \] Next, we need to determine the value of \(a\). We can do this by using the point \(\left(7, -\frac{61}{4}\right)\) that the parabola passes through. We substitute \(x = 7\) and \(f(x) = -\frac{61}{4}\) into the equation: \[ -\frac{61}{4} = a(7 + 7)^2 - 3 \] This simplifies to: \[ -\frac{61}{4} = a(14)^2 - 3 \] Calculating \(14^2\): \[ -\frac{61}{4} = 196a - 3 \] Next, we add 3 to both sides. To do this, we convert 3 to a fraction with a denominator of 4: \[ 3 = \frac{12}{4} \] So we have: \[ -\frac{61}{4} + \frac{12}{4} = 196a \] This simplifies to: \[ -\frac{49}{4} = 196a \] Now, we solve for \(a\) by dividing both sides by 196: \[ a = -\frac{49}{4} \cdot \frac{1}{196} \] Calculating \(\frac{1}{196}\): \[ 196 = 49 \cdot 4 \implies \frac{1}{196} = \frac{1}{49 \cdot 4} = \frac{1}{49} \cdot \frac{1}{4} \] Thus: \[ a = -\frac{49}{4} \cdot \frac{1}{196} = -\frac{49}{4 \cdot 49 \cdot 4} = -\frac{1}{16} \] Now we have \(a = -\frac{1}{16}\). We can substitute \(a\) back into the vertex form of the quadratic function: \[ f(x) = -\frac{1}{16}(x + 7)^2 - 3 \] Thus, the equation of the quadratic function in vertex form is: \[ \boxed{f(x) = -\frac{1}{16}(x + 7)^2 - 3} \]