A parabola opening up or down has vertex \( (2,1) \) and passes through \( \left(7,-\frac{21}{4}\right) \). Write its equation in vertex form. Simplify any fractions.

The vertex form of a parabola is given by the equation: \[ y = a(x - h)^2 + k \] where \((h, k)\) is the vertex of the parabola. In this case, the vertex is \((2, 1)\), so we can substitute \(h = 2\) and \(k = 1\): \[ y = a(x - 2)^2 + 1 \] Next, we need to find the value of \(a\). We know that the parabola passes through the point \(\left(7, -\frac{21}{4}\right)\). We can substitute \(x = 7\) and \(y = -\frac{21}{4}\) into the equation to find \(a\): \[ -\frac{21}{4} = a(7 - 2)^2 + 1 \] Calculating \(7 - 2\): \[ 7 - 2 = 5 \] Now substituting this back into the equation: \[ -\frac{21}{4} = a(5)^2 + 1 \] Calculating \(5^2\): \[ 5^2 = 25 \] So we have: \[ -\frac{21}{4} = 25a + 1 \] Next, we isolate \(25a\) by subtracting \(1\) from both sides: \[ -\frac{21}{4} - 1 = 25a \] To combine the fractions, we convert \(1\) to a fraction with a denominator of \(4\): \[ 1 = \frac{4}{4} \] Thus: \[ -\frac{21}{4} - \frac{4}{4} = 25a \] This simplifies to: \[ -\frac{21 + 4}{4} = 25a \] \[ -\frac{25}{4} = 25a \] Now, we solve for \(a\) by dividing both sides by \(25\): \[ a = -\frac{25}{4} \cdot \frac{1}{25} = -\frac{1}{4} \] Now we have \(a = -\frac{1}{4}\). We can substitute this back into the vertex form of the equation: \[ y = -\frac{1}{4}(x - 2)^2 + 1 \] Thus, the equation of the parabola in vertex form is: \[ \boxed{y = -\frac{1}{4}(x - 2)^2 + 1} \]