A parabola opening up or down has vertex \( (-5,2) \) and passes through \( \left(-13,-\frac{6}{5}\right) \). Write its equation in vertex form. Simplify any fractions. Submit

To find the equation of a parabola in vertex form, we use the formula: \[ y = a(x - h)^2 + k \] where \((h, k)\) is the vertex of the parabola. Given the vertex \((-5, 2)\), we have \(h = -5\) and \(k = 2\). Thus, the equation becomes: \[ y = a(x + 5)^2 + 2 \] Next, we need to determine the value of \(a\). We can do this by using the point \(\left(-13, -\frac{6}{5}\right)\) that the parabola passes through. We substitute \(x = -13\) and \(y = -\frac{6}{5}\) into the equation: \[ -\frac{6}{5} = a(-13 + 5)^2 + 2 \] Calculating \(-13 + 5\): \[ -13 + 5 = -8 \] Now we substitute this back into the equation: \[ -\frac{6}{5} = a(-8)^2 + 2 \] Calculating \((-8)^2\): \[ (-8)^2 = 64 \] So we have: \[ -\frac{6}{5} = 64a + 2 \] Next, we isolate \(a\). First, subtract 2 from both sides: \[ -\frac{6}{5} - 2 = 64a \] To combine the fractions, we convert 2 to a fraction with a denominator of 5: \[ 2 = \frac{10}{5} \] Thus: \[ -\frac{6}{5} - \frac{10}{5} = 64a \] This simplifies to: \[ -\frac{16}{5} = 64a \] Now, divide both sides by 64 to solve for \(a\): \[ a = -\frac{16}{5} \cdot \frac{1}{64} = -\frac{16}{320} = -\frac{1}{20} \] Now we have \(a = -\frac{1}{20}\). We can substitute \(a\) back into the vertex form equation: \[ y = -\frac{1}{20}(x + 5)^2 + 2 \] Thus, the equation of the parabola in vertex form is: \[ \boxed{y = -\frac{1}{20}(x + 5)^2 + 2} \]