Find an equation for the parabola with vertex \( (1,4) \) that passes through the point \( (3,8) \).

To find the equation of the parabola, we can use the vertex form of a quadratic equation, which is given by: \[ y = a(x - h)^2 + k \] where \( (h, k) \) is the vertex of the parabola. In this case, the vertex is \( (1, 4) \), so we have: \[ y = a(x - 1)^2 + 4 \] Next, we can use the point \( (3, 8) \) to find the value of \( a \). Substituting \( x = 3 \) and \( y = 8 \) into the equation: \[ 8 = a(3 - 1)^2 + 4 \] \[ 8 = a(2)^2 + 4 \] \[ 8 = 4a + 4 \] Now, subtract 4 from both sides: \[ 4 = 4a \] Dividing both sides by 4 gives: \[ a = 1 \] Now we can substitute \( a \) back into our vertex form: \[ y = 1(x - 1)^2 + 4 \] This simplifies to: \[ y = (x - 1)^2 + 4 \] Therefore, the equation of the parabola is: \[ y = (x - 1)^2 + 4 \] Now go out and graph this beauty! You'll find its characteristic U-shape, opening upwards, centered around the vertex with all its glory!