To write the equation of a parabola given the vertex, axis of symmetry, y-intercept, and x-intercepts, we can use the vertex form of a quadratic equation:
\[ h(t) = a(t - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola.
Given the vertex \((0.5, 9)\), we have:
\[ h(t) = a(t - 0.5)^2 + 9 \]
Next, we need to determine the coefficient \(a\). We can use one of the intercepts to find \(a\). Let's use the y-intercept \((0, 5)\):
\[ h(0) = a(0 - 0.5)^2 + 9 \] \[ 5 = a(0.5)^2 + 9 \] \[ 5 = a(0.25) + 9 \] \[ 5 - 9 = 0.25a \] \[ -4 = 0.25a \] \[ a = -4 / 0.25 = -16 \]
Now we have determined \(a = -16\). Thus, the equation becomes:
\[ h(t) = -16(t - 0.5)^2 + 9 \]
This is the equation of the parabola with the given properties. If desired, we can expand this into standard form:
- Expand the square: \[ (t - 0.5)^2 = t^2 - t + 0.25 \]
- Substitute it back into the equation: \[ h(t) = -16(t^2 - t + 0.25) + 9 \]
- Distribute \( -16 \): \[ h(t) = -16t^2 + 16t - 4 + 9 \]
- Combine like terms: \[ h(t) = -16t^2 + 16t + 5 \]
So the equation in standard form is:
\[ h(t) = -16t^2 + 16t + 5 \]