Final answer:
To find the value of 'a' in the vertex form equation using the given points (0, 5) and (2, -7), we can solve a system of equations by substituting the x and y values into the equation. Rearrange the equation to solve for the value of 'a' using algebraic manipulations.
Step-by-step explanation:
The vertex form equation of a quadratic function is given by f(x) = a(x - h)^2 + k. To find the value of a using the given points (0, 5) and (2, -7), we first need to substitute the x and y values into the equation. For the point (0, 5), we have 5 = a(0 - h)^2 + k, which simplifies to 5 = ah^2 + k. For the point (2, -7), we have -7 = a(2 - h)^2 + k, which simplifies to -7 = a(4 - 4h + h^2) + k.
To solve this system of equations, we can use the method of substitution. Let's solve the first equation for k: k = 5 - ah^2. Now, substitute this expression for k in the second equation: -7 = a(4 - 4h + h^2) + 5 - ah^2. Simplify the equation to obtain: -7 = 4a - 4ah + ah^2 + 5 - ah^2. Combine like terms to get: -7 = 4a - 4ah + 5. Finally, rearrange the equation to solve for a: a = (-7 - 5 + 4ah) / (4h - 4).