Question

-3[f(a-1)] = -3a^2 + 18

Answer 1

`\( f(a) = -(1)/(3)(a^2 - 6) \`) showing a horizontal shift to the right by 1 unit and a vertical compression by a factor of
`\( (1)/(3) \).`

Explanation:

To solve the equation \(
`3[f(a-1)] = -3a^2 + 18 \),` first, isolate
`\( f(a-1) \`) by dividing both sides by 3. This yields
`\( f(a-1) = -a^2 + 6 \).` Next, substitute \( a-1 \) for \( a \) in the function \( f(a) \) to find \( f(a-1) \). Doing so, we get \( f(a-1) = -\frac{1}{3}(a^2 - 6) \). Therefore,
`\( f(a) = -(1)/(3)(a^2 - 6) \).`

This equation represents a transformation of the function
`\( f(a) \)`in terms of ( a-1 ). The given equation expresses \( f(a-1) \) in relation to \( a \) by isolating \( f(a-1) \). Substituting ( a-1 ) back into the original function allows us to represent \( f(a) \) in terms of \( a \). The result indicates that \( f(a) \) is equal to a scaled and translated version of the original function \( -\frac{1}{3}(a^2 - 6) \), showing a horizontal shift to the right by 1 unit and a vertical compression by a factor of
`\( (1)/(3) \).`

This process illustrates a way to manipulate functions using algebraic operations, allowing us to understand the transformational relationship between different forms of a function based on a given equation. Understanding such transformations aids in analyzing the behavior and properties of functions across varying inputs and outputs.