Review question

# What's the area between $f_k(x) = x(x-k)(x-2)$ and the $x$-axis? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8130

## Question

Let $0< k <2$. Below is sketched a graph of $y=f_k(x)$ where $f_k(x) = x(x-k)(x-2)$. Let $A(k)$ denote the area of the shaded region.

1. Without evaluating them, write down an expression for $A(k)$ in terms of two integrals.

2. Explain why $A(k)$ is a polynomial in $k$ of degree $4$ or less. [You are not required to calculate $A(k)$ explicitly.]

3. Verify that $f_k (1+t) = -f_{2-k}(1-t)$ for any $t$.

4. How can the graph of $y=f_k(x)$ be transformed to the graph of $y = f_{2-k}(x)$?

Deduce that $A(k)=A(2-k)$.

5. Explain why there are constants $a, b, c$ such that $A(k) = a(k-1)^4 + b(k-1)^2 + c.$ [You are not required to calculate $a,b, c$ explicitly.]