function analysis

Practise function analysis with an original VectorPaths question, worked explanation and related VCE Methods practice.

1 practice marks

Original practice question

A rectangle has vertices at the origin $O$, the point $A(b, 0)$, the point $B(b, b^2 - 4b)$ and the point $C(0, b^2 - 4b)$. Explain why the value of $b$ must satisfy $0 < b < 4$ for the area of the rectangle $OABC$ to be positive.

Worked explanation

• Analyze area conditions

  \[b > 0 \text{ and } b^2 - 4b < 0 \implies b(b-4) < 0 \implies 0 < b < 4\]

For the area to be positive, both the width $b > 0$ and the height $b^2 - 4b < 0$ (since the rectangle extends downward). From $b(b-4) < 0$ we get $0 < b < 4$.

Original VectorPaths practice; not an official VCAA question or marking rubric.