3.5 Chapter Review
3.5.1 Chapter summary
Vertex form reveals the turning point of a quadratic, and completing the square converts standard form to vertex form. A polynomial’s leading term controls end behaviour, while zeros and multiplicities control axis crossings. Rational functions require careful distinction among holes, vertical asymptotes, and horizontal asymptotes. A reliable graph combines algebraic features, signs, and physical restrictions.
3.5.2 Common mistakes
- Reading the vertex sign incorrectly from \((x-h)^2\).
- Completing the square without accounting for a leading coefficient.
- Assuming every zero produces an axis crossing.
- Treating a cancelled denominator factor as a vertical asymptote.
- Forgetting original domain restrictions.
- Assuming a horizontal asymptote can never be crossed.
3.5.3 Exercises
- Rewrite \(2x^2+8x-3\) in vertex form.
Check Your Work
\(2(x+2)^2-11\). - Describe the end behaviour of \(-5x^6+2x^3\).
Check Your Work
Both ends approach \(-\infty\). - Describe the zeros of \((x+3)^2(x-1)^3\).
Check Your Work
It touches at \(-3\) and crosses at 1. - Find the hole and asymptote of \((x^2-9)/(x^2-x-6)\).
Check Your Work
Hole at \(x=3\) and vertical asymptote at \(x=-2\). - Find the horizontal asymptote of \((4x^2+1)/(5x^2-2x)\).
Check Your Work
\(y=4/5\). - A basin-area model is \(A(w)=-2w^2+32w\). Find the maximum.
Check Your Work
\(A=-2(w-8)^2+128\), so the maximum is 128 m².