Additional Mathematics – Quadratic functions | e-Consult

1. Vertex

For a quadratic \(ax^{2}+bx+c\), the x‑coordinate of the vertex is \(-\dfrac{b}{2a}\).

\(a=-2,\; b=8\) so

\(x_{v}= -\dfrac{8}{2(-2)} = 2\).

Substituting \(x=2\) into \(f(x)\):

\(f(2)= -2(2)^{2}+8(2)-3 = -8+16-3 = 5\).

Thus the vertex is \((2,\;5)\).

2. Maximum or Minimum

Since \(a=-2maximum point.

3. Sketch description

• Vertex: \((2,5)\)
• Axis of symmetry: \(x=2\)
• y‑intercept: set \(x=0\) → \(f(0)=-3\) → point \((0,-3)\)
• x‑intercepts: solve \(-2x^{2}+8x-3=0\) → \(x=\dfrac{8\pm\sqrt{64-24}}{4}= \dfrac{8\pm\sqrt{40}}{4}= \dfrac{8\pm2\sqrt{10}}{4}=2\pm\frac{\sqrt{10}}{2}\).

The graph is a downward‑opening parabola passing through the points above.

4. Range

Because the maximum value is \(5\) and the parabola extends indefinitely downwards, the range is

\(\displaystyle \{\,y\mid y\le 5\,\}\) or \((-\infty,5]\).