parameter equation

Practise parameter equation with an original VectorPaths question, worked explanation and related VCE Methods practice.

3 practice marks

Original practice question

The function $f(x) = kx^2 - 4x + 3$, where $k$ is a positive constant, has a minimum value of $-1$. Find the value of $k$.

Worked explanation

• Find x-coordinate of vertex

  \[x_{\text{vertex}} = \frac{-(-4)}{2k} = \frac{2}{k}\]

• Express minimum value

  \[f\left(\frac{2}{k}\right) = k \cdot \frac{4}{k^2} - 4 \cdot \frac{2}{k} + 3 = \frac{4}{k} - \frac{8}{k} + 3 = 3 - \frac{4}{k}\]

• Solve the equation

  \[3 - \frac{4}{k} = -1 \implies \frac{4}{k} = 4 \implies k = 1\]

$k = 1$

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