Transformations of Functions · Paper 1 — Free Maths AI HL Practice Questions

Determine the coordinates of the vertex of $g(x)$. [2 marks]

The graph of $g(x)$ is translated by $\begin{pmatrix} 4 \\ -3 \end{pmatrix}$. Write down the equation of the resulting function $h(x)$ in the form $y = a(x - h)^2 + k$, and hence state the range of $h(x)$. [2 marks]

Write down the coordinates of the vertex of the graph of $h$. [1 mark]

The graph of $h$ is translated by the vector $\begin{pmatrix} 0 \\ k \end{pmatrix}$ so that the resulting graph has exactly one $x$-intercept. Determine the value of $k$. [1 mark]

The function $g$ is defined by $g(x) = -f(x+2) + 3$. Determine the coordinates of the minimum point of $g$. [2 marks]

The graph of $f$ is transformed as follows: - reflected in the $y$-axis, - translated by the vector $\begin{pmatrix} 1 \\ -2 \end{pmatrix}$, - stretched vertically by scale factor $4$. Determine the equation of the resulting function $h(x)$, writing your answer in the form $h(x) = a\ln(x^2 + bx + c) + d$, where $a,\, b,\, c,\, d \in \mathbb{R}$. [4 marks]