Combined stretch and translation - Function Transformations

CAIE P1 2023 November Q6

8 marks Moderate -0.3

The equation of a curve is $y = x^2 - 8x + 5$.

Find the coordinates of the minimum point of the curve. [2]

The curve is stretched by a factor of 2 parallel to the $y$-axis and then translated by $\begin{pmatrix} 4 \\ 1 \end{pmatrix}$.

Find the coordinates of the minimum point of the transformed curve. [2]
Find the equation of the transformed curve. Give the answer in the form $y = ax^2 + bx + c$, where $a$, $b$ and $c$ are integers to be found. [4]

OCR C3 2010 January Q8

11 marks Standard +0.8

The curve $y = \sqrt{x}$ can be transformed to the curve $y = \sqrt{2x + 3}$ by means of a stretch parallel to the $y$-axis followed by a translation. State the scale factor of the stretch and give details of the translation. [3]
It is given that $N$ is a positive integer. By sketching on a single diagram the graphs of $y = \sqrt{2x + 3}$ and $y = \frac{N}{x}$, show that the equation $$\sqrt{2x + 3} = \frac{N}{x}$$ has exactly one real root. [3]
A sequence $x_1, x_2, x_3, \ldots$ has the property that $$x_{n+1} = N^{\frac{1}{2}}(2x_n + 3)^{-\frac{1}{4}}.$$ For certain values of $x_1$ and $N$, it is given that the sequence converges to the root of the equation $$\sqrt{2x + 3} = \frac{N}{x}.$$
1. Find the value of the integer $N$ for which the sequence converges to the value 1.9037 (correct to 4 decimal places). [2]
2. Find the value of the integer $N$ for which, correct to 4 decimal places, $x_3 = 2.6022$ and $x_4 = 2.6282$. [3]

SPS SPS FM 2024 October Q4

3 marks Moderate -0.8

The curve $y = \sqrt{2x - 1}$ is stretched by scale factor $\frac{1}{4}$ parallel to the $x$-axis and by scale factor $\frac{1}{2}$ parallel to the $y$-axis. Find the resulting equation of the curve, giving your answer in the form $\sqrt{ax - b}$ where $a$ and $b$ are rational numbers. [3]