IB Mathematics Applications & Interpretation Question Type 2: Finding Parameter Values for Changes in Transformations Exercises

The graph of $y=f(x)$ has a point $(2,3)$. It is transformed into $y=g(x)$ by a horizontal scale factor $k$ (about the $y$-axis) and a vertical scale factor $m$ (about the $x$-axis), sending $(2,3)$ to $(8,15)$. Find $k$ and $m$.

A point $P(3,2)$ is carried to $Q(18,16)$ by an enlargement of factor $k$ at the origin, then a horizontal stretch of factor $k$, and finally a vertical stretch of factor $m$, where $k>0$. Determine $k$ and $m$.

A point $P(1,2)$ is mapped to $Q(4,12)$ by an enlargement of scale factor $k$ from the origin, followed by a horizontal stretch of factor $k$, then a vertical stretch of factor $m$. Find the values of $k$ and $m$.

A rectangle with vertices $(1,1),(3,1),(3,2),(1,2)$ is enlarged from centre $(1,1)$ by factor $k$, then stretched vertically by factor $m$ about the $x$-axis. Its image has vertices $(1,2.5),(7,2.5),(7,10),(1,10)$.

Find the value of $k$ and the value of $m$.

The graph of $y=\sin x$ is transformed to $y=a\sin (bx)$ which has amplitude $4$ and period $\pi$. Find $a$ and $b$.

A triangle $ABC$ with $A(1,2)$, $B(3,4)$, $C(5,0)$ is enlarged from the origin to triangle $A^{\prime }{B}^{\prime }{C}^{\prime }$ with $A^{\prime }(2,4)$, $B^{\prime }(6,8)$, $C^{\prime }(10,0)$. Find the scale factor $k$.

The graph of $y=f(x)=x^2$ is transformed to the graph of $y=g(x)=af(x+b)$, where $a,b\in \mathbb{R}$ and $b>0$.

Given that $g(0)=4$ and $g(1)=9$, find the value of $a$ and the value of $b$.

Point $P(2,1)$ is mapped to $Q(8,9)$ by a horizontal stretch of factor $k$ about the $y$-axis, followed by a vertical stretch of factor $m$ about the $x$-axis, then an enlargement of factor $k$ from the origin. Find $k$ and $m$.

A parallelogram with vertices $(0,0),(1,2),(4,3),(3,1)$ undergoes a horizontal stretch of factor $k$ about the $y$–axis and a vertical stretch of factor $m$ about the $x$–axis, mapping to the parallelogram with vertices $(0,0),(3,4),(12,6),(9,2)$. Determine $k$ and $m$.

A circle $x^2+y^2=1$ is transformed by a horizontal stretch of factor $k$ and a vertical stretch of factor $m$ into the ellipse with equation $$\frac{x^2}{9}+\frac{y^2}{4}=1.$$ Determine $k$ and $m$.

Triangle with vertices $A(1,0)$, $B(2,1)$ and $C(1,2)$ is mapped to $A^{\prime }(3,0)$, $B^{\prime }(6,2)$ and $C^{\prime }(3,4)$ by a horizontal stretch of factor $k$ about the $y$-axis followed by a vertical stretch of factor $m$ about the $x$-axis. Find $k$ and $m$.