IB MYP Year 4-5: Standard Mathematics : Unit 4: Geometry -Movement on a plane MYP Style Questions

Ans:

To identify the stages that transform $f(x)$ into $g(x)$, let’s compare the two functions and determine the steps needed to get from $f(x)$ to $g(x)$.

Transformation 1: Reflection across the x-axis
This transformation changes the sign of the function, resulting in $-f(x)$. It can be achieved by multiplying $f(x)$ by $-1$. So, the first transformation is the reflection across the x-axis.

Transformation 2: Vertical stretch/compression
The function $g(x)$ is compressed vertically by a factor of 4 compared to $-f(x)$. To achieve this, we multiply $-f(x)$ by $\frac{1}{4}$.

Transformation 3: Horizontal shift to the left
The function $g(x)$ is a result of shifting $-\frac{1}{4}f(x)$ to the left by 2 units. This horizontal shift can be obtained by replacing $x$ with $x+2$ in the equation $-\frac{1}{4}f(x)$.

Transformation 4: Vertical shift upward
Finally, the function $g(x)$ is obtained by shifting $-\frac{1}{4}f(x+2)$ upwards by 1 unit. This vertical shift can be achieved by adding 1 to the expression $-\frac{1}{4}f(x+2)$.

Putting it all together, the stages that transform $f(x)$ into $g(x)$ are as follows:

Transformation 1: Reflection across the x-axis: $-f(x)$
Transformation 2: Vertical stretch/compression: $-\frac{1}{4}f(x)$
Transformation 3: Horizontal shift to the left: $-\frac{1}{4}f(x+2)$
Transformation 4: Vertical shift upward: $-\frac{1}{4}f(x+2)+1$

Therefore, the stages that transform $f(x)$ onto $g(x)$ are as described above.