Question Type 5: Drawing the effect of multiple transformations on a graph Exercises

Reflect the graph of $y=x^2$ in the $x$-axis, then translate it left 2 units and up 3 units. Write the equation of the resulting graph.

Find the equation of the graph obtained by reflecting $y=x^2-2$ in the $y$-axis followed by a translation by the vector $\begin{pmatrix} 0 \\ 4 \end{pmatrix}$.

Starting from $y=(x+1)^2+3$, perform a horizontal stretch by a factor of $2$ (from the $y$-axis), then reflect in the $x$-axis. Give the final equation and its vertex.

Describe the sequence of transformations that maps the graph of $y=x^2$ to the graph of $y=4(x-1)^2-6$.

Sketch the graph of $y=4(x-1)^2-6$, labelling its vertex and axis of symmetry.

Describe the transformations that map $y=x^2$ to $y=2(x+3)^2-4$.

Given $f(x)=x^2-2$, find the equation of the function after applying a vertical stretch by factor 4, then a horizontal shift right by 3, then a vertical shift down by 6. Identify its vertex.

Sketch the graph of $y=x^2-2$, labelling its vertex and axis of symmetry.

Express $g(x)=4(x-1)^2-6$ as a transformation of $f(x)=x^2$. Then find the image of the point $(5,10)$ under the inverse transformation (from $g$ back to $f$).

Write the equation of the function obtained by applying the following transformations in order to $y=x^2$: shift left 2 units, vertical stretch by factor 3, then shift up 5 units.

Sketch the graph of the parabola $y=-3(x-2)^2+1$, labelling its vertex and axis of symmetry.