Question Type 1: Drawing the effect of single transformations on a graph Exercises

For $f(x)=x^2-3$, sketch the graph of the function after a vertical stretch by factor $5$. Label the vertex and two more points.

Determine whether applying a horizontal stretch of factor $\tfrac14$ to $f(x)=x^2$ yields the same graph as a vertical stretch by factor 16. Provide a short proof. [4 marks]

Sketch the graph of the function obtained by a horizontal compression of factor 3 of $f(x)=x^2$. Label the vertex and two other symmetric points.

Given $f(x)=x^2+1$, sketch the graph of the function obtained by applying a vertical stretch of factor 3 to $f$. Label the new vertex and two other points.

Sketch the graph of $g(x)=4(x^2-2)$. Clearly label the vertex and two additional points on each side of the axis of symmetry.

Determine whether the following two transformations of $f(x)=x^2$ produce the same graph:

1. A horizontal stretch by factor 4
2. A vertical stretch by factor 16

Explain your reasoning.

For $f(x)=x^2-1$, determine whether a horizontal stretch by a scale factor of 5 of the graph of $f$ is equivalent to a vertical stretch by a scale factor of 25. Show all steps.

Let $f(x)=x^2-4$. Draw the graph of the function obtained by a horizontal stretch of factor 2 applied to $f$. Label the vertex and two other points.

Given $f(x)=x^2+2$, draw the graph of the function after reflecting $f$ in the $x$-axis. Label key points.

Determine whether a horizontal compression by a factor of 3 of $f(x)=x^2$ is equivalent to a vertical stretch by a factor of 9. Justify your answer.

Is a horizontal compression by factor 2 of $f(x)=x^2$ the same transformation as a vertical stretch by factor 4? Justify your answer.

Sketch the graph of $g(x)=2|x|$ by applying a vertical stretch of factor 2 to $f(x)=|x|$. Label the vertex and two other points.

For $f(x)=x^2$, determine whether a horizontal stretch by factor $\frac{1}{2}$ and a vertical stretch by factor 4 produce the same transformed graph. Justify your answer.