Transformations of Graphs

Section 1.4 Transformations of Graphs

Subsection 1.4.1 Recommended Tutorials

Before starting on these exercises, you should familiarize yourself with the material covered in the following tutorials:

🔗

Subsection 1.4.2 Introduction

In this activity, you will plot multiple functions at once to investigate basic transformations of functions.

🔗

Exercises 1.4.3 Exercises

1.

Consider the graphs of \(\sin(x)\) and \(\cos(x)\text{.}\) Can we apply a transformation of one graph to obtain the other?

🔗

Aside

(a)

Plot the graphs of \(\sin (x)\) and \(\cos (x)\) on the same set of axes. Use red for \(\sin (x)\) and blue for \(\cos (x)\text{.}\)

🔗

(b)

By what amount do you need to shift the graph of \(\sin (x)\) to the left (negative \(x\) direction) so that it coincides with the graph of \(\cos (x)\text{?}\) Answer the question in sentence form by using the

🔗

button to create a new paragraph after the current line.

🔗

(c)

By what amount do you need to shift the graph of \(\sin (x)\) to the right (positive \(x\) direction) so that it coincides with the graph of \(\cos (x)\text{?}\) Answer the question in sentence form using a new paragraph.

🔗

2.

The graphs of \(e^x\) and \(\ln(x)\) should appear to be reflected over the line \(y=x\) because they are inverses. Plot \(e^x\text{,}\) \(\ln(x)\text{,}\) and \(x\) together with the option linestyle=[solid,solid,dash] so that the line of reflection is dashed.

🔗

Hint 1.

Remember when typing the exponential function, use exp(x) or \(e\) from the palettes toolbar.

🔗

Hint 2.

An example of plotting transformations of a function can be found in Example 9.2.

🔗

3.

Assign the function \(f(x) = \sqrt{-x^2 + 4x + 21}\) using the assignment operator, :=. Plot all of the following functions together on the same set of axes:

\begin{equation*} y = f(x), \, y = f(2x), \, y = 3f(x), \, y = f(-x), \, y = -f(x)\text{.} \end{equation*}

Make sure that the graph is displayed with constrained scaling (1:1). Describe each transformation using a new paragraph after your plot.

🔗

Hint 1.

When plotting a graph of any function in the form \(y=f(x)\) using the plot() command, you should omit the \(y=\text{.}\)

🔗

Hint 2.

An example of plotting transformations of a function can be found in Example 9.2.

🔗

Hint 3.

It is always a good practice to specify the colours of multiple graphs in order to tell them apart. A list of plot colours can be found by typing ?colours on a new Maple input.

🔗

4.

Plot each of the following functions and describe how the absolute value transforms the graph.

🔗

(a)

\(y = \sin(x)\)

🔗

(b)

\(y = \sin(|x|)\)

🔗

Hint.

The absolute value function \(|~|\) in Maple is denoted as abs( ).

🔗

(c)

\(y = |\sin(x)|\)

🔗

Prev Top Next