Problem 38 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

\(f(x)=2 \tan x\), translation \(\pi\) units right, followed by a horizontal shrink by a factor of \(\frac{1}{3}\)

Step-by-Step Solution

Answer

The final transformed function after applying the translations is \(f_2(x) = 2 \tan(3x - \pi)\)

Step 1: Apply Horizontal Translation

The horizontal translation of \(\pi\) units to the right means that instead of \(x\) we need to take \((x - \pi)\) everywhere in the function. So, we get the function \(f_1(x) = 2 \tan(x - \pi)\).

Step 2: Apply Horizontal Shrink

To apply a horizontal shrink to the function by a factor of \(\frac{1}{3}\), we replace \(x\) with \(3x\) in the translated function \(f_1(x)\). This gives us the final transformed function \(f_2(x) = 2 \tan(3x - \pi)\). Note, when we shrink the function horizontally by a factor of \(\frac{1}{3}\), we are essentially stretching it out.

Key Concepts

Function Transformation

Function transformation is a method used to modify the basic form of a function. With transformation, you change the graph of a function in various ways, either by shifting, stretching, compressing, or reflecting it. These alterations help in better understanding and depicting real-world scenarios using functional models.

• Vertical Transformation: Moves the graph up or down.
• Horizontal Transformation: Moves the graph left or right.
• Stretching and Shrinking: Alters the graph's width or height.

Function transformations aren't limited to just one type. Several transformations can be applied in sequence, like in the given exercise, where both translation and shrink operations are performed.

Horizontal Translation

Horizontal translation involves shifting the function along the x-axis. In the case of trigonometric functions like the tangent function, this translation means moving the graph left or right.
To shift the function horizontally, every term involving x is adjusted. In general terms, if you want to move a function right by 'a' units, replace 'x' with '(x - a)' in the function. Conversely, to shift it left by 'a' units, replace 'x' with '(x + a)'.
In the exercise, the function underwent a rightward translation of \(\pi\) units, adjusting the function from\(2 \tan x\) to\(2 \tan(x - \pi)\). This shift relocates the function's behavior on the x-axis by \(\pi\) units.

Horizontal Shrink

Horizontal shrink changes the width of the graph of a function by modifying how quickly the function's cycles repeat as you move along the x-axis. When you apply a horizontal shrink by a factor of \(\frac{1}{b}\), you replace 'x' with 'bx'.
Contrary to what the name suggests, a horizontal "shrink" actually stretches the graph when \(b > 1\). This transformation compresses the x-values of the function, making the function appear to cycle faster.
In the exercise, we applied a horizontal shrink by a factor of \(\frac{1}{3}\), modifying the equation from\(2 \tan(x - \pi)\) to\(2 \tan(3x - \pi)\). This results in a faster repetition of the tangent function across the x-axis.

Tangent Function

The tangent function, expressed as \( \tan x \), is one of the basic trigonometric functions. It relates the angle in a right triangle to the ratio of the opposite side over the adjacent side.
A key feature of \( \tan x \) is its periodicity, with a period of \( \pi \), meaning its pattern repeats every \( \pi \) units along the x-axis.

• It has vertical asymptotes, or undefined points, whenever the cosine function is zero.
• The tangent function is odd, meaning it's symmetric about the origin.
• Within one period, it increases from \(-\infty\) to \(+\infty\).

In transformations, changing parameters of \( \tan x \) can significantly affect its graph. The specific transformations in the exercise manipulate the primary properties but maintain the main characteristics, adjusting how rapidly and where it oscillates.