Problem 40 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Describe the transformation of the graph of \(f\) represented by the function \(g\). \(f(x)=\cos x, g(x)=\cos 6(x-\pi)+9\)

Step-by-Step Solution

Answer

The function \(g(x)=\cos 6(x-\pi)+9\) has three transformations compared to \(f(x)=\cos x\): a horizontal shift to the right by \(\pi\), a vertical shift upwards by \(9\), and a change in period to \(\frac{\pi}{3}\).

Step 1: Identify the Horizontal Shift

The horizontal transformation of the function is represented by what is inside the cosine parentheses. The function \(g(x)=\cos 6(x-\pi)\), \(x-\pi\) signifies a horizontal shift. As the term \(\pi\) is subtracted from \(x\), a shift to the 'right' has occurred by a magnitude of \(\pi\)

Step 2: Identify the Vertical Shift

The vertical transformation of the function is represented by the value being added or subtracted from the cosine function. In the function \(g(x)=\cos 6(x-\pi)+9\), the term \(+9\) signifies a vertical shift. As the term \(9\) is added to the cosine function, a shift 'upwards' has occurred by a magnitude of \(9\)

Step 3: Identify the Change in Period

The period of a function is the interval over which its graph completes one full pattern. The original period for the cosine function is \(2\pi\). From the given function \(g(x)=\cos 6(x-\pi)+9\), we see a 6 is being multiplied with \(x-\pi\) which changes the period of the function. The new period will be \(\frac{2\pi}{6}=\frac{\pi}{3}\)

Key Concepts

Horizontal Shift

In graph transformations, a horizontal shift involves moving the entire graph of a function left or right along the x-axis. It's indicated by changes inside the function's parentheses. For instance, if you have the function \(g(x) = \cos 6(x-\pi)\), the expression \(x-\pi\) indicates a horizontal shift. In this case, the function shifts to the right by \(\pi\).

• Consider the inside of the parenthesis: if you subtract a number, it shifts to the right.
• If you add a number, it shifts to the left.

This shift doesn't alter the shape or period of the graph, only its position along the x-axis. By understanding horizontal shifts, you can easily predict the new graph's starting position.

Vertical Shift

A vertical shift involves moving the entire graph of a function up or down the y-axis. It is determined by the number added to or subtracted from the function. In the cosine transformation \(g(x) = \cos 6(x-\pi) + 9\), we observe a "+9".

• Adding a number (like "+9") shifts the graph upwards by that amount.
• Subtracting a number would shift it downwards.

This type of transformation affects the y-values of the graph, elevating or lowering the entire graph without altering the period or the x-values. Vertical shifts are straightforward and significantly change the visual position of the graph.

Period of a Function

The period of a function refers to the distance over which the function’s pattern repeats. For standard trigonometric functions such as the cosine function, the period is \(2\pi\). However, when this function is altered, the period can change.
In transformed functions, the period can be calculated by considering the coefficient of \(x\) inside the function. For example, in \(g(x) = \cos 6(x-\pi) + 9\), a 6 is multiplied by \(x-\pi\). This coefficients alters the period to \(\frac{2\pi}{6} = \frac{\pi}{3}\).

• The period is determined using the formula: \(\frac{2\pi}{|b|}\) where \(b\) is the coefficient of \(x\).
• A larger \(|b|\) reduces the period, meaning the graph completes its cycles faster.

Understanding changes in the period is essential for analyzing how quickly functions repeat their cycles. This is particularly significant in fields like physics and engineering where predictability over intervals is necessary.