Problem 36 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

USING EQUATIONS Graph one period of each function. Describe the transformation of the graph of its parent function. a. \(g(x)=\sec x+3\) b. \(g(x)=\csc x-2\) c. \(g(x)=\cot (x-\pi)\) d. \(g(x)=-\tan x\)

Step-by-Step Solution

Answer

The functions \(g(x)=\sec x+3\) and \(g(x)=\csc x-2\) undergo vertical translations of 3 units upward and 2 units downward, respectively. The function \(g(x)=\cot (x-\pi)\) undergoes a horizontal shift of \(\pi\) units to the right. Lastly, \(g(x)=-\tan x\) is a reflection of the graph of \(y=\tan x\) over the x-axis.

Step 1: Graphing \(g(x)=\sec x+3\)

Start by graphing the parent function \(y=\sec x\). This graph resembles the reciprocal of the cosine function, with vertical asymptotes at odd multiples of \(\pi/2\). Carry out a vertical shift of three units upward which gives the graph of \(g(x) = \sec x + 3\).

Step 2: Graphing \(g(x)=\csc x-2\)

Start with the parent function \(y=\csc x\), which resembles the reciprocal of the sine function, with vertical asymptotes at multiples of \(\pi\). Perform a vertical shift of two units downward to obtain the graph of \(g(x) = \csc x - 2\).

Step 3: Graphing \(g(x)=\cot (x-\pi)\)

Start with the parent function \(y=\cot x\), which is a periodic function with period \(\pi\) and has vertical asymptotes at multiples of \(\pi\). Perform a horizontal shift of \(\pi\) units to the right to obtain the graph of \(g(x) = \cot (x - \pi)\).

Step 4: Graphing \(g(x)=-\tan x\)

Start with the parent function \(y=\tan x\), which is a periodic function with period \(\pi\) and has vertical asymptotes at odd multiples of \(\pi/2\). Apply a reflection over the x-axis to get the graph of \(g(x) = -\tan x\).

Key Concepts

Graph Transformations

Graph transformations allow us to modify the position and shape of trigonometric functions in a graphical representation. By applying transformations, we can stretch, compress, shift, or reflect the parent functions, which are the basic forms of trigonometric functions.

• Vertical shifts involve moving the graph up or down by adding or subtracting a number to the function.
• Horizontal shifts involve moving the graph left or right by adding or subtracting from the variable inside the function.
• Reflections flip the graph over the x-axis or y-axis by multiplying the function by -1.
• Stretches or compressions occur by multiplying the function by a constant greater or less than one.

By recognizing these transformations, we can quickly sketch and understand new versions of basic trigonometric functions.

Secant Function

The secant function, denoted as \(\sec x\), is the reciprocal of the cosine function. This means that \(\sec x = \frac{1}{\cos x}\). The graph of the secant function exhibits vertical asymptotes where the cosine function is zero because division by zero is undefined. These asymptotes occur at odd multiples of \(\frac{\pi}{2}\).
To graph a transformed secant function, such as \(g(x) = \sec x + 3\), you apply a vertical shift upward by 3 units. This transformation keeps the shape of the secant graph the same while moving each point three units higher on the y-axis. The vertical asymptotes remain unchanged, but all other points are shifted accordingly.

Cosecant Function

The cosecant function, symbolized as \(\csc x\), is the reciprocal of the sine function. Thus, \(\csc x = \frac{1}{\sin x}\). Like the secant function, it has vertical asymptotes where the sine function is zero, occurring at integer multiples of \(\pi\).
For the transformed function \(g(x) = \csc x - 2\), you perform a vertical shift downward by 2 units. Each point on the original graph moves two units down, altering the position but not the shape of the graph.
Vertical asymptotes at multiples of \(\pi\) remain, while all other points are lowered by two units, giving a new position for the function on the coordinate plane.

Cotangent Function

The cotangent function, written as \(\cot x\), is the reciprocal of the tangent function, hence \(\cot x = \frac{1}{\tan x}\). The graph of the cotangent function features vertical asymptotes at integer multiples of \(\pi\). Its period is \(\pi\), meaning the pattern of the graph repeats every \(\pi\) units.
When graphing a transformed cotangent function such as \(g(x) = \cot (x - \pi)\), a horizontal shift of \(\pi\) units to the right is performed. This transformation shifts the entire graph to the right without altering its shape. Asymptotes and all graph features move to the right by \(\pi\) units.

Tangent Function

The tangent function, denoted as \(\tan x\), is defined as the ratio of the sine function to the cosine function: \(\tan x = \frac{\sin x}{\cos x}\). This function has vertical asymptotes at odd multiples of \(\frac{\pi}{2}\), where the cosine is zero. Its period is \(\pi\), so the graph repeats every \(\pi\) units.
For \(g(x) = -\tan x\), a reflection over the x-axis is applied. This changes the direction of the graph, flipping it upside down. The period and location of the vertical asymptotes do not change, but every point on the graph moves to the opposite side of the x-axis, reversing the slopes of the tangent waves.