Problem 41 - Algebra 2

Question

Graph \(y=5+\tan \left(\theta+\frac{\pi}{4}\right) \cdot\) Describe the transformation to the parent graph \(y=\tan \theta .\)

Step-by-Step Solution

Answer

The graph of \(y=\tan \theta\) is shifted left by \(\frac{\pi}{4}\) and up by 5 units, with vertical asymptotes at \(\theta=\frac{\pi}{4}+n\pi\).

Step 1: Identify the Parent Function

The parent graph for this problem is given as the function \(y = \tan \theta\). This graph has vertical asymptotes at \(\theta = \frac{\pi}{2} + n\pi\) where \(n\) is an integer, and it passes through the origin \((0,0)\). The period of the tangent function is \(\pi\).

Step 2: Analyze Horizontal Shift

In the given function \(y = 5 + \tan\left(\theta + \frac{\pi}{4}\right)\), the expression \(\theta + \frac{\pi}{4}\) inside the tangent function indicates a horizontal shift. The graph of \(y = \tan(\theta)\) is shifted to the left by \(\frac{\pi}{4}\) units, since adding a positive constant results in a leftward shift.

Step 3: Analyze Vertical Shift

The +5 outside the tangent function indicates a vertical shift. The graph of the function is shifted up by 5 units. Each point on the original \(y = \tan(\theta)\) graph moves upward by 5 units, including its horizontal asymptotes, which do not appear visibly as lines.

Step 4: Determine the New Asymptotes

Since the graph is shifted horizontally, the new vertical asymptotes will be at \(\theta = \frac{\pi}{2} - \frac{\pi}{4} + n\pi\). Simplifying, this results in vertical asymptotes at \(\theta = \frac{\pi}{4} + n\pi\), where \(n\) is an integer.

Step 5: Graph the Function

To graph, start by plotting the new vertical asymptotes and center the tangent curve between them. The curve will pass through the point \((-\frac{\pi}{4}, 5)\) since originally \((0, 0)\), and the curve follows its typical "S" shape between each pair of asymptotes, but shifted up by 5 units.

Key Concepts

Graph Transformations

Graph transformations involve altering the appearance of a graph by altering its position, size, or shape. There are different types of graph transformations that can be applied to functions:

• Translation: Shifts the graph either horizontally or vertically or both.
• Reflection: Flips the graph over a certain axis.
• Stretching and Shrinking: Alters the size of the graph in either the vertical or horizontal direction or both.

Understanding how each of these transformations affects the original graph of a function helps in sketching the transformed graph without plotting numerous points. In our case, the function undergoes both horizontal and vertical translations. It's important to visualize these shifts in connecting the original behavior of the tangent function to its transformed form. This involves identifying changes to key features like asymptotes and intercepts.

Tangent Function

The tangent function, denoted as \(y = \tan \theta\), is a fundamental trigonometric function. It has several characteristic traits:

• Periodicity: The tangent function is periodic, meaning it repeats its shape at regular intervals. Specifically, it has a period of \(\pi\).
• Vertical Asymptotes: The graph has vertical asymptotes where the function is undefined at \(\theta = \frac{\pi}{2} + n\pi\), where \(n\) is an integer.
• Symmetry: It is an odd function, so it exhibits symmetry about the origin.
• Behavior: The function increases from negative infinity to positive infinity within each interval between asymptotes, forming a characteristic "S" shape.

When graphing the tangent function, focusing on its asymptotes and behavior within one period provides a clear picture of its structure, making it easier to apply transformations.

Horizontal Shift

A horizontal shift in a graph occurs when the entire graph of a function is moved left or right without changing its shape. This alteration happens due to the addition or subtraction of a constant within the function's argument. Here is how it affects the tangent function:

• Left Shift: Adding a constant \(c\) to \(\theta\) (e.g., \(\theta + c\)) results in shifting the graph to the left by \(c\) units.
• Right Shift: Subtracting a constant \(c\) from \(\theta\) (e.g., \(\theta - c\)) causes a shift to the right by \(c\) units.

In the function \(y = 5 + \tan\left(\theta + \frac{\pi}{4}\right)\), the term \(\theta + \frac{\pi}{4}\) shifts the graph to the left by \(\frac{\pi}{4}\) units. Consequently, the original positions of the graph's features, such as its vertical asymptotes and zeros, move accordingly. Understanding this shift ensures that key characteristics like the "S" shape and periodicity maintain their functional integrity.

Vertical Shift

A vertical shift occurs when every point of a graph is moved up or down by adding or subtracting a constant to the function's output (i.e., outside of the function's argument). Here's how it specifically affects a function:

• Upward Shift: Adding a constant \(k\) to a function \(f(\theta)\) results in a shift upwards by \(k\) units, such that the new function is \(f(\theta) + k\).
• Downward Shift: Subtracting a constant \(k\) shifts the graph downwards by \(k\) units.

In the transformed tangent function \(y = 5 + \tan\left(\theta + \frac{\pi}{4}\right)\), the \(+5\) translates the entire graph upwards by 5 units. Consequently, each point, including those on asymptotes and critical points like zeros, rises by 5. Such alterations do not influence the shape or period, ensuring the original form of the graph is preserved while its position is modified. Recognizing these adjusts assists in accurately sketching the graph and analyzing any vertical transformations.