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Chapter 12: Problem 29

Graph each generalized square root function. $$ \frac{y}{3}=\sqrt{1+\frac{x^{2}}{9}} $$

Short Answer

Expert verified
Graph the function \( y = 3 \sqrt{1 + \frac{x^2}{9}} \) with key points such as \( (0, 3) \) and symmetry around the y-axis.

Step by step solution

01

- Isolate the Square Root

First, multiply both sides of the equation by 3 to isolate the square root on one side. This yields: \( y = 3\text{sqrt}(1 + \frac{x^2}{9}) \).
02

- Simplify the Equation

Notice that the square root inside can be simplified further: \( y = 3 \sqrt{1 + \left(\frac{x}{3}\right)^2} \).
03

- Define the Domain

The square root function must always have a non-negative argument. Since \( 1 + \left( \frac{x}{3} \right)^2 \) is always positive for all real \( x \), the domain is \( (-\infty, \infty) \).
04

- Identify Key Points

To understand the shape of the graph, calculate a few key points. For example: - When \( x = 0 \), \( y = 3 \sqrt{1} = 3 \). - When \( x = 3 \), \( y = 3 \sqrt{1 + 1} = 3 \sqrt{2} \). - When \( x = -3 \), \( y = 3 \sqrt{1 + 1} = 3 \sqrt{2} \).Plot these on the graph.
05

- Sketch the Graph

Now plot the points calculated and sketch the curve. Notice the curve is symmetric with respect to the y-axis and increases as \( x \) gets farther from zero in either direction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating Square Roots
To start graphing square root functions, we first need to isolate the square root. Given the equation: \(\frac{y}{3} = \sqrt{1 + \frac{x^2}{9}}\), we multiply both sides by 3 to get the square root on one side: \(y = 3\sqrt{1 + \frac{x^2}{9}}\). This step is important because it simplifies the equation and makes it easier to handle. Isolating the square root allows us to focus on the core part of the function and understand its behavior.
Defining Domain
Next, let's define the domain of the function. The domain of a function is the set of all possible input values (x-values) that will produce a valid output (y-value). For a square root function, the expression under the square root must be non-negative. Here, we have: \(\sqrt{1 + \left( \frac{x}{3} \right)^2}\). Since \(1 + \left( \frac{x}{3} \right)^2\) is always greater than or equal to zero for all real numbers, the domain of the function is: \( (-\infty, \infty) \). This means that the function is defined for all real numbers.
Key Points Calculation
To understand the graph's shape and behavior, it's helpful to calculate some key points. These points give us critical information about the function's values at specific x-coordinates. Here, we calculate a few key points: - When \( x = 0 \), \( y = 3 \sqrt{1} = 3 \). - When \( x = 3 \), \( y = 3 \sqrt{1 + 1} = 3 \sqrt{2} \). - When \( x = -3 \), \( y = 3 \sqrt{1 + 1} = 3 \sqrt{2} \). By plotting these points on the graph, we can start to see the function's shape and understand how it behaves at these key positions.
Symmetric Graphs
Finally, we notice that the graph of this function is symmetric. A graph is symmetric with respect to the y-axis if for every point \((x, y)\) on the graph, there's also a point \((-x, y)\). In this case, if we examine the equation \( y = 3 \sqrt{1 + \left( \frac{x}{3} \right)^2} \), we see that replacing \(x\) with \(-x\) does not change the output: \( y = 3 \sqrt{1 + \left( \frac{-x}{3} \right)^2} = 3 \sqrt{1 + \left( \frac{x}{3} \right)^2} \). This demonstrates that the graph is symmetric with respect to the y-axis. This symmetry helps us sketch the graph more easily since you only need to graph one side and then reflect it over the y-axis.

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