Problem 18 - Algebra 2

Question

Identify the type of function represented by each equation. Then graph the equation. \(y=\sqrt{9 x}\)

Step-by-Step Solution

Answer

It's a square root function. Graph starts at the origin and curves upward.

Step 1: Identify the Type of Function

The given equation is \( y = \sqrt{9x} \). To understand the function type, notice that there is a square root involved, which means it's a square root function. Square root functions are a type of radical function.

Step 2: Simplify the Equation

First, simplify the expression under the square root. The equation is \( y = \sqrt{9x} \). The square root of 9 can be simplified, so the equation becomes \( y = 3\sqrt{x} \). This is the simplified form of the function.

Step 3: Determine the Domain of the Function

Square root functions have a specific domain as you cannot take the square root of a negative number in the real number system. Therefore, the domain for \( y = 3\sqrt{x} \) is \( x \geq 0 \), meaning \( x \) must be zero or a positive number.

Step 4: Identify Key Points for Plotting the Graph

Let's find key points that can be used to sketch the graph. Calculate \( y = 3\sqrt{x} \) for some values of \( x \).- For \( x = 0 \), \( y = 3\sqrt{0} = 0 \)- For \( x = 1 \), \( y = 3\sqrt{1} = 3 \)- For \( x = 4 \), \( y = 3\sqrt{4} = 6 \)These points, (0,0), (1,3), and (4,6), will help in sketching the curve.

Step 5: Sketch the Graph

Using the points from Step 4, plot them on a coordinate plane. The graph of \( y = 3\sqrt{x} \) starts at the origin (0,0), goes through the point (1,3), and continues through (4,6), creating a curve that rises to the right. The graph's shape is a typical square root curve, which begins from the origin and gradually increases.

Key Concepts

Radical Function

A radical function is a function that contains a radical expression with the independent variable. The most common type of radical function includes square roots and cube roots. In the exercise you have been working on, you are given the equation \( y = \sqrt{9x} \), indicating that it is a square root function. This type of function takes the form of \( y = \sqrt{a} \), where \( a \) can be any number or expression involving a variable.Square root functions belong to a larger family of radical functions. These functions exhibit unique properties, such as restricted domains because of the nature of square roots, which cannot accept negative inputs for real numbers. In order to handle these functions correctly, it's key to understand how to simplify them. For instance, \( y = \sqrt{9x} \) can be rewritten as \( y = 3\sqrt{x} \), showcasing that simplifying inside the radical first can make the function easier to analyze and graph.

Function Graphing

Graphing functions is an essential skill in mathematics that visualizes how a function behaves over a range of values. When graphing a square root function like \( y = 3\sqrt{x} \), it's helpful to calculate specific points that will guide your sketch.

Determining Key Points

To decide which points to plot, substitute various non-negative values for \( x \) into the equation:

• For \( x = 0 \), \( y = 3\sqrt{0} = 0 \).
• For \( x = 1 \), \( y = 3\sqrt{1} = 3 \).
• For \( x = 4 \), \( y = 3\sqrt{4} = 6 \).

Use these points (0,0), (1,3), and (4,6) to plot on a coordinate plane.Once plotted, connect the points smoothly. The curve will begin at the origin (0, 0), indicating where the function "starts," and will move through these points, showing how the function grows. The graph of a square root function tends to rise more gradually compared to linear or exponential functions.

Domain of a Function

The domain of a function refers to all the possible input values (usually \( x \)) that will result in valid outputs in the context of the function. For the square root function \( y = 3\sqrt{x} \), determining the domain is crucial because square roots are undefined in the real number system for negative numbers.

Understanding the Domain

For \( y = 3\sqrt{x} \):

• Identify the limitation that \( x \) must satisfy: \( x \geq 0 \).
• This means the function's domain is all non-negative real numbers — essentially, zero and all positive values.

From this understanding, you can identify that input values lower than zero would result in imaginary numbers. Therefore, they are not included in the real-world application of the function. Always ensure that when working with radicals, you properly determine and interpret their domain to avoid undefined parts of the graph.