/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 20 Graph using a graphing calculato... [FREE SOLUTION] | 91影视

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Chapter 15: Problem 20

Graph using a graphing calculator. \(y=\sqrt{3-x}\)

Short Answer

Expert verified
Graph the function \( y = \sqrt{3 - x} \) over the domain \( x \leq 3 \). Use key points like (0, \( \sqrt{3} \)) and (3, 0).

Step by step solution

01

Understand the Function

Identify the function to be graphed: \[ y = \sqrt{3 - x} \]This is a square root function, which means it will only have real values where the expression inside the square root is non-negative.
02

Determine the Domain

Find where the expression inside the square root is non-negative:\[ 3 - x \geq 0 \]Solve for \(x\): \[ x \leq 3 \]The domain is \( x \in (-\infty, 3] \).
03

Find Key Points

Identify key points on the graph by plugging in values of \( x \) into \( y = \sqrt{3 - x} \):1. \( x = 3 \rightarrow y = \sqrt{3 - 3} = 0 \)2. \( x = 0 \rightarrow y = \sqrt{3 - 0} = \sqrt{3} \)
04

Draw the Graph Using Calculator

Enter the function \( y = \sqrt{3 - x} \) into the graphing calculator. Use the graphing calculator to plot the function over the domain \( x \leq 3 \). The graph will start at the point (3, 0) and descend as \( x \) decreases.

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Key Concepts

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

Square Root Function
A square root function includes a square root symbol, typically written as \(y = \sqrt{x}\). Understanding this type of function can help in many algebraic and graphical problem areas. A square root function graph often starts at a point and spreads outward, because the values of y only make sense for non-negative inputs within the root.
For our exercise function, \[y = \sqrt{3 - x}\], the expression inside the square root (3-x) needs to be non-negative to provide real-valued outputs. This shapes how the graph looks.
Domain of a Function
The domain of a function is every possible value of the independent variable (usually x) for which the function is defined.
For example, in our square root function \[y = \sqrt{3 - x}\], the domain is determined by the expression inside the square root needing to be non-negative. Solve \[3 - x \geq 0\] to find the domain:
\[- x \geq -3 \rightarrow x \leq 3\]
So, the domain of our function is \(( - \infty, 3 ]\), meaning x can be any value less than or equal to 3. This information helps us correctly graph the function.
Graphing Calculator
A graphing calculator is a powerful tool for visualizing functions, especially when they involve square roots or other complex expressions.
To graph our function \[y = \sqrt{3 - x}\] using the calculator, follow these steps:
  • Enter the function into the calculator's graphing mode.
  • Ensure the calculator's settings align with the domain \(( - \infty, 3 ] \). This means adjusting the window settings to include x-values up to 3 (since beyond this point y-values become imaginary).
  • Plot the function and look for the curve that starts at the point (3,0) and moves downwards as x decreases.

This visualization helps to get a more intuitive understanding of the function's behavior.

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