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Chapter 0: Problem 52

The Technology Connection heading indicates exercises designed to provide practice using a graphing calculator. Graph. \(y=\sqrt{23-7 x}\)

Short Answer

Expert verified
The domain is \((-infty, \frac{23}{7}]\) and the range is \(y \geq 0\). The graph is a downward-opening arc.

Step by step solution

01

Understand the Function

The function we need to graph is given by \( y=\sqrt{23-7x} \). This represents a square root function, which is defined for the values of \(x\) which make the expression inside the square root non-negative, i.e., \(23-7x \geq 0\).
02

Determine the Domain

To find the domain, solve the inequality \(23 - 7x \geq 0\). This simplifies to \(7x \leq 23\) and further to \(x \leq \frac{23}{7}\). Thus, the domain of the function is \(x \in (-\infty, \frac{23}{7}]\).
03

Find the Range

The range of a square root function is always non-negative (since square roots are non-negative for real numbers). Therefore, the range is \(y \geq 0\).
04

Plot Key Points

Determine a few key points to help with the graphing process. Start by finding the value of the function at the endpoints: - If \(x = 0\), then \(y = \sqrt{23} \approx 4.79\)- If \(x = \frac{23}{7}\), then \(y = \sqrt{0} = 0\).Optional: Choose additional points such as \(x = 1\) and \(x = 2\) and calculate \(y\).
05

Graph the Function

Using a graphing calculator, plot the key points and sketch the graph. The curve should start at \(y = 0\) when \(x = \frac{23}{7}\) and approach \(y \approx 4.79\) when \(x = 0\). The graph is an arc that opens downward.

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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 typically involves the mathematical operation of finding the square root of a given expression. In this case, the expression under the square root is given by \( 23 - 7x \). Square root functions have distinct characteristics:
  • They start at a specific point called the "endpoint," which can be found by setting the expression inside the square root to zero.
  • These functions usually have one side that stretches endlessly while the other side terminates at an endpoint.
  • The graph of a square root function always represents a curve, not a straight line, which begins at the endpoint and extends in one direction.
The function \( y = \sqrt{23 - 7x} \) is a variation of the square root function, and its graph will comprise a downward-opening arc as described. Studying the standard form of the square root function \( y = \sqrt{x} \) helps grasp how transformations affect the graph, such as horizontal shifts depending on the values inside the radical.
Domain and Range
Understanding the domain and range of a function is crucial when graphing it, especially for functions involving square roots.
  • **Domain**: The domain consists of all possible values of \(x\) for which the function is defined and real. For our square root function \( y = \sqrt{23 - 7x} \), this means the expression inside the square root must be non-negative: \(23 - 7x \geq 0\). Solving this inequality, we find \(x \leq \frac{23}{7}\), so the domain is \(x \in (-\infty, \frac{23}{7}]\).
  • **Range**: The range includes all possible \(y\)-values the function can take. Since square roots of non-negative numbers are zero or positive, \(y \geq 0\) is our range. Thus, the range is \(y \geq 0\).
Understanding these concepts aids in plotting the function correctly on a graph and ensures all parts of the function are covered within the graphing boundaries.
Graphing Techniques
Graphing a function, such as \( y = \sqrt{23 - 7x} \), involves specific techniques that help accurately represent the behavior of the function. Here are steps to ensure precision when using a graphing calculator:
  • Determine and plot the domain endpoint. For this problem, it occurs at \(x = \frac{23}{7}\) and \(y = 0\). Begin the graph at this endpoint.
  • Select additional key points within the domain for greater accuracy. Calculating \(y\) values for intermediate \(x\) values, like \(x = 0\), yields \(y \approx 4.79\).
  • Use the graphing calculator to plot these points and draw a smooth, curving arc that connects them. The curve should show a decrease in \(y\) as \(x\) increases.
  • Ensure your graph reflects the expected behavior as determined by domain and range constraints, such as starting at the endpoint and continuing downwards.
Utilizing these techniques aids in producing an accurate visual representation of the square root function, providing clarity about its behavior across the domain.

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Most popular questions from this chapter

The annual interest rate \(r,\) when compounded more than once a year, results in a slightly higher yearly interest rate; this is called the annual (or effective) yield and denoted as Y. For example, \$1000 deposited at 5\%, compounded monthly for 1 yr \((12\) months \(),\) will have a value of \(A=1000\left(1+\frac{0.05}{12}\right)^{12}=\$ 1051.16 .\) The interest earned is \(\$ 51.16 / \$ 1000,\) or \(0.05116,\) which is \(5.116 \%\) of the original deposit. Thus, we say this account has a yield of \(Y=0.05116,\) or \(5.116 \% .\) The formula for annual yield depends on the annual interest rate \(r\) and the compounding frequency \(n:\) \(Y=\left(1+\frac{r}{n}\right)^{n}-1.\) For Exercises 41-48, find the annual yield as a percentage, to two decimal places, given the annual interest rate and the compounding frequency. Annual interest rate of \(3.75 \%,\) compounded weekly

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