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Chapter 2: Problem 17

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ h(x)=\sqrt{-x+2} $$

Short Answer

Expert verified
The graph of the function \(h(x)=\sqrt{-x+2}\) is obtained by transforming the base graph \(f(x)=\sqrt{x}\) by reflecting it in the y-axis and shifting it 2 units to the right.

Step by step solution

01

Graph the base function

Start by sketching the graph of the function \(f(x)=\sqrt{x}\). This is a curve that starts at point (0,0) and slowly rises to the right.
02

Perform a horizontal reflection

The negative sign in front of 'x' in the function \(h(x) = \sqrt{-x+2}\) causes a reflection across the y-axis. This means the graph is 'flipped' horizontally, flipping the base graph of \(f(x)=\sqrt{x}\) over the y-axis so that it starts on the y-axis (0,0), and falls to the left.
03

Perform a horizontal shift

The '+2' inside the square root in the function \(h(x)=\sqrt{-x+2}\) indicates a horizontal shift. It shifts the graph 2 units to the right. So, every point on the curve of \(f(x)=\sqrt{-x}\) is moved 2 units to the right to get the final graph of \(h(x)=\sqrt{-x+2}\).
04

Verify

Check that the graph of \(h(x)=\sqrt{-x+2}\) is a valid transformation from the graph of \(f(x)=\sqrt{x}\). The graph should start from the point (2,0) and go downwards towards the left as \(x\) decreases from 2.

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Key Concepts

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

Graph Transformations
Transforming graphs is a method to modify the appearance of a graph using various techniques. When dealing with the square root function, like the base function \(f(x) = \sqrt{x}\), transformations can change where the graph is positioned on a coordinate plane, its shape, or its orientation. Such transformations allow us to create new functions from the base function.
  • Start with the basic graph of \(f(x) = \sqrt{x}\), which begins at the origin (0,0) and curves upward to the right.
  • By applying transformations, such as shifting and reflecting, this graph can be moved around the plane or flipped to create different versions of the square root function.
  • Such transformations include horizontal shifts, vertical shifts, reflections over the axes, and compressions/stretching both horizontally and vertically.
Understanding these modifications is crucial as they help us create the graph of any altered square root equation, like \(h(x) = \sqrt{-x+2}\), starting from the simpler base form of \(\sqrt{x}\).
Horizontal Reflection
A horizontal reflection involves flipping a graph across a vertical line, typically the y-axis. In mathematical terms, when you see a negative sign in front of the x-variable within a function, this reflects the graph horizontally. For example, in the function \(h(x) = \sqrt{-x+2}\), the presence of \(-x\) indicates such a reflection.
  • This transformation flips the graph of the base square root function \(f(x) = \sqrt{x}\) over the y-axis.
  • Originally, \(\sqrt{x}\) rises to the right; however, after the reflection to create \(\sqrt{-x}\), the graph now falls towards the left.
  • After a horizontal reflection, the starting point remains the same on the y-axis as long as no other transformations affect the function.
This flipping is essential when plotting functions to immediately identify and adjust the direction a graph heads once reflected.
Horizontal Shift
A horizontal shift moves a graph left or right on the coordinate plane. Inside a function, if you have \(+2\) as in \(\sqrt{-x+2}\), it results in a shift. However, the understanding is opposite to how it might appear; it moves the graph 2 units to the right rather than the left.
  • A positive value like \(+2\) causes a rightward shift, indicating each point of the graph moves 2 units to the right.
  • After performing a horizontal reflection, applying a horizontal shift ensures that you place the graph correctly around the x-axis based on the given function.
  • For the function \(h(x) = \sqrt{-x+2}\), this means starting the new reflected graph at (2,0) instead of (0,0).
By understanding horizontal shifts, we can accurately graph functions such as \(h(x) = \sqrt{-x+2}\), ensuring an accurate interpretation from its base form \(f(x) = \sqrt{x}\). This shift is typically one of the final transformation steps to finalize the function's graph on the plane.

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Most popular questions from this chapter

Begin by graphing the absolute value function, \(f(x)=|x| .\) Then use transformations of this graph to graph the given function. $$ h(x)=|x+4|-2 $$

Give an example of an equation that does not define \(y\) as a function of \(x\) but that does define \(x\) as a function of \(y .\)

The graph shows the amount of money, in billions of dollars, of new student loans from 1993 through 2000 . (graph can't copy) The data shown can be modeled by the function \(f(x)=6.75 \sqrt{x}+12,\) where \(f(x)\) is the amount, in billion of dollars, of new student loans \(x\) years after 1993 . a. Describe how the graph of \(f\) can be obtained using transformations of the square root function \(f(x)=\sqrt{x} .\) Then sketch the graph of \(f\) over the interval \(0 \leq x \leq 9 .\) If applicable, use a graphing utility to verify your hand-drawn graph. b. According to the model, how much was loaned in \(2000 ?\) Round to the nearest tenth of a billion. How well does the model describe the actual data? c. Use the model to find the average rate of change, in billions of dollars per year, between 1993 and 1995 Round to the nearest tenth. d. Use the model to find the average rate of change, in billions of dollars per year, between 1998 and 2000 . Round to the nearest tenth. How does this compare with you answer in part (c)? How is this difference shown by your graph? e. Rewrite the function so that it represents the amount, \(f(x),\) in billions of dollars, of new student loans \(x\) years after 1995

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