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Chapter 1: Problem 10

Starting with the graph of \(y=\sqrt{x},\) the graph of \(y=1 / x,\) and the graph of \(y=\sqrt{1-x^{2}}\) (the upper unit semicircle), sketch the graph of each of the following functions: $$f(x)=4+2 \sqrt{1-(x-5)^{2} / 9}$$

Short Answer

Expert verified
The graph of \(f(x)\) is a semicircle centered at (5, 4) with horizontal radius 3 (from 2 to 8) and vertical range [4, 6].

Step by step solution

01

Understand the base graph

The base graph is given as \(y = \sqrt{1 - x^2}\), which is the upper semicircle of the unit circle. This graph includes all points \((x, y)\) where \(-1 \leq x \leq 1\) and \(0 \leq y \leq 1\).
02

Applying horizontal shift

The function \( f(x) = 4 + 2 \sqrt{1 - (x-5)^2 / 9} \) involves a shift inside the square root. This corresponds to a horizontal shift. The term \((x - 5)^2\) indicates a shift to the right by 5 units. So, the center of the semicircle is now at \(x = 5\).
03

Adjust for the horizontal stretch/compression

The given function includes a factor of \( \frac{1}{9} \) inside the square root, implying a horizontal stretch. A factor of \(1/k^2\) causes the function to stretch by a factor of \(k\). Therefore, our function has been horizontally stretched by a factor of 3, since \( \frac{1}{9} = \left(\frac{1}{3}\right)^2 \). This means the semicircle now stretches from \(x = 2\) to \(x = 8\).
04

Understand vertical stretch and shift

The function \(2 \sqrt{1 - (x-5)^2 / 9} \) applies a vertical stretch by a factor of 2. Therefore, the maximum height of the semicircle is now 2 instead of 1. Additionally, the entire graph is shifted up by 4 units, as indicated by the \(+4\) outside the radical. Thus, the range of \(f(x)\) is \([4, 6]\).
05

Sketch the graph

Incorporating all the transformations: the graph of \(f(x) = 4 + 2 \sqrt{1 - (x-5)^2 / 9}\) is a semicircle centered at \((5, 4)\) with a radius of 3 horizontally and a radius stretched to 2 vertically. It ranges from \(x = 2\) to \( x = 8 \) on the x-axis and from \(y = 4\) to \(y = 6\) on the y-axis.

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Key Concepts

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

Horizontal Shift
A horizontal shift in a graph involves moving every point of a function a certain distance left or right on the x-axis. It alters the position of the graph without changing its shape. Consider the expression inside a function, such as \((x - 5)\), which indicates a horizontal shift.
  • If the expression is \((x - c)\), the graph moves to the right by \(c\) units.
  • Conversely, \((x + c)\) shifts the graph left by \(c\) units.
In the example given, \((x - 5)^2\) shows a shift to the right by 5 units. This modification relocates the center of the semicircle to \(x = 5\) on the x-axis, repositioning the base graph to fit this new center.
Horizontal Stretch
Horizontal stretching alters the width of a graph. It makes the graph either broader or narrower without affecting its symmetry or peak height. This is determined by multiplying the x-coordinates of the graph by a certain factor.

In the function\(\sqrt{1 - (x-5)^2/9}\), the fraction \(\frac{1}{9}\) implies a horizontal stretch. Specifically:
  • A term of \(\frac{1}{k^2}\) stretches the graph horizontally by a factor of \(k\).
  • Thus,\(\frac{1}{9}\) indicates a stretch by a factor of 3 (since \(\left(\frac{1}{3}\right)^2 = \frac{1}{9}\)).
This transformation means the semicircle, initially spanning \(-1\) to \(1\), now stretches from \(x = 2\) to \(x = 8\)on the x-axis, making the original unit semicircle three times as wide.
Vertical Stretch
Vertical stretching involves scaling a graph along the y-axis, affecting its height without altering its width. The transformation alters the y-values by multiplying them by a given factor.

In the equation \(2\sqrt{1 - (x-5)^2/9}\), the factor outside the square root, which is \(2\), indicates a vertical stretch.
  • This multiplier affects the peak height of the base graph.
  • In this case, the height reaches 2 rather than the original height of 1.
Consequently, the modified semicircle maintains its half-circle shape but extends vertically, doubling in height.
Vertical Shift
A vertical shift moves a graph up or down on the y-axis without altering its shape. This adjustment depends on adding or subtracting a constant to the main equation, which alters all y-values uniformly.

In the function \(4 + 2\sqrt{1 - (x-5)^2/9}\), the \(+ 4\) indicates a vertical shift.
  • This positive addition shifts the entire graph vertically upward by 4 units.
  • Thus, every point in the semicircle moves from its original position at y = 0 to 4, and the peak moves from 2 to 6.
This transformation results in adjusting the graph's position along the y-axis, affecting the range from \([0, 2]\) to \([4, 6]\).

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Most popular questions from this chapter

Starting with the graph of \(y=\sqrt{x},\) the graph of \(y=1 / x,\) and the graph of \(y=\sqrt{1-x^{2}}\) (the upper unit semicircle), sketch the graph of each of the following functions: $$f(x)=2+\sqrt{1-(x-1)^{2}}$$

Determine whether the lines \(3 x+6 y=7\) and \(2 x+4 y=5\) are parallel. \(\Rightarrow\)

Starting with the graph of \(y=\sqrt{x},\) the graph of \(y=1 / x,\) and the graph of \(y=\sqrt{1-x^{2}}\) (the upper unit semicircle), sketch the graph of each of the following functions: $$y=f(x)=x /(1-x)$$

Graph the circle \(x^{2}+y^{2}+10 y=0\).

Change the equation \(3=2 y\) to the form \(y=m x+b\), graph the line, and find the \(y\) -intercept and \(x\) -intercept. \(\Rightarrow\)
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