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Chapter 2: Problem 31

Exer. \(23-34\) : Sketch the graph of the circle or semicircle. $$y=-\sqrt{16-x^{2}}$$

Short Answer

Expert verified
The graph is the lower half of a circle with radius 4 centered at (0,0).

Step by step solution

01

Recognize the Equation Form

The given equation is in the form of \( y = -\sqrt{16-x^2} \). This represents the lower half of a circle centered at the origin with a radius of 4, because \( y = \pm \sqrt{16-x^2} \) is the equation for a circle with radius 4 centered at (0,0) . The negative sign indicates the lower semicircle.
02

Identify the Radius and Center

The equation \( 16 - x^2 \) inside the square root implies that the circle is centered at \((0,0)\). The expression \( x^2 + y^2 = 16 \) is extracted from this, which is the equation of a circle with radius 4 centered at the origin.
03

Determine the Domain and Range

The domain of \( y = -\sqrt{16-x^2} \) is the set of \( x \) values for which the expression inside the square root is non-negative: \( -4 \leq x \leq 4 \). The range for \( y \) is from \(-4\) to \(0\) because \( y \) represents the negative values of the square root.
04

Plot Key Points

Determine key points on the graph. When \( x = 0 \), \( y = -4 \). At \( x = \pm 4 \), \( y = 0 \). These points (0,-4), (4,0), and (-4,0) help shape the semicircle graph.
05

Sketch the Graph

Using the center at (0,0) and radius 4, sketch the lower semicircle. Start from (-4,0) to (4,0) along the x-axis and draw the arc down to meet at (0,-4), completing the lower half of the circle.

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Key Concepts

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

Equation of a Circle
The equation of a circle is one of the fundamental concepts in geometry. A typical equation for a circle with a center at \( (h, k) \) and radius \( r \) is given by:
  • \( (x - h)^2 + (y - k)^2 = r^2 \)
This indicates that every point \( (x, y) \) on the circle is at a distance \( r \) from the center. In the exercise given, we have the equation \( y = -\sqrt{16-x^2} \), which corresponds to the semicircle derived from the circle equation \( x^2 + y^2 = 16 \). The appearance of \( y = \pm \sqrt{16-x^2} \) denotes both the upper and lower halves of a circle. However, in our problem, the negative sign specifies the inclusion of only the lower half.
Domain and Range
Understanding the domain and range is crucial when graphing functions. The domain of a function is the complete set of possible values of the independent variable, which is \( x \) in this context. For the equation \( y = -\sqrt{16-x^2} \), the domain is determined by the values for which the expression under the square root is non-negative. \( 16-x^2 \) must be greater than or equal to zero, leading us to:
  • \( -4 \leq x \leq 4 \)
The range is the complete set of possible values of the dependent variable, \( y \). Since the equation represents the lower half of a circle, \( y \) takes on values from the maximum negative, which is -
  • \( y: -4 \leq y \leq 0 \)
Center and Radius
A circle's center and radius give insight into its position and size. From the circle equation \( x^2 + y^2 = 16 \), we infer the circle is centered at the origin \( (0,0) \). This is because there are no additional constants added or subtracted from \( x \) or \( y \), which means both \( h \) and \( k \) are zero. The expression equates to the circle's bottom quadrant due to its negative square root. The radius is derived from the constant on the right-hand side of the circle equation, which is 16:
  • Taking the square root yields a radius of \( 4 \).
Key Points on a Graph
Identifying key points is a stepping stone to accurately sketching graphs. For the semicircle \( y = -\sqrt{16-x^2} \), let's find the values that define its shape:
  • When \( x = 0 \), \( y = -4 \). This is the lowest point of the semicircle.
  • When \( x = 4 \), \( y = 0 \), marking the rightmost endpoint.
  • When \( x = -4 \), \( y = 0 \), marking the leftmost endpoint.
These points \( (0, -4) \), \( (4, 0) \), and \( (-4, 0) \) guide the semicircle's arc along the x-axis. Sketch from the left at \( (-4, 0) \) to the right at \( (4, 0) \), passing through the lowest point at \( (0, -4) \). Visualizing these points helps plot an accurate graceful curve of the semicircle.

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Most popular questions from this chapter

A proposed energy tax \(T\) on gasoline, which would affect the cost of driving a vehicle, is to be computed by multiplying the number \(x\) of gallons of gasoline that you buy by \(125,000\) (the number of BTUs per gallon of gasoline) and then multiplying the total BTUs by the tax \(-34.2\) cents per million BTUs. Find a linear function for \(T\) in terms of \(x .\)

(a) Sketch the graph of \(f\) on the given interval \([a, b] .\) (b) Estimate the range of \(f\) on \([a, b] .\) (c) Estimate the intervals on which \(f\) is increasing or is decreasing. $$\begin{aligned} &f(x)=x^{5}-3 x^{2}+1\\\ &[-0.7,1.4] \end{aligned}$$

A particular graphing calculator screen is 95 pixels wide and 63 pixels high. Find the total number of pixels in the screen. If a function is graphed in dot mode, determine the maximum number of pixels that would typically be darkened on the calculator screen to show the function.

One section of a suspension bridge has its weight uniformly distributed between twin towers that are 400 feet apart and rise 90 feet above the horizontal roadway (see the figure). A cable strung between the tops of the towers has the shape of a parabola, and its center point is 10 feet above the roadway. Suppose coordinate axes are introduced, as shown in the figure. (figure can't copy) (a) Find an equation for the parabola. (b) Nine equally spaced vertical cables are used to support the bridge (see the figure). Find the total length of these supports.

(a) Sketch the graph of \(f\) on the given interval \([a, b] .\) (b) Estimate the range of \(f\) on \([a, b] .\) (c) Estimate the intervals on which \(f\) is increasing or is decreasing. $$f(x)=x^{4}-0.4 x^{3}-0.8 x^{2}+0.2 x+0.1 ; \quad[-1,1]$$
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