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Chapter 2: Problem 28

Exer. \(23-34\) : Sketch the graph of the circle or semicircle. $$x^{2}+(y-2)^{2}=25$$

Short Answer

Expert verified
The graph is a circle with center (0, 2) and radius 5.

Step by step solution

01

Identify the Equation Type

The given equation is \(x^2 + (y-2)^2 = 25\). This equation represents a circle in the standard form \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center of the circle, and \(r\) is the radius.
02

Determine the Center of the Circle

The standard equation for a circle is \((x-h)^2 + (y-k)^2 = r^2\). Comparing with \(x^2 + (y-2)^2 = 25\), we can identify \(h=0\) and \(k=2\). Thus, the center of the circle is \((0, 2)\).
03

Calculate the Radius

The right side of the equation \(x^2 + (y-2)^2 = 25\) represents \(r^2\). So, \(r^2 = 25\). Taking the square root, we find the radius \(r = \sqrt{25} = 5\).
04

Sketch the Graph

To sketch the circle, plot the center at \((0, 2)\) on the coordinate plane. From this center, draw a circle with a radius of 5. This circle will intersect the x-axis at \((-5, 2)\) and \((5, 2)\), and the y-axis at \((0, -3)\) and \((0, 7)\).

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Key Concepts

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

Equation of a Circle
When dealing with the equation of a circle, it is key to recognize its standard form, which is \[(x-h)^2 + (y-k)^2 = r^2\]Here, \(h,k\) denote the coordinates of the circle's center, and \(r\) represents the radius.
Understanding this format allows you to quickly determine both the position and the size of the circle on a graph.
In the example given, the equation \(x^2 + (y-2)^2 = 25\) naturally fits this standard form.
  • The \(x\) part, \(x-h\), lacks an \(h\), implying the \(h\) value is 0.
  • The \(y\) part, \(y-2\), gives us the \(k\) value of 2.
  • The number 25 on the right represents \(r^2\), so \(r = \sqrt{25} = 5\).
This information makes graphing and understanding circle placements in coordinate geometry straightforward and efficient.
Graphing Techniques
Graphing a circle involves a few clear and structured steps. First, identify the circle's center, \(h,k\), from its equation as discussed.
Next, determine its radius from the \(r^2\) term. Once you know these, you can plot the center on a coordinate grid. Knowing the center \( (0,2) \) and radius \(5\) for this specific circle, the following points make contact on the axes:
  • On the x-axis, at points \(-5, 2\) and \(5, 2\).
  • On the y-axis, at points \((0, -3)\) and \((0, 7)\).
Drawing lines through these boundary points will enable you to sketch an accurate circle. These graphing techniques can serve you well as they apply to any circle-related question.
Coordinate Geometry
In coordinate geometry, a combination of algebra and geometry is used to describe and find properties of geometric shapes through coordinates.
Circles, being special geometric shapes, fit perfectly into this framework, allowing precise graphical representations and analytical solutions.
A circle, defined through its equation, not only provides insight into where it sits on a plane, but also how it relates to other shapes or points in that space. Key coordinates in solving circle problems often include:
  • The circle's center \(h,k\).
  • Any intercepts with the x-axis or y-axis, useful for sketching.
  • Any symmetry or connection with other geometric figures.
Using these points, coordinate geometry transforms visual data into numerical data, enhancing understanding with even more complex shapes and scenarios.

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Most popular questions from this chapter

Algebraic methods were used to find solutions to each of the following equations. Now solve the equation graphically by assigning the expression on the left side to \(Y_{1}\) and the number on the right side to \(\mathbf{Y}_{2}\) and then finding the \(x\) -coordinates of all points of intersection of the two graphs. (a) \(x^{45}=-27\) (b) \(x^{2 / 3}=25\) (c) \(x^{43}=-49\) (d) \(x^{3 / 2}=27\) (e) \(x^{3 / 4}=-8\)

The diagonal \(d\) of a cube is the distance between two opposite vertices. Express \(d\) as a function of the edge \(x\) of the cube. (Hint: First express the diagonal \(y\) of a face as a function of \(x\) )

A spherical balloon is being inflated at a rate of \(\frac{9}{2} \pi \mathrm{ft}^{3} / \mathrm{min}\). Express its radius \(r\) as a function of time \(t\) (in minutes), assuming that \(r=0\) when \(t=0\)

(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. $$f(x)=x^{4}-0.4 x^{3}-0.8 x^{2}+0.2 x+0.1 ; \quad[-1,1]$$

Find the standard equation of a parabola that has a vertical axis and satisfies the given conditions. \(x\) -intercepts \(-3\) and \(5,\) highest point has \(y\) -coordinate 4
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