/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 Use the center and the radius to... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 30

Use the center and the radius to graph each circle. $$ (x-1)^{2}+(y+3)^{2}=16 $$

Short Answer

Expert verified
The center of the circle is (1,-3), and it has a radius of 4. The circle is plotted on the graph accordingly.

Step by step solution

01

Identify the center and the radius

From the equation \((x-1)^2+(y+3)^2=16\), the value of (h,k) can be identified where h=1 and k=-3. This is because the formula is \((x-h)^2+(y-k)^2 = r^2\). The value of r(radius) can be inferred from the equation which is 鈭�16=4.
02

Plotting the center

The center of the circle is at the point (1,-3). This point is plotted on the graph.
03

Draw the circle using the radius

From the center (1,-3), count outwards 4 units in all directions (up, down, right and left) due to radius being 4. Plot these points. Draw the circle through these points to complete the graph.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Center-Radius Form
A widely used method to represent circles in math is the center-radius form. It is a way of expressing the equation of a circle to easily identify its center and radius. It typically looks like \[(x - h)^2 + (y - k)^2 = r^2\] where:
  • \( h \) and \( k \) are the coordinates of the center of the circle.
  • \( r \) represents the radius.
To read it, you just need to match these components with the equation you're dealing with. In our exercise, the equation \[(x-1)^2+(y+3)^2=16\]can be compared with the general form. Here, you can see:- The center is at \((h, k) = (1, -3)\),- The radius \( r \) is \( \sqrt{16} = 4 \). Understanding the center-radius form allows you to quickly graph the circle or make modifications.
Equation of a Circle
An equation of a circle defines all the points that make up the circle's boundary. The general formula, specifically when using the center-radius form, \[(x - h)^2 + (y - k)^2 = r^2\]reveals crucial information about the circle:
  • The left-hand side, \((x - h)^2 + (y - k)^2\), measures how far any point \((x, y)\) is from the center \((h, k)\).
  • When this distance equals \( r^2 \), the point lies on the circle.
For example, consider \[(x - 1)^2 + (y + 3)^2 = 16\]This tells us:
  • The circle has a center at \((1, -3)\).
  • It has a radius of \(4\) {since \(\sqrt{16} = 4\)}.
This mathematical framework allows us to visualize and map circles consistently.
Plotting Coordinates
Plotting coordinates involves locating specific points on a graph, which is critical in graphing circles. For the given equation \[(x-1)^2+(y+3)^2=16\]we start with the center \((1, -3)\) by marking it on the graph. This point acts as the reference for drawing the circle. Once the center is set:
  • From the center, count \(4\) units (the radius) upwards, downwards, rightwards, and leftwards to find boundary points.
  • These points are key: they ensure the circle remains equidistant from the center in all directions.
Finally, connect these boundary points with a smooth curve to complete the circle. This method underscores how coordinate plotting helps visualize mathematical relationships like circles effectively.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Simplify each expression. What are the restrictions on the variable? $$ \frac{3 x}{6 x^{2}-9 x^{5}} $$

Expand each binomial. $$ (p+q)^{6} $$

Draw an ellipse by placing two tacks in a piece of graph paper laid over a piece of cardboard. Place a loop of string around the tacks. With your pencil keeping the string taut, draw around the tacks. Mark the center of your ellipse \((0,0)\) and draw the \(x\) - and \(y\) -axes. a. Where are the vertices and co-vertices of your ellipse? b. Where are the foci? c. Write the equation of your ellipse.

Find the foci for each equation of an ellipse. $$ 25 x^{2}+4 y^{2}=100 $$

Expand each binomial. $$ (x-2)^{4} $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.