Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

8. Conic Sections

Parabolas

College Algebra

8. Conic Sections

Parabolas

Previous problemNext problem

0:58 minutes

Problem 67

Textbook Question

In Exercises 63–68, find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. x = (y + 2)^2 - 1(x - 2)^2 + (y + 2)^2 = 1

Verified step by step guidance

Step 1: Graph the first equation, x = (y + 2)^2 - 1. This equation represents a parabola that opens to the right with vertex at (-1, -2).

Step 2: Graph the second equation, (x - 2)^2 + (y + 2)^2 = 1. This equation represents a circle with center at (2, -2) and radius 1.

Step 3: Identify the points of intersection between the parabola and the circle by observing where the graphs overlap. These points are the solutions to the system of equations.

Step 4: Check each point of intersection by substituting the x and y values back into both original equations to verify that they satisfy both equations.

Step 5: List all the points that satisfy both equations as the solution set of the system.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

58s

Play a video:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Graphing Equations

Graphing equations involves plotting points on a coordinate system to visualize the relationship between variables. For the given equations, one is a quadratic function and the other represents a circle. Understanding how to graph these shapes accurately is crucial for identifying their points of intersection, which represent the solutions to the system.

Points of Intersection

Points of intersection are the coordinates where two graphs meet on the coordinate plane. In the context of a system of equations, these points represent the solutions that satisfy both equations simultaneously. Finding these points requires solving the equations either algebraically or graphically, and they are essential for determining the solution set.

Checking Solutions

Checking solutions involves substituting the found intersection points back into the original equations to verify their validity. This step ensures that the identified points are indeed solutions to both equations, confirming their accuracy. It is a critical part of the problem-solving process in algebra to ensure that no extraneous solutions have been included.