In Problems 65-70, graph each equation and find the point(s) of intersection, if any. The circle

In Problems 65-70, graph each equation and find the point(s)...

Key Concepts

Conic Sections

Conic sections are curves obtained by slicing a cone with a plane, and include circles, parabolas, ellipses, and hyperbolas. Understanding their standard forms and geometric properties is essential for graphing and analyzing their behavior.

Graphing Equations

Graphing equations involves plotting all the points that satisfy a given equation on the coordinate plane. This helps visualize the shapes and positions of the curves and is key to identifying their intersections.

Solving Systems of Equations

Finding the intersection points of curves involves solving a system of equations, where each equation represents a curve. This process, typically accomplished through substitution or elimination methods, determines the coordinates common to all the curves involved.

Standard Form Conversion

Converting equations to their standard forms is a crucial step in analyzing curves. For example, rewriting the equations of conic sections in standard form makes it easier to identify key features like centers, vertices, and orientations, which in turn aids in accurate graphing.

Transcript

00:01 In this problem, we're asked to solve the system of equations by graphing it and finding their points of intersection.

00:09 So for the first equation, because it's a linear equation, i need to write it as y equals.

00:14 So i need to subtract x on both sides.

00:17 So therefore, 2y equals to 0 minus x is negative x, and then divided by 2.

00:23 So y equals to negative 1 over 2x.

00:30 And so i'm going to go ahead and grab this equation.

00:33 The y intercept is 0, so you're going to start at 0 -0.

00:37 And then the slope is negative 1 1 1.

00:39 So you're going down 1, right 2, down 1, right 2.

00:47 And then if you want to go to the other direction, you're going to go up 1, left 2, up 1, left 2.

00:56 So now we're going to connect these points together.

00:59 So that's the line for that equation.

01:03 Okay, so to graph this equation of the circle, we need to identify the center, and the center is located inside the squares here.

01:20 Okay, so it's not going to be negative one, negative one.

01:23 Make sure you change the sign for the center, so negative one becomes positive one, negative one is positive one.

01:29 And then the radius is always equal to r squares.

01:32 So to find the radius, you have to take the square root of five, which is around 2 .24.

01:39 So from the center 1 -1, you're going to go right 2 .24 times, up 2 .24 times, left 2 .24 times, and then down 2 .24 times.

01:56 And now you're going to draw a circle that goes through all these four points.

02:12 So as you can see, these two equations have two intersections, so i need to find 2.

02:19 Intersection points here.

02:21 Okay, so the first one is pretty obvious.

02:24 It's located at two negative one.

02:27 Now the second point is not so much, so therefore we need to solve it to find exactly where they intersected.

02:34 So we know what y equals to, which is negative one have x.

02:38 We're going to substitute this into our second equation where the y is.

02:43 So therefore, our second equation is going to become x minus one squared plus the y value is negative one half x minus one squared equals to five.

03:00 So now we're going to foil the x minus one square, which gives you x squared minus 2x plus one, plus now we're going to foil the negative one half x minus one squared.

03:13 So that gives you positive one -fourth x squared...

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