Matrix of a Parabola? (It's my first day)

In summary, the conversation revolves around a question from a linear algebra textbook about "showing" that a given set of equations satisfies the system of equations. The participants discuss different strategies for solving the problem and provide helpful tips for understanding the question. Ultimately, the solution involves substituting the given points into the equation of a parabola and using the fact that the points lie on the parabola to prove that the equations are satisfied.
  • #1
Hi guys, I started linear algebra just yesturday. I've read the first section of the first chapter and now I'm trying to do some problems from this section and I'm stuck on #7. I scanned the page from my math book (here's a link: [Broken]). After reading this section of my book, I'm pretty sure I understand what an "augmented matrix" is. I understand what "consistent" means, and also understand the basic row operations.

In an attempt to figure out what the question is asking, I wrote down their augmented matrix in equation form, and I wrote the following.

y1 = ax1^2 + bx1 + c
y2 = ax2^2 + bx2 + c
y3 = ax3^2 + bx3 + c

Even if this isn't in the right direction to solving the problem, did I do this correctly?

I'm confused as to what exactly the question is asking, and what to "show". I've always been bad at the questions that ask to "show" something is true or false. Please help!
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  • #2
Looks like you almost have it. Yes, those equations are the ones corresponding to the given augmented matrix. Now, do you see that those are precisely the equations stating that (x1,y1), (x2, y2), (x3,y3) are on the parabola y= ax2+ bx+ c?
  • #3
HallsofIvy, I do see and understand that, although I'm having trouble understanding why or how that answers the question in the second phrase. Could you explain to me what the second phrase is asking me to do? Sorry for all the questions. This whole thing is totally new to me.
  • #4

I've run into the same problem from the same book. I understand that the stated equations are for each x,y point, however I'm not sure how to proceed!
  • #5
What "second question in the secnd phrase" are you talking about?
  • #6
when the question asks to show that (a,b,c) satisfies the system of equations, it's asking you to show that it satisfies _each_ of the equations in your system of equations. You have three equations. Each of the equations has 3 unkowns (x,y,z). One way to proceed is to show that if you substitute (a,b,c) for (x,y,z) in any of the equations, then you have a solution to that equation. To do this you must use the fact that the points (x1,y1),(x2,y2), and (x3,y3) each lie on the parabola. Also, you must use the fact that the equation of the parabola is given in the statement of the problem. I hope I haven't just confused you more. In any case, soon you will be able to answer a similar problem (if you can't already): Given three distinct points in the x-y plane, find the equation of a parabola that passes through each of them simultaneously. Perhaps when you (or if you could go ahead and) solve this related problem, #7 may seem much more simple.

1. What is a parabola?

A parabola is a type of curve that is formed when a plane intersects a cone at a specific angle. It is a symmetrical curve with one axis of symmetry and is often described as a "U" shape.

2. How is a parabola represented mathematically?

A parabola can be represented using the quadratic function y = ax^2 + bx + c, where a, b, and c are constants and x is the variable. This equation is known as the standard form of a parabola.

3. What is the vertex of a parabola?

The vertex of a parabola is the point where the curve changes direction. It is the highest or lowest point on the parabola depending on the direction of the curve. In the standard form, the vertex can be found by calculating (-b/2a, c-b^2/4a).

4. What is the focus and directrix of a parabola?

The focus of a parabola is a fixed point inside the curve that is equidistant from every point on the parabola. The directrix is a fixed line outside the curve that is perpendicular to the axis of symmetry and is also equidistant from every point on the parabola. Both the focus and directrix are used to define the shape of the parabola.

5. How is the matrix of a parabola used?

The matrix of a parabola, also known as the quadratic form, is a 2x2 matrix that represents the coefficients of the standard form of the parabola. It is used in linear algebra to solve systems of equations and can also be used to determine the properties of the parabola, such as its vertex, focus, and directrix.

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