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Magazine Chapter 5: Problem 70
Use a translation of axes to put the conic in standard position. Identify the graph, give its equation in the translated coordinate system, and sketch the curve. $$x^{2}+10 x-3 y=-13$$
Short Answer
Expert verified
The conic is a parabola \(X^2 = 3Y\) with vertex at \((-5, -4)\).
Step by step solution
01
Identify the Type of Conic
First, we need to recognize the type of conic represented by the equation. The given equation is \(x^2 + 10x - 3y = -13\). Observe that there is an \(x^2\) term but no \(y^2\) term, which indicates that this is a parabolic equation.
02
Move Terms to One Side
To make it easier to complete the square, rewrite the equation by moving the constant term to the other side: \(x^2 + 10x = 3y - 13\).
03
Complete the Square for x
To complete the square for the \(x\) terms, take half of the coefficient of \(x\), square it, and add and subtract that number inside the equation: \(x^2 + 10x + 25 - 25 = 3y - 13\). This results in \((x + 5)^2 - 25 = 3y - 13\).
04
Simplify the Equation
Add \(25\) to both sides to balance the equation: \((x + 5)^2 = 3y + 12\). Now the equation is \((x + 5)^2 = 3(y + 4)\).
05
Apply Translation of Axes
Set new axes \(X, Y\) where \(X = x + 5\) and \(Y = y + 4\). Substituting these, the equation becomes \(X^2 = 3Y\). This represents a standard form of a parabola.
06
Identify and Sketch the Parabola
The transformed equation \(X^2 = 3Y\) is a standard parabola that opens upwards with vertex at the origin of the new axes. Sketch this parabola, remembering the translation used to set it in standard position: Shift the vertex to \((-5, -4)\) in the original coordinate system.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Parabola
A parabola is a fascinating shape in the field of mathematics, specifically in the study of conic sections. It is a curve where each point is equidistant from a fixed point (called the focus) and a fixed line (called the directrix). The most common form you'll see in algebra is the quadratic equation, typically written as \(y = ax^2 + bx + c\). Parabolas have distinct characteristics:
- They are symmetric around a central axis called the axis of symmetry.
- The highest or lowest point on the parabola is known as the vertex, which serves as a key feature in defining its position and shape.
- A parabola can open upwards or downwards if it is a vertical parabola, and right or left if it is horizontal.
Translation of Axes
Translation of axes is a process that can make equations simpler and easier to analyze or solve. This is done by shifting the origin of the coordinate system to a new point without rotating the axes. Here's how it works:
- Suppose the translation is defined by \(X = x - h\) and \(Y = y - k\), where \((h, k)\) is the point to which we wish to move the origin.
- Substituting these into the original equation essentially re-centers the graph around a new origin.
Completing the Square
Completing the square is a neat algebraic technique used to simplify quadratic expressions. The purpose is to rewrite a quadratic expression so that it is readily recognizable in its vertex form. Let's break it down:
- First, identify the quadratic and linear terms鈥攊t鈥檚 often given as \(ax^2 + bx\).
- Next, find the value that makes a perfect square trinomial. It is calculated by taking half of the linear coefficient (\(b\)), squaring it, and then adding and subtracting this value inside the equation.
- The expression \((x + \frac{b}{2})^2 - \left(\frac{b}{2}\right)^2\) results, forming a complete square.
Standard Form of a Parabola
The standard form of a parabola is a simple yet powerful way to represent parabolic equations. For parabolas that open vertically, this form is \(y = ax^2 + bx + c\), but horizontal parabolas are more easily understood using the vertex form. Let's see why it's important:
- The standard form provides a clear view of the parabola's direction and vertex.
- It commonly appears as \((X - h)^2 = 4p(Y - k)\) for vertical parabolas, revealing the distance \(p\) from the vertex to the focus, while horizontal parabolas use a similar concept with the roles of \(X\) and \(Y\) interchanged.
