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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: \(y = \frac{(x+5)^2}{3} - 4\); vertex at \((-5, -4)\).
Step by step solution
01
Identify the type of conic
The given equation is \(x^2 + 10x - 3y = -13\). To identify the conic type, notice that there is only one squared term \(x^2\), which suggests a parabola.
02
Rewrite the equation for translation
Rewrite the equation in a form that shows the parabola's orientation. The equation can be rearranged as \(x^2 + 10x = 3y - 13\). This represents a parabola that opens vertically.
03
Completing the square
Translate the equation by completing the square for the \(x\) terms. The equation \(x^2 + 10x\) can be rewritten by completing the square: \(x^2 + 10x = (x+5)^2 - 25\). Substitute this back into the equation to get \((x+5)^2 - 25 = 3y - 13\).
04
Simplify translated equation
Solve \((x+5)^2 - 25 = 3y - 13\) for \(y\): \((x+5)^2 - 25 + 13 = 3y\). Simplify to get \((x+5)^2 - 12 = 3y\). Divide by 3: \(y = \frac{(x+5)^2}{3} - 4\). This is the standard form \(y = a(x-h)^2 + k\) representing a parabola.
05
Determine the graph characteristics
The vertex of the parabola in the translated coordinate system is \((-5, -4)\). Since \(a = \frac{1}{3}\), the parabola opens upwards and is wider than the standard parabola \(y = x^2\).
06
Sketch the translated conic
Plot the vertex at \((-5, -4)\). Since \(a = \frac{1}{3}\), the parabola is wider with its axis of symmetry being vertical. Indicate the opening direction and mark a few additional points around the vertex to capture the curve's shape.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Parabola
A parabola is one of the primary types of conic sections. It is a curve where each point is equidistant from a fixed point, known as the focus, and a fixed line called the directrix. In this exercise, the equation given, \(x^2 + 10x - 3y = -13\), represents a parabola since it has only the squared term \(x^2\). This indicates that we are dealing with a parabola with a vertical orientation. This makes it easy for us to translate and plot the parabola on a coordinate plane. The general formula for a vertical parabola is usually expressed as \(y = a(x-h)^2 + k\), where \((h, k)\) is the vertex and \(a\) dictates the direction and width of the opening.
Translation of Axes
Translation of axes is a technique used in algebra to simplify equations, particularly those involving conic sections, by eliminating linear terms. In essence, it's like moving the entire graph horizontally and vertically to a new position. The purpose of this translation is to center the conic, such as a parabola, around a new point on the graph to make it easier to analyze or simplify.
- For a parabola, this usually means finding the vertex, or the "tip" of the curve, and moving the entire graph to align with this point.
- The equation becomes simpler after translation, often allowing for easier calculation of other properties like roots or range.
Completing the Square
Completing the square is a method used to transform a quadratic expression into a perfect square trinomial, which makes it easier to graph or further simplify the equation. This algebraic technique is useful in transforming equations into their vertex form.
Here's a quick rundown of how it works:
Here's a quick rundown of how it works:
- Consider the expression \(x^2 + 10x\).
- Focus on the \(x\) coefficient, which is 10. Halve it to get 5, and then square it, resulting in 25.
- Rewrite the expression as \((x + 5)^2 - 25\).
Vertex Form
The vertex form of a parabola is a convenient way to express its equation, highlighting the vertex's position. This form, given as \(y = a(x-h)^2 + k\), makes it straightforward to identify the parabola's characteristics:
- \((h, k)\) is the vertex, the highest or lowest point on the parabola.
- The coefficient \(a\) shows whether the parabola opens upwards (if \(a>0\)) or downwards (if \(a<0\)), and influences the width of the parabola.
