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Chapter 13: Problem 98

How can you distinguish parabolas from other conic sections by looking at their equations?

Short Answer

Expert verified
The equation of a parabola can be differentiated from other conic sections by the presence of only one squared term (either in x or y) and absence of 'xy' term in the equation. Other conic sections, such as ellipses, circles, and hyperbolas, include two squared terms in their standard equations.

Step by step solution

01

Understand the General form of equations

Recognize the general form of equations for different conic sections. The general form of the equation of a parabola is \(y = ax^2 + bx + c\) or \(x = ay^2 + by + c\), where a, b, and c are constants. On the other hand, the ellipses, hyperbolas, and circles generally have the quadratic form \(Ax^2 + By^2 + Cxy + Dx + Ey + F = 0\), where A equals B for circles and A not equals B for ellipses and hyperbola.
02

Identify the Quadratic Terms

Look for the quadratic (squared) terms in the equation. A parabola's equation will have only one squared term, either \(x^2\) or \(y^2\). If the equation of the conic section has two squared terms, it is not a parabola.
03

Check for Absence of xy term

Ensure there is no product of x and y present. Parallel or vertical parabolas will not have an 'xy' term in their equation. If there is an 'xy' term, the conic section is not a parabola. If an equation has only one squared term (either x or y) and does not contain any 'xy' term then it represents a parabola.

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Most popular questions from this chapter

Use the vertex and intercepts to sketch the graph of each equation. If needed, find additional points on the parabola by choosing values of y on each side of the axis of symmetry. $$x=-2(y+6)^{2}+2$$

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function? $$y=-x^{2}-4 x+4$$

The equation of a parabola is given. Determine: a. if the parabola is horizontal or vertical. b. the way the parabola opens. c. the vertex. $$y=2(x-3)^{2}+1$$

Use the vertex and intercepts to sketch the graph of each equation. If needed, find additional points on the parabola by choosing values of y on each side of the axis of symmetry. $$x=(y-2)^{2}-4$$

Indicate whether the graph of each equation is a circle, an ellipse, a hyperbola, or a parabola. Then graph the conic section. $$4 x^{2}+y^{2}=16$$
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