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Vertex x0 = b/(2a) y0 = b 2/(4a) + c
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Figure 13-10
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Graph of the parabola for y = ax2 + bx + c
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Standard Equations
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and y0 = ax02 + bx0 + c = b2/(4a) + c If a > 0, the parabola opens upward, and the vertex represents the absolute minimum value of y If a < 0, the parabola opens downward, and the vertex represents the absolute maximum value of y In the graph of Fig 13-10, the parabola opens upward, so we know that a > 0 in its equation
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Equations for hyperbolas The standard-form general equation of a hyperbola in the Cartesian xy plane, as shown in Fig 13-11, is
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(x x0)2/a2 ( y y0)2/b2 = 1 where x0 and y0 are real constants that tell us the coordinates (x0,y0) of the center
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(x0, y0)
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Graph of the hyperbola for (x x0)2/a2 (y y0)2/b2 = 1
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Conic Sections
The dimensions of a hyperbola are harder to define than the dimensions of a circle or an ellipse Suppose that D is a rectangle whose center is at (x0,y0), whose vertical edges are tangent to the hyperbola, and whose corners lie on the asymptotes When we define D this way, then a is the distance from (x0,y0) to D along a line parallel to the x axis, and b is the distance from (x0,y0) to D along a line parallel to the y axis We call a the width of the horizontal semi-axis, and we call b the height of the vertical semi-axis For hyperbolas centered at the origin, we have x0 = 0 and y0 = 0, so the general equation becomes x2/a2 y2/b2 = 1 The simplest possible case is the unit hyperbola whose equation is x 2 y2 = 1
Are you astute
You might imagine that the above-mentioned standard forms are not the only ways that the equations of conic sections can present themselves If that s what you re thinking, you re right! However, you can always convert the equation of a conic section to its standard form For example, suppose you encounter 49x2 + 25y2 = 1225 You say, This looks like it might be the equation for an ellipse, but it s not in the standard form for any known conic section Then you notice that 1225 is the product of 49 and 25 When you divide the whole equation through by 1225, you get 49x2/1225 + 25y2/1225 = 1225 / 1225 which simplifies to x2/25 + y2/49 = 1 which can also be written as x2/52 + y2/72 = 1 Now you know that the equation represents an ellipse centered at the origin whose horizontal semi-axis is 5 units wide, and whose vertical semi-axis is 7 units tall
Here s a challenge!
Whenever we have an equation that can be reduced to the standard form y = ax2 + bx + c
Practice Exercises
we get a parabola that opens either upward or downward, and that represents a true function of x How can we write the standard-form general equation of a parabola that opens to the right or the left Does such a parabola represent a true function of x
Solution
We can simply switch the variables to get x = ay2 + by + c If a > 0, we have a parabola that opens to the right If a < 0, we have a parabola that opens to the left If we define x as the independent variable and y as the dependent variable as is usually done in Cartesian xy coordinates, then vertical-line tests reveal that these parabolas do not represent true functions of x
Practice Exercises
This is an open-book quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find worked-out answers in App B The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it! 1 At the beginning of this chapter, we learned that the intersection between a plane and a double right circular cone is always a circle, an ellipse, a parabola, or a hyperbola, as long as the plane doesn t pass through the point where the apexes of the two cones meet What happens if the plane does pass through that point 2 Figure 13-12 shows an ellipse in the Cartesian xy plane with some dimensions labeled The lower focus is at the origin (0,0) The lower vertex is at (0, 2) Both foci and both vertices lie on the y axis The ellipse is taller than it is wide What is its eccentricity 3 Recall the formulas relating the parameters of an ellipse when plotted in the manner of Fig 13-12: u = f + f /e + y and x2 + y2 = (eu)2 Based on these formulas, the information provided in the figure, and the solution you worked out to Problem 2, determine a relation between x and y that describes our ellipse The equation should include only the variables x and y, but it doesn t have to be in the standard form 4 What are the coordinates of the upper vertex of the ellipse shown in Fig 13-12 What are the coordinates of the upper focus of the ellipse shown in Fig 13-12
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