Conic Sections

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Upper vertex

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Upper focus Point (x, y) Lower focus eu

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f u = f + f /e + y Lower vertex is at (0, 2)

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Equation of directrix is y = 6

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Figure 13-12

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Illustration for Problems 2 through 5

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5 What are the coordinates of the center of the ellipse shown in Fig 13-12 What is the length of the vertical semi-axis What is the length of the horizontal semi-axis Based on these results, write down the standard-form equation for the ellipse 6 Determine the type of conic section the following equation represents, and then draw its graph: x2 + 9y2 = 9 7 Determine the type of conic section the following equation represents, and then draw its graph: x2 + y2 + 2x 2y + 2 = 4 8 Determine the type of conic section the following equation represents, and then draw its graph: x2 y2 + 2x + 2y = 4

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Practice Exercises

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9 Determine the type of conic section the following equation represents, and then draw its graph: x2 3x y + 3 = 1 10 Following is an equation in the standard form for a hyperbola: (x 1)2/4 ( y + 2)2/9 = 1 First, find the coordinates (x0,y0) of the center point Then determine the length a of the horizontal semi-axis and the length b of the vertical semi-axis Next, sketch a graph of the curve Finally, work out the equations of the lines representing the asymptotes Here s a hint: Use the point-slope form of the equation for a straight line in the xy plane If it has slipped your memory, the general form is y y0 = m(x x0) where m is the slope of the line, and (x0,y0) represents a known point on the line

CHAPTER

Exponential and Logarithmic Curves

In your algebra courses, you learned about exponential functions and logarithmic functions If you need a refresher, the basics are covered in Chap 29 of Algebra Know-It-All Let s look some graphs that involve these functions

Graphs Involving Exponential Functions

An exponential function of a real variable x is the result of raising a positive real constant, called the base, to the xth power The base is usually e (an irrational number called Euler s constant or the exponential constant) or 10 The value of e is approximately 271828

Exponential: example 1 When we raise e to the xth power, we get the natural exponential function of x When we raise 10 to the xth power, we get the common exponential function of x Figure 14-1 shows graphs of these functions At A, we see the graph of

y = ex over the portion of the domain between 25 and 25 At B, we see the graph of y = 10x over the portion of the domain between 1 and 1 The curves have similar contours When we tailor the axis scales in a certain relative way (as we do here), the two curves appear almost identical In the overall sense, both of these functions have domains that include all real numbers, because we can raise e or 10 to any real-number power and always get a real-number as the result However, the ranges of both functions are confined to the set of positive reals No matter what real-number exponent we attach to e or 10, we can never produce an output that s equal to 0, and we can never get an output that s negative

Graphs Involving Exponential Functions

Figure 14-1 Graphs of the natural

exponential function (at A) and the common exponential function (at B)

x 2 1 0 y 10 1 2

x 1 0 1

Exponential: example 2 Let s see what happens to the graphs of the foregoing functions when we take their reciprocals and then graph them over the same portions of their domains as we did before Figure 14-2A is a graph of

y = 1/ex Figure 14-2B is a graph of y = 1/10x

Exponential and Logarithmic Curves

Figure 14-2 Graphs of the

reciprocals of the natural exponential (at A) and the common exponential (at B)