# qr code vb.net open source Part Two 411 in .NET Encode Code 39 Full ASCII in .NET Part Two 411

Part Two 411
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How can we tell whether that vertex point represents the absolute minimum value of y or the absolute maximum value of y
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Answer 13-9
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We can find the coordinates (x0,y0) of the vertex point based on the constants in the standardform equation of the parabola The x value is x0 = b/(2a) The y value is y0 = b2/(4a) + c If a > 0, the parabola opens upward, so the vertex represents the absolute minimum value of y on the curve If a < 0, the parabola opens downward, so the vertex represents the absolute maximum value of y on the curve
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Question 13-10
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What s the standard-form general equation for a hyperbola that opens toward the right and left in a Cartesian xy plane where the x axis is horizontal and the y axis is vertical
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Answer 13-10
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The standard-form general equation is (x x0)2/a2 (y y0)2/b2 = 1 where x0 and y0 are real-number constants that tell us the coordinates (x0,y0) of the center of the hyperbola, a is a positive real-number constant that tells us the length of the horizontal semi-axis, and b is a positive real-number constant that tells us the length of the vertical semi-axis
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14
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Question 14-1
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How can we informally describe the graph of the function y = ex in the Cartesian xy plane
Answer 14-1
The graph is a smooth, continually increasing curve that crosses the y axis at the point (0,1) The domain is the set of all real numbers, and the range is the set of all positive real numbers The curve is entirely contained within the first and second quadrants As we move to the left (in the negative x direction), the curve approaches, but never reaches, the x axis As we move to the right (in the positive x direction), the graph rises at an ever-increasing rate
Review Questions and Answers Question 14-2
How can we informally describe the graph of the function y = e x in the Cartesian xy plane
Answer 14-2
The graph is a smooth, continually decreasing curve that crosses the y axis at (0,1) The domain is the set of all real numbers, and the range is the set of all positive real numbers The curve is entirely contained within the first and second quadrants As we move to the right, the curve approaches the x axis but never quite reaches that axis As we move to the left, the graph rises at an ever-increasing rate In fact, the curve for the function y = e x has exactly the same shape as the curve for the function y = ex but is reversed left-to-right around the y axis, so the two graphs are horizontal mirror images of each other
Question 14-3
How can we informally describe the graphs of the functions y = 10x and y = 10 x in the Cartesian xy plane
Answer 14-3
The graphs of these functions are curves that closely resemble the graphs of the functions y = ex and y = e x respectively Both base-10 graphs cross the y axis at (0,1), just as the base-e graphs do However, the contours differ The base-10 curves are somewhat steeper than the base-e curves
Part Two 413 Question 14-4
How can we visually and qualitatively compare the graphs of the four functions described in Questions 14-1 through 14-3
Answer 14-4
We can graph them all together on a generic rectangular-coordinate grid such as the one shown in Fig 20-7
Question 14-5
How can we informally describe the graph of the function y = ln x in the Cartesian xy plane
Answer 14-5
The graph is a smooth, continually increasing curve that crosses the x axis at the point (1,0) The domain is the set of positive real numbers, and the range is the set of all real numbers The curve is entirely contained within the first and fourth quadrants As we move to the left (in the negative x direction) from the point (1,0), the curve blows up negatively, approaching the y axis but never reaching it As we move to the right from (1,0), the graph rises at an everdecreasing rate
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