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# 150ch2_4-5 - Math 150 Fall 2007 c Benjamin Aurispa Chapter...

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Unformatted text preview: Math 150, Fall 2007, c Benjamin Aurispa Chapter 2, Continued 2.4 Transformations of Functions This section tells how transformations change the graph of a function. It is very useful in graphing functions, when you know its general shape. 1. Vertical Shifts Suppose we are given y = f (x) and c > 0. (a) To graph y = f (x) + c, shift the graph of y = f (x) up by c. (b) To graph y = f (x) - c, shift the graph of y = f (x) down by c. Examples: Sketch the graphs of the following functions. f (x) = x2 + 4 f (x) = x3 - 3 p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 2. Horizontal Shifts Suppose we are given y = f (x) and c > 0. (a) To graph y = f (x - c), shift the graph of y = f (x) to the right by c. (b) To graph y = f (x + c), shift the graph of y = f (x) to the left by c. Examples: Sketch the graphs of the following functions. f (x) = (x - 1)2 f (x) = x+2 p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 1 Math 150, Fall 2007, c Benjamin Aurispa 3. Reflections (a) To graph y = -f (x), reflect the graph of y = f (x) across the x-axis. (b) To graph y = f (-x), reflect the graph of y = f (x) across the y-axis. Examples: f (x) = -|x| p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p f (x) = p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p -x p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 4. Vertical Stretching and Shrinking To graph y = cf (x): (a) If c > 1, stretch the graph of y = f (x) vertically by a factor of c. (b) If 0 < c < 1, shrink the graph of y = f (x) vertically by a factor of c. Examples: p p p p p p p p p p p 1 f (x) = 2 x2 p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p f (x) = 2x3 p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 5. Horizontal Scaling To graph y = f (cx): (a) If c > 1, shrink the graph of y = f (x) horizontally by a factor of 1/c. (b) If 0 < c < 1, stretch the graph of y = f (x) horizontally by a factor of 1/c. Examples: f (x) = |3x| p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p f (x) = p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 1 4x p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 2 Math 150, Fall 2007, c Benjamin Aurispa You can also combine many of these transformations on a function. Examples: 3 - 2(x + 1)2 p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p -3x - 2 p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p Even and Odd Functions Let f be a function. 1. f is even if f (-x) = f (x) for all x in the domain. The graph of an even function is symmetric about the y-axis. 2. f is odd if f (-x) = -f (x) for all x in the domain. The graph of an odd function is symmetric about the origin. 3 Math 150, Fall 2007, c Benjamin Aurispa Determine whether the following functions are even, odd, or neither. f (x) = x + 1 x f (x) = x3 - x2 + 1 f (x) = x4 - 4x2 2.5 Quadratic Functions; Maxima and Minima A quadratic function is a function of the form f (x) = ax2 + bx + c where a = 0. A quadratic function always has a parabola as its graph. To determine information about the parabola, we must write the quadratic function in standard form, f (x) = a(x - h)2 + k, by completing the square. Once we have written the quadratic function in standard form, then the graph of f is a parabola with vertex (h, k). The parabola opens upward if a > 0 and opens downward if a < 0. We can see this by using methods of Section 2.4. Example: Write the quadratic function in standard form, find the vertex of the parabola, and sketch the graph. f (x) = x2 - 2x + 2 4 Math 150, Fall 2007, c Benjamin Aurispa f (x) = -2x2 - 12x - 10 Let f be a quadratic function written in standard form, f (x) = a(x - h)2 + k. The maximum or minimum value of f occurs at x = h (the vertex). If a > 0, then the minimum value of f is f (h) = k. If a < 0, then the maximum value of f is f (h) = k. Example: What are the maximum/minimum value of the examples we did above? Example: Find a function whose graph is a parabola with vertex (3, 4) and that passes through the point (1, -8). There is another way of determining maximum/minimum of a quadratic function. b The maximum or minimum value of a quadratic function f (x) = ax2 + bx + c occurs at x = - 2a . b If a > 0, then the minimum value is f (- 2a ). b If a < 0, then the maximum value is f (- 2a ). b b So, essentially, h = - 2a and k = f (- 2a ). 5 Math 150, Fall 2007, c Benjamin Aurispa Example: Find the maximum or minimum value of the quadratic function f (x) = 2x2 + 8x + 11. What is the vertex? Example: A keychain vendor finds that if he sells x keychains in one day, his profit is given by P (x) = -2x2 + 16x - 25. What is his maximum profit per day, and how many keychains must he sell to reach maximum profit? The domain of a parabola is always all real numbers. However, the range of a parabola depends on where the vertex of the parabola is and whether it opens upward or downward. If a > 0, then the range of f is [k, ), also written as {y | y k}. If a < 0, then the range of f is (-, k], also written as {y | y k}. Example: What is the range of the quadratic function, f (x) = 3x2 - 6x - 5? If a function has many maxima or minima, depending on what part of the domain we are looking at, then these maxima/minima are called local maxima/minima. 6 ...
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