150ch2_4-5 - Math 150, Fall 2007, c Benjamin Aurispa...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

This note was uploaded on 10/08/2009 for the course MATH 150 taught by Professor Unknown during the Fall '08 term at Texas A&M.

Ask a homework question - tutors are online