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August, 2009

Math 1 Unit 1. Functions and Their Graphs. August, 2009. Functions and Function Notation. A function is a relation between two sets X and Y is a set of ordered pairs, each in the form (x,y) where every x is associated with only one value for y.

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August, 2009

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  1. Math 1 Unit 1 Functions and Their Graphs August, 2009

  2. Functions and Function Notation • A function is a relation between two sets X and Y is a set of ordered pairs, each in the form (x,y) where every x is associated with only one value for y. • For example {(1,5), (2, 10), (3, 15)} is a relation that is a function, because each x is unique. • If an x is used with more than one y, then the relation is not a function. • For example {(2,5), (2, 10), (3, 15)} is a relation that is not a function.

  3. Functions and Function Notation Functions can be specified in a variety of ways. For example, x² + 2y = 1 is called the implicit or standard form. You are more used to another form, the explicit form. In this form, y is on one side of the equal sign. y = ½(1 - x²) For explicit form, the variable x is the independent variable and y is the dependent variable.

  4. Many real life situations can be modeled by functions. For instance, the area A of a circle is a function of the circle's radius r. A = r² In this case, r is the independent variable and A is the dependent variable. The input variable is the domain and the output variable is the range.

  5. Function Notation • Another representation is function notation. • y is replaced by f(x). f(x) = ½(1 - x²) • This last form is the one we will use f(x) for a large amount of work in the class. In this way, we can find the value f(3), the value of the function when we substitute x = 3. • f(x) = 2x² - 4x +1 • f(3) = 2(3)² - 4(3) + 1 • =18 - 12 + 1 • = 7 • Find f(-3) for this function. Note that (-3)² = 9.

  6. Remember: Although we normally use f(x) for functions, we can use any variable for x. f(s), f(t), f(h) are all function names. Also, we do not have to use f(x). We can use g(x) or h(x). They all mean the same thing: What value do I replace in the function on the other side of the equal sign for the variable represented in the parenthesis. f(x) = x² +7Find the following: a). f(3)b) f(-3)c). f(b-1) f(-3) = (-3)² + 7 = 9 + 7 = 16 f(b - 1) = (b - 1)² + 7 = b² - 2b + 1 + 7 = b² - 2b + 8 f(3) = (3)² + 7 = 9 + 7 = 16

  7. The test to see if a relationship between x and y is a function is the vertical line test. If there is no more than 1 point for y for any value of x, then the relationship is a function. If there is more than 1 point on y for any x, then the relationship is not a function. On the next pages, there are some basic graphs whose family of graphs you will need to KNOW and recognize immediately for the remainder of this course.

  8. Transformations of Functions For some families of graphs, there is a basic shape we have used in past math classes. For f(x) = x², it is a basic parabola, symmetric to the y axis. If we have the equation f(x) = (x-2)², we will move the vertex of the parabola 2 units to the right. For the equation f(x) = (x+1)² + 3, we would move the vertex 1 unit to the left, and 3 units up from the x axis. For the equation f(x) = -2(x - 2)² - 1, we would move the vertex 2 units to the right, down 1 unit, invert the opening to move downward, and become narrower.

  9. Basic Tranformations Original graph:y = f(x) Horizontal shift c units to the right:y = f(x-c) Horizontal shift c units to the left:y = f(x+c) Vertical shift c units downward:y = f(x) - c Vertical shift c units upward:y = f(x) + c Reflection (about the x axis):y = -f(x) Reflection (about the y axis):y = f(-x) Reflection (about the origin):y = -f(-x)

  10. Polynomial Expressions The most common type of algebraic function is the polynomial expression where the positive integer n is the degree of the expression. The numbers ai are the coefficients, with an as the leading coefficient and a0 as the constant term. Here are the names of some common polynomial functions: Zeroth degree: f(x) = a(constant function) First degree:f(x) = ax + b(linear function) Second degree:f(x) = ax² + bx + c(quadratic functioin) Third degree:f(x) = ax³ + bx² + cx + d (cubic function)

  11. Domain of a Function These are different forms of what you will use to find the domain of a function. If it is stated, you must use that domain. If it is not stated, you will use all possible numbers where the function is defined. Explicit Given to you, you don't have to worry about where it is defined or not defined Implicit Not given to you, but you know where it is defined or not

  12. Odd and Even Functions A function is an odd function if it has origin symmetry A function is an even function if it has y axis symmetry Graph the following functions and see if you can spot a way to decide if a function is odd, even, or neither just by looking at the function:

  13. Determine if the following functions are even, odd, or neither. Then, find the zeros of the function.

  14. CREDITS Ron Larson, Robert P. Hostetler, Bruce Edwards, David Heyd Calculus of a Single Variable, 7th Ed.; Houghton Mifflin Company

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