#### Transcript x - Math KSU

A graph is a diagram of a relationship of (at least) two variables with changing values. The current value of each variable is represented as a distance from the origin. The coordination of the variables is represented as a point on the graph All points on the graph are equally important. A graph can just be a set of discrete points {(0,1), (1.5,3), (2.3,π), (5,0.7), (5+π,6), (9,2)} Or a shape… The equation of a graph • The algebraic relationship between variables on a graph • The equation is test: If you pick a pair of values that makes the equation true, that point is on the graph. If you pick a pair that is not true, the point is not on the graph. • Graphing is finding all the points that make the equation true, and changing their color. How would we find the equation of this graph? Tool: The distance between two points Tool: The distance between two points Horizontal distance between points is x-a Vertical distance between points is y-b We have a right triangle. Pythagorean Theorem (x - a) + (y - b) = ? 2 2 2 (x - a) + (y - b) = ? 2 2 Distance Formula • The distance between (a,b) and (x,y) is (x - a) + (y - b) 2 2 Back to a circle problem Circle • A circle is the set of all points a given distance (the radius) from a given point (the center). Our circle is all the points 4 units away from (1,2) All points 4 units away from (1,2) • For any point (x,y) • The distance between (1,2) and (x,y) is 4 • Using the Pythagorean Theorem… (x -1) + ( y - 2) = 4 2 • Is the equation of the circle 2 2 Equation of a circle • For radius r and center (a,b) (x - a) + (y - b) = r 2 2 2 Consider a circle with equation x2 + (y+4)2 = 289. The center and radius are given by: A) B) C) D) E) (0,4), radius=289 (0,-4), radius=289 (4,0), radius=289 (-4,0), radius=289 None of the above. Consider a circle with equation x2 + (y+4)2 = 289. The center and radius are given by: (x - 0) + (y - -4) = (17) 2 2 2 center (0, -4) radius 17 E Midpoint formula • The point halfway between (a,b) and (x,y) is æ x +a y+bö , ç ÷ è 2 2 ø Functions A function is a relationship between two changing variables • An “input” variable • An “output” variable – The result of “doing” the function to the output variable • Both variables change so that the “input” variable always tells you exactly what the “output” variable is. – You never get two outputs for the same input. Function Output Any time I know the input, that’s enough information to tell me the output. Example: when input is 2, output is 20. input Not a Function Input Just knowing the input is not enough to tell me the output Example: when input is 20, the output could be 2, 3.6, or 6.9. Output Function output input Not a Function output input Function notation • • • • For a function named ƒ And an input variable named x The output variable is named ƒ(x). ƒ(x) is the number that is the result of doing the action ƒ to the number x WARNING • ƒ(x) DOES NOT MEAN ƒ*x – You can only multiply numbers. f is NOT a number. f is the name of a relationship. – x and ƒ(x) are the numbers. • Brangelina is not a person. – Brangelina is the name of a relationship – Brad and Angelina are the people How to do a function to the input number • Algebra: Substitute f ( x) = x - 3 Function definition x=2 Input variable x has value 2 2 f ( 2) = 2 - 3 2 Write 2 anywhere you see an x f (2) = 2 - 3 = 1 Simplify 2 f (2) = 1 Output variable has value 1 How to do a function to the input number ƒ(x) ƒ x Find your input value on the x axis. Here the input value is 5.5 How to do a function to the input number ƒ(x) ƒ x Go up and over to Find your output value on the ƒ(x) axis. Here the output value is 25 How to do a function to the input number ƒ(x) ƒ x Input 5.5, output 25, name of function ƒ. ƒ(5.5)=25. Given the function f(x)=x2 -2Mx, where M is some parameter, find f(3). A) B) C) D) E) 3 3M 9-6M 6-9M None of the above. Given the function f(x)=x2 -2Mx, where M is some parameter, find f(3). f ( x ) = x - 2Mx Function definition x=3 Input variable x has value 3 2 f ( 3) = 3 - 2M 3 Write 3 anywhere you see an x 2 f (3) = 3 - 2M 3 = 9 - 6M Simplify 2 f (3) = 9 - 6M Output variable has value 9-6M C Domain and Range • Function is a relationship between a changing input variable and a changing output variable. • The domain is a description of all the values that the changing input variable takes on. • The range is a description of all the value that the changing output variable takes on. Example ƒ(x) Domain: [0,9] Range: [0,30] ƒ x State the range of the function whose graph is pictured here. Select the best answer! A) [3,1] [1,3] B) [3,1) [1,3] C) [3,1] [2,3] D) [3,1) [2,3] E) None of the above State the range of the function whose graph is pictured here. Select the best answer! First, the output changes from -1 to -3 (excluding -1). The output can be any value between -3 and -1, but can’t be 1. Range: [-3,-1) Later, the output changes from 2 to 3. The output can be any value between 2 and 3, including 2 and 3. Additional Range: [2,3] [3,1) [2,3] D