Section 9.3: Functions and their Graphs

Transcription

1 Section 9.: Functions and their Graphs Graphs provide a wa of displaing, interpreting, and analzing data in a visual format. In man problems, we will consider two variables. Therefore, we will need to have two aes one for the variable and another for the variable. Together these aes will form the Rectangular Coordinate Sstem, or Cartesian Coordinate Sstem. The horizontal ais is the ais and the vertical ais is the ais. These two aes divide the plane into four quadrants and the intersection of the two aes is called the origin. See the following diagram. Quadrant II 0, 0 Quadrant I 0, 0 Quadrant III 0, 0 Quadrant IV 0, 0 Ordered pair: Each point in the plane is called an ordered pair and is denoted p, q. The first number indicates the point s horizontal location with respect to the -ais, and the second number indicates the point s vertical location with respect to the -ais. Hence, the origin is labeled p0, 0q.

2 SECTION 9.: FUNCTIONS AND THEIR GRAPHS Eample : Plot the following points on the same set of aes: A p, q, B p, q, C p, q, and D p, 5q The -intercept of a graph is the point where the graph crosses the -ais. This point is pa, 0q where to find a, we let 0 and solve for. The -intercept of a graph is the point where the graph crosses the -ais. This point is p0, bq where to find b, we let 0 and solve for. Eample : Find the -intercept and -intercept for 7. Function: A function is a rule or correspondence that assigns to each element of one set, called the domain, eactl one element of a second set, called the range. A function ma be defined b a set of ordered pairs, a table, a graph, or an equation. Domain: The domain of a function is the set of all inputs. If is an element in the domain, then is called the independent variable. Range: The range of a function is the set of all outputs. If represents an output of the function f from an input, then is called the dependent variable and is denoted b f pq.

3 SECTION 9.: FUNCTIONS AND THEIR GRAPHS Eample : Determine which of the following are eamples of functions. For each function, determine the domain and range. (a) tp, q, p, 6q, p6, 8q, p9, q, p, 5qu (b) (c) 9 (d) 9

4 SECTION 9.: FUNCTIONS AND THEIR GRAPHS The graph of a function is a set of points p, q in the -plane such that f pq. The Vertical Line Test: A set of points in the -plane is the graph of a function if and onl if no vertical line intersects the set of points more than once. Eample : Determine if each of following curves is the graph of a function. Linear Functions: Linear functions are functions whose graphs are lines. A linear function has the algebraic form f pq m b where m and b are constants. In the function f pq, m and b. Eample 5: Make a table of at least five values for f pq and sketch the graph.

5 SECTION 9.: FUNCTIONS AND THEIR GRAPHS 5 Quadratic functions: A quadratic function is a function of the form f pq a b c, where a, b, and c are constants and a 0. The graph of a quadratic function is a parabola. Below of some graphs for quadratic functions. NOTE: When a 0, the parabola opens up. When a 0, the parabola opens down.

6 6 SECTION 9.: FUNCTIONS AND THEIR GRAPHS Eponential functions: The function f pq a, where is a real number, a 0 and a, is called an eponential function with base a. If a, then the function is increasing; if 0 a, then the function is decreasing. Below are two graphs of eponential functions. Cubic functions: A cubic function is a function of the form f pq a b c d, where a, b, c, and d are constants and a 0. Below are two graphs of cubic functions.

7 SECTION 9.: FUNCTIONS AND THEIR GRAPHS 7 Step functions: A step function is a function that increases or decreases from one constant value to another. One of the most common step functions is the greatest integer function f pq vw which is defined to be the greatest integer that is less than or equal to. For eample, v.8w and v.w. Below are two graphs of step functions. Eample 6: Find the following. (a) v.w (b) v8.5w (c) : Eample 7: Determine which tpe of function best fits each of the following graphs: linear, quadratic, eponential, cubic, or step?

8 8 SECTION 9.: FUNCTIONS AND THEIR GRAPHS Eample 8: For the function f graphed below, find the following: (a) f p 5q (d) f pq (b) f p q (e) f p5q (c) f pq (f) f p6q