Chpt 1. Functions and Graphs. 1.1 Graphs and Graphing Utilities 1 /19

Transcription

1 Chpt 1 Functions and Graphs 1.1 Graphs and Graphing Utilities 1 /19

2 Chpt 1 Homework , 18, 22, 24, 28, 42, 46, 52, 54, 56, 78, 79, 80, 82 2 /19

3 Objectives Functions and Graphs Plot points in the rectangular coordinate system. Graph equations in the rectangular coordinate system. Interpret information about a graphing utility s viewing rectangle or table. Use a graph to determine intercepts. Interpret information given by graphs. 3 /19

4 The Rectangular Coordinate System The coordinate plane (Cartesian Plane) provides identification for every point on a plane. The horizontal line is the x-axis. The vertical line is the y-axis. The point of intersection for these axes is the point (0, 0), known as the origin. 4/19

5 The Rectangular Coordinate System Positive values are represented to the right of the origin and above the origin. Negative values are represented to the left of the origin and below the origin. 5/19

6 Plotting Points in the Rectangular Coordinate System Every point on the plane is identified by an ordered pair (x, y) The first number in each pair, called the x-coordinate, denotes the distance and direction from the origin along the x- axis. The second number in each pair, called the y-coordinate, denotes the vertical distance and direction from the origin along the y-axis. (x, y) 6/19

7 Example: Plotting Points in the Rectangular Coordinate System Plot the point ( 2, 4). To plot the point ( 2, 4), we move 2 units to the left of the origin... (-2, 4) and 4 units up. 7/19

8 Example: Plotting Points in the Rectangular Coordinate System (continued) Plot the point (4, 2). To plot the point (4, -2), move 4 units to the right... and 2 units down. (4, -2) 8/19

9 Graphs of Equations A relationship between two quantities can be expressed as an equation in two variables, such as y = 4 x 2 A solution of an equation in two variables, x and y, is an ordered pair of real numbers with the following property When the x-coordinate is substituted for x and the y-coordinate is substituted for y in the equation, we obtain a true statement. 9/19

10 Solutions So to repeat, the solutions to an equation in two variables are... ordered pair, (x, y) (POINTS) 10/19

11 Example: Graphing an Equation By Plotting Points y = x +1 Graph. x y = x + 1 y solution To graph an equation means to draw a picture of all of the points that are solutions (satisfy) for that equation y= -4+1 =3 y= -3+1 =2 3 2 (-4,3) (-3,2) To do that necessitates finding some of those points y= -2+1 =1 y= -1+1 =0 1 0 (-2,1) (-1,0) We start by selecting integer values for our x coordinates. Let us start with -4 and continue to y= 0+1 =1 y= 1+1 =2 1 2 (0,1) (1,2) 2 y= 2+1 =3 3 (2,3) Find the appropriate (corresponding y-value. The resulting table of values gives us points to plot showing the shape. 11/19

12 Example: Graphing an Equation Using the Point- Plotting Method (continued) Graph y = x +1. x y solution We plot the points in our table of values and connect them to draw the graph (-4,3) (-3,2) -2 1 (-2,1) -1 0 (-1,0) 0 1 (0,1) 1 2 (1,2) 2 3 (2,3) 12/19

13 Intercepts An x-intercept of a graph is the x-coordinate of the point when the corresponding y-coordinate is 0. It is also the x value of the point where the graph intersects the x-axis. The y-coordinate corresponding to an x-intercept is always zero. A y-intercept of a graph is the y-coordinate of a point (or points) when the corresponding x-coordinate is 0. It is also where the graph intersects the y-axis. The x-coordinate corresponding to a y-intercept is always zero. 13/19

14 Example: Identifying Intercepts Identify the x- and y-intercepts. The value of x when y = 0 is -3 and the graph crosses the x-axis at (-3, 0). The x-intercept is -3. The value of y when x = 0 is 5 and the graph crosses the y-axis at (0, 5). The y-intercept is 5. 14/19

15 Example: Interpret Information Given by Graphs Divorce rates are considerably higher for couples who marry in their teens. The equation d = 4n + 5 models the percentage, d, of marriages that end in divorce after n years if the wife is under 18 at the time of marriage. Determine the percentage of marriages ending in divorce after 15 years when the wife is under 18 at the time of the marriage. To determine the percentage of marriages ending in divorce after 15 years (when the wife is under 18 at the time of the marriage), we simply substitute 15 for n. d = 4n + 5 = 4(15) + 5 = 65 Based on our model, 65% of marriages, when the wife is under 18 at the time of marriage, end in divorce after 15 years. 15/19

16 Example: Example: Interpret Interpret Information Information Given Given by Graphs by (continued) Graphs The graph of d = 4n + 5 is shown below. (15, 65) We calculated that 65% of marriages would end in divorce after 15 years. We can confirm our answer with the graph. Our solution is the point (15, 65) and that point is on the graph. 16/19

17 TI-84 Layout Window Key 2 nd TBLSET Y = Key 2 nd STAT PLOT 2nd Key X (variable) x 2 Key 2 nd Trace Key 2 nd Calc Graph Key 2 nd TABLE Scrolling arrows 2 nd Screen contrast Stat Key 2 nd LIST Carat Key +,,, Keys On Key 2 nd OFF Enter Key 17/19

18 Graphing on the TI-84 Graph y = x +1. Now we enter the equation and let the calculator graph. Y= MATH NUM 1:abs( X,T,θ,n + 1 ) zoom 5 You should see the graph 18/19

19 Graphing Utilities To zoom in on the graph we can set different window dimensions. window Xmin= -10 Xmax= 10 Xscl= 1 Ymin= -3 Ymax= 10 Yscl= 1 Xres= 1 Xres = ignore ΔX= ignore The minimum x-value is 10. The maximum x-value is 10. The distance between consecutive tic marks on the x-axis is 1. The minimum y-value is 3. The maximum y-value is 10. The distance between consecutive tic marks on the y-axis is 1. trace You should also notice the cursor. 19/19