(4) If the fuel consumption is 39 gallons/hour, the air speed is mph.

35. Examples of graphs.-a. The graphs that will be used and constructed will consist of a succession of points plotted with reference to coordinate axes and connected by a smooth line forming a straight line or a curve. The coordinates are obtained either from a formula

or from empirical data (that obtained by observation).

b. Graphing from an equation.-(1) In algebraic work, various expressions suitable for graphing are employed, especially the equations. Consider the equation y-2x=5, or y=2x+5, and draw its graph. Since it is of the type ax+by=c (a, b, c some known constants), the graph of which is always a straight line, the graph of y=2x+5 should be a straight line. If a number, for example 3, is substituted for x in the quantity 2x+5, then the quantity 2x+5 (or y) takes on the value 11. In a similar manner, many more pairs of numbers can be obtained which satisfy the above equation. For example:

when x= 3, y= 11, 2(3)

+5=11

Construct a table of these values, set up a pair of coordinate axes with suitable scales, and plot the points. Always remember that the two numbers describing the location of a point will satisfy the equation being graphed. After the points are plotted, notice that Y

(2,9)

(2x+5)

they all lie in a straight line. Draw a straight line through the points. Every point on this line has a Y-value and an X-value satisfying the equation.

(2) A freely falling object will fall a distance d feet in t seconds as given by:

1

d:

=2912

Use 32 for g, and the formula then becomes: d=16ť2. This formula is of a different type from those discussed above. After calculating a few values which satisfy this equation and drawing a graph representing the formula, figure 35 is obtained.

Exercise. From the graph

How far will an object fall in 21⁄2 seconds?

Check by calculating this distance from the formula.

c. Exercises.-Plot the following points on a coordinate system of graph paper:

(1) (2, 3); (5, 7); (8, 0); (0, 5).

(2) (-2, 5); (−7, 8); (—5, 0); (−2, 3).

(3) (−2, −4); (0, −5); (−4, −7); (−2, −3).

(4) (1, −5); (2, −4); (5, −8); (2, −7).

(5) Three corners of a rectangle are at (1, 4), (4, 8), and (9, −2). Find the coordinates of the other corner.

(6) Compute a table of values and draw a graph of the following: (a) y=2x; y-3x=5.

(7) From the graph in figure 35, read off—

(a) The distance that an object will fall in 11⁄2 seconds and in 3%1⁄2 seconds.

(b) How long it will take an object to fall 50 feet? 100 feet? 225 feet?

(8) The velocity of sound in air depends on the temperature of the air. By use of the following data, draw a graph showing how the velocity varies with the temperature.

From the graph, find the velocity if the temperature is 35°; 10.5°; -25°; 120°.

1095; 1070.5; 1035; 1180

Answers.

(9) The effective disk area of a propeller depends on the diameter of the propeller. By use of the table below, draw a graph showing how the effective area A in square feet varies with the diameter in feet.

From the graph, find the area if the diameter is 9 feet; 12.5 feet; 144 feet.

36. Graphic solution of algebraic equations containing two unknowns.—a. It is sometimes necessary to find a pair of numbers that will satisfy two equations at the same time. One method of doing this is to graph each equation on one set of coordinate axes and find the intersection of the curves.

Example: Find the values of x and y which will satisfy the following two equations simultaneously:

Solution: By graphing each equation separately on the same pair of coordinate axes, the line AB is obtained, every point of which has coordinates satisfying the equation 2x-y=3; and the line CD is obtained, every point of which has coordinates satisfying the equation 3x+2y=8.

The intersection of CD and AB is the point P the coordinates of which (2, 1) are the only ones satisfying both of the given equations. b. This graphical method usually gives approximate results only,

because of errors in measuring line segments when determining certain coordinates.

c. If the lines are parallel, it is obvious that no coordinate values will satisfy both of the given equations.

d. Exercises. Solve graphically—

(1)x+y=3

x-2y=0

(2) 2y-3x=0

4y+3x=-18

(3) x+2y=4

3x-y=6

(4) 5y-3=0

10y+3x=4

(-2,-3) Answer.

(-%, %) Answer.

e. Under certain circumstances, curves other than straight lines are plotted in pairs and their intersection found.

Example: Find graphically the values of x and y which satisfy the following two equations simultaneously:

2y=x2 y+x=4

Solution: By graphing each equation separately on the same pair of coordinate axes, a curve is obtained, every point of which has coordinates satisfying the equation 2y=x2; and the line AB is obtained, every point of which has coordinates satisfying the equation y+x=4.

The intersections of the curve and the straight line are the points P and Q the coordinates of which (2, 2) and (-4, 8) are the only ones satisfying both of the given equations.