Graphs of Equations MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011
Objectives In this lesson we will learn to: sketch the graphs of equations, find the x- and y-intercepts of the graphs of equations, use symmetry to sketch graphs, find the equations and circles and sketch their graphs, applying graphing to real-world problems.
Graph of an Equation The relationship between two variables can often be expressed in terms of an equation. For example, x + 3y = 7. An ordered pair (a, b) is called a solution to the equation if when x is replaced by a and y is replaced by b, the equation is a true statement. For example, (1, 2) is a solution of the equation above since 1 + (3)(2) = 7. The graph of an equation is the set of all points in the plane whose ordered pairs of coordinates are solutions to the equation.
Sketching Graphs by Plotting Points Given an equation, 1 If possible, rewrite the equation so that one of the variables is isolated on one side of the equation. 2 Make a table of values containing several solution points. 3 Plot these points on a rectangular coordinate system. 4 Connect the points with a smooth curve or line.
Example Sketch the graph of y = 5 2x.
Example Sketch the graph of y = 5 2x. x y = 5 2x (x, y) 2 9 ( 2, 9) 1 7 ( 1, 7) 0 5 (0, 5) 1 3 (1, 3) 2 1 (2, 1)
Graph 10 10 8 8 6 6 4 4 2 2 3 2 1 0 1 2 3 x 3 2 1 1 2 3 x
Example Sketch the graph of y = 5 x 2.
Example Sketch the graph of y = 5 x 2. x y = 5 x 2 (x, y) 3 4 ( 3, 4) 2 1 ( 2, 1) 1 4 ( 1, 4) 0 5 (0, 5) 1 4 (1, 4) 2 1 (2, 1)
Graph 6 6 4 4 2 2 4 3 2 1 1 2 3 x 4 3 2 1 1 2 3 x 2 2 4 4
Intercepts of a Graph Often the graph of an equation will cross one or both of the coordinate axes. The crossing points are called intercepts. If a graph crosses the x-axis, then coordinate y = 0 and the x-intercept has coordinates (x, 0). If a graph crosses the y-axis, then coordinate x = 0 and the y-intercept has coordinates (0, y).
Intercepts of a Graph Often the graph of an equation will cross one or both of the coordinate axes. The crossing points are called intercepts. If a graph crosses the x-axis, then coordinate y = 0 and the x-intercept has coordinates (x, 0). If a graph crosses the y-axis, then coordinate x = 0 and the y-intercept has coordinates (0, y). Finding Intercepts 1 To find x-intercepts, set y = 0 and solve the equation for x. 2 To find y-intercepts, set x = 0 and solve the equation for y.
Example Find the x- and y-intercepts of the graph of the equation y 2 = x + 4.
Example Find the x- and y-intercepts of the graph of the equation y 2 = x + 4. x-intercept: y-intercept: 0 = x + 4 x = 4 y 2 = 0 + 4 y 2 4 = 0 (y + 2)(y 2) = 0 y = 2 or y = 2
Example Find the x- and y-intercepts of the graph of the equation y 2 = x + 4. x-intercept: ( 4, 0) 0 = x + 4 x = 4 y-intercept: (0, 2) and (0, 2) y 2 = 0 + 4 y 2 4 = 0 (y + 2)(y 2) = 0 y = 2 or y = 2
Symmetry Some equations will have graphs which exhibit symmetry with respect to the x-axis, the y-axis, or the origin. x,y x,y x x x, y x,y x,y x-axis symmetry y-axis symmetry origin symmetry x x, y
Graphical Tests for Symmetry 1 A graph is symmetric with respect to the x-axis if, whenever (x, y) is on the graph, (x, y) is also on the graph. 2 A graph is symmetric with respect to the y-axis if, whenever (x, y) is on the graph, ( x, y) is also on the graph. 3 A graph is symmetric with respect to the origin if, whenever (x, y) is on the graph, ( x, y) is also on the graph.
Algebraic Tests for Symmetry 1 The graph of an equation is symmetric with respect to the x-axis if replacing y by y yields an equivalent equation. 2 The graph of an equation is symmetric with respect to the y-axis if replacing x by x yields an equivalent equation. 3 The graph of an equation is symmetric with respect to the origin if replacing x by x and y by y yields an equivalent equation.
Example Determine whether the graph of the following equation is symmetric with respect to the x-axis, the y-axis, or the origin. y = 3x 3 4x
Example Determine whether the graph of the following equation is symmetric with respect to the x-axis, the y-axis, or the origin. y = 3x 3 4x x-axis: y-axis: origin: y = 3x 3 4x y = 3x 3 4x y = 3x 3 4x y = 3( x) 3 4( x) y = 3x 3 + 4x y = 3x 3 4x y = 3( x) 3 4( x) y = 3x 3 + 4x y = 3x 3 4x Not equivalent. Not equivalent. Equivalent.
Example Use the symmetry of the graph of the equation below to help sketch its graph. y = 1 x
Example Use the symmetry of the graph of the equation below to help sketch its graph. y = 1 x Since y = 1 x = 1 x the graph is symmetric about the y-axis. We need only plot points whose x coordinates are greater than or equal to 0 and reflect them across the y-axis to create the entire graph.
Example Use the symmetry of the graph of the equation below to help sketch its graph. y = 1 x Since y = 1 x = 1 x the graph is symmetric about the y-axis. We need only plot points whose x coordinates are greater than or equal to 0 and reflect them across the y-axis to create the entire graph. x y = 1 x (x, y) ( x, y) 0 1 (0, 1) (0, 1) 1 0 (1, 0) ( 1, 0) 2 1 (2, 1) ( 2, 1) 3 2 (3, 2) ( 3, 2) 4 3 (4, 3) ( 4, 3)
Graph 2 1 2 1 1 2 3 4 5 x 4 2 2 4 x 1 1 2 2 3 3 4 4
Circles A circle is the set of all points in the plane a constant distance called the radius of the circle from a fixed point called the center of the circle. If the center of the circle is at the point with coordinates (h, k), if the radius of the circle is r, and if (x, y) is any point on the circle, then by the Distance Formula (x h) 2 + (y k) 2 = r.
Circles A circle is the set of all points in the plane a constant distance called the radius of the circle from a fixed point called the center of the circle. If the center of the circle is at the point with coordinates (h, k), if the radius of the circle is r, and if (x, y) is any point on the circle, then by the Distance Formula (x h) 2 + (y k) 2 = r. Standard Form of the Equation of a Circle The point (x, y) lies on the circle of radius r and center (h, k) if and only if (x h) 2 + (y k) 2 = r 2.
Example The point ( 1, 2) lies on a circle whose center is at (3, 4). Find the standard form of the equation of this circle.
Example The point ( 1, 2) lies on a circle whose center is at (3, 4). Find the standard form of the equation of this circle. The radius of the circle is the distance from ( 1, 2) to (3, 4). r = ( 1 3) 2 + (2 4) 2 = 16 + 4 = 20 If (h, k) = (3, 4) then the equation of the circle is (x h) 2 + (y k) 2 = r 2 (x 3) 2 + (y 4) 2 = ( 20) 2 (x 3) 2 + (y 4) 2 = 20.
Application The resistance y (measured in ohms) of 1000 feet of solid copper wire at 68 F can be approximated by the equation y = 107700 x 2 0.37 where x is the diameter of the wire in millimeters. 1 Sketch a graph of the equation to show the relationship between the diameter of the wire and its resistance. 2 Describe the general relationship between the diameter of the wire and its resistance. 3 Use the graph to estimate the resistance when x = 85 mm.
Solution x y 10 1076.63 20 268.88 1000 30 119.297 800 40 66.9425 50 42.71 600 60 29.5467 70 21.6096 400 80 16.4581 200 90 12.9263 100 10.4 20 40 60 80 100 From the table or the graph we estimate the resistance when x = 85 mm to be approximately 14.5 ohms. x
Homework Read Section 1.2. Exercises: 1, 5, 9, 13,..., 77, 81, 83