4 CHAPTER 1 Graphs 1 Graph Equations by Plotting Points An equation in two variables, say x and y, is a statement in which two expressions involving x and y are equal. The expressions are called the sides of the equation. Since an equation is a statement, it may be true or false, depending on the values of the variables. Any values of x and y that result in a true statement are said to satisfy the equation. For example, the following are all equations in two variables x and y: x y x y y x x y 5 2 6 2 5 2 2 2 + = − = = + = The first of these, x y 5, 2 2 + = is satisfied for x y 1, 2, = = since 1 2 5. 2 2 + = Other choices of x and y, such as x y 1, 2, = − = − also satisfy this equation. It is not satisfied for x 2 = and y 3, = since 2 3 4 9 13 5. 2 2 + = + = ≠ The graph of an equation in two variables x and y consists of the set of points in the xy-plane whose coordinates x y , ( ) satisfy the equation. Graphs play an important role in helping us to visualize the relationships that exist between two variables or quantities. Figure 8 shows the relation between the level of risk in a stock portfolio and the average annual rate of return. The graph shows that, when 30% of a portfolio of stocks is invested in foreign companies, risk is minimized. Lonnie Johnson (1949-present) Lonnie Johnson is mathematician, engineer, and inventor who holds over 120 patents. He worked for the U.S. Air Force, where he helped develop the stealth bomber. He later worked for NASA’s Jet Propulsion program. Johnson is also known for inventing the Super Soaker water gun and a “toy projectile gun,” which eventually became the Nerf Gun. Credit: PJF Military Collection/ Alamy Stock Photo Determining Whether a Point Is on the Graph of an Equation Determine whether each of the following points is on the graph of the equation x y 2 6. − = (a) 2, 3 ( ) (b) 2, 2 ( ) − Solution EXAMPLE 1 (a) For the point 2, 3 , ( ) check to see whether x y 2, 3 = = satisfies the equation x y 2 6. − = x y 2 2 2 3 4 3 1 6 − = ⋅ − = − = ≠ The equation is not satisfied, so the point 2, 3 ( ) is not on the graph of x y 2 6. − = (b) For the point 2, 2 , ( ) − x y 2 2 2 2 4 2 6 ( ) − = ⋅ − − = + = The equation is satisfied, so the point 2, 2 ( ) − is on the graph of x y 2 6. − = Figure 8 Source: T. Rowe Price 18.5 18.5 19.5 17.5 17.5 16.5 16.5 15.5 15.5 14.5 14.5 13.5 13.5 18 18 19 20 17 17 16 16 15 15 14 14 Average Annual Return (%) Level of Risk (%) 10% 40% 50% 30% 60% 70% (30% foreign/70% U.S.) 0% (100% U.S.) 80% 90% 100% (100% foreign) 20% Now Work PROBLEM 27
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