Graphing Quadratic Equations

Transcription

1 .4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph first-degree equations. Similar methods will allow ou to graph quadratic equations of the form a b c a The first thing ou will notice is that the graph of an equation in this form is not a straight line. The graph is alwas the curve called a parabola. Here are some eamples: McGraw-Hill Companies To graph quadratic equations, start b finding solutions for the equation. We begin b completing a table of values. This is done b choosing an convenient values for. Then use the given equation to compute the corresponding values for, as Eample illustrates. Eample Completing a Table of Values If, complete the ordered pairs to form solutions. Then show these results in a table of values. (, ), (, ), (, ), (, ), (, ) 777

2 778 CHAPTER QUADRATIC EQUATIONS For eample, to complete the pair (, ), substitute for in the given equation. ( ) 4 NOTE Remember that a solution is a pair of values that makes the equation a true statement. So (, 4) is a solution. Substituting the other values for in the same manner, we have the following table of values for : 4 4 CHECK YOURSELF If, complete the ordered pairs to form solutions and form a table of values. (, ), (, ), (, ), (, ), (, ) We can now plot points in the cartesian coordinate sstem that correspond to solutions to the equation. Eample Plotting Some Solution Points Plot the points from the table of values corresponding to from Eample. 4 4 (, 4) (, ) (, ) (, ) Notice that the ais acts as a mirror. Do ou see that an point graphed in quadrant I will be reflected in quadrant II? (, 4) McGraw-Hill Companies

3 GRAPHING QUADRATIC EQUATIONS SECTION CHECK YOURSELF Plot the points from the table of values formed in Check Yourself. The graph of the equation can be drawn b joining the points with a smooth curve. Eample 3 Completing the Graph of the Solution Set Draw the graph of. We can now draw a smooth curve between the points found in Eample to form the graph of. NOTE As we mentioned earlier, the graph must be the curve called a parabola. NOTE Notice that a parabola does not come to a point. CHECK YOURSELF 3 Draw a smooth curve between the points plotted in the Check Yourself eercise. McGraw-Hill Companies

4 78 CHAPTER QUADRATIC EQUATIONS You can use an convenient values for in forming our table of values. You should use as man pairs as are necessar to get the correct shape of the graph (a parabola). Eample 4 Graphing the Solution Set Graph. Use values of between and 3. First, determine solutions for the equation. For instance, if, () () 3 then (, 3) is a solution for the given equation. Substituting the other values for, we can form the table of values shown below. We then plot the corresponding points and draw a smooth curve to form our graph. The graph of. NOTE An values can be substituted for in the original equation CHECK YOURSELF 4 Graph 4. Use values of between 4 and. McGraw-Hill Companies

5 GRAPHING QUADRATIC EQUATIONS SECTION.4 78 Choosing values for is also a valid method of graphing a quadratic equation that contains a constant term. Eample 5 Graphing the Solution Set Graph. Use values of between and 3. We ll show the computation for two of the solutions. If : If 3: ( ) ( ) You should substitute the remaining values for into the given equation to verif the other solutions shown in the table of values below. The graph of CHECK YOURSELF 5 Graph 4 3. Use values of between and 4. McGraw-Hill Companies In Eample 6, the graph looks significantl different from previous graphs.

6 78 CHAPTER QUADRATIC EQUATIONS Eample 6 Graphing the Solution Set Graph 3. Use values between and. Again we ll show two computations. NOTE ( ) 4 If : If : ( ) 3 () Verif the remainder of the solutions shown in the table of values below for ourself. The graph of 3. 3 There is an important difference between this graph and the others we have seen. This time the parabola opens downward! Can ou guess wh? The answer is in the coefficient of the term. If the coefficient of is positive, the parabola opens upward. The coefficient of is positive. McGraw-Hill Companies

7 GRAPHING QUADRATIC EQUATIONS SECTION If the coefficient of is negative, the parabola opens downward. The coefficient of is negative. CHECK YOURSELF 6 Graph. Use values between 3 and. There are two other terms we would like to introduce before closing this section on graphing quadratic equations. As ou ma have noticed, all the parabolas that we graphed are smmetric about a vertical line.this is called the ais of smmetr for the parabola. The point at which the parabola intersects that vertical line (this will be the lowest or the highest point on the parabola) is called the verte. You ll learn more about finding the ais of smmetr and the verte of a parabola in our net course in algebra. The ais of smmetr McGraw-Hill Companies The verte

8 784 CHAPTER QUADRATIC EQUATIONS CHECK YOURSELF ANSWERS McGraw-Hill Companies

9 Name.4 Eercises Section Date Graph each of the following quadratic equations after completing the given table of values.. ANSWERS McGraw-Hill Companies 785

10 ANSWERS McGraw-Hill Companies 786

11 ANSWERS McGraw-Hill Companies

12 ANSWERS McGraw-Hill Companies 788

13 ANSWERS Match each graph with the correct equation on the right (a) (b) (c) 4.. (d) (e) 3 (f) (g) (h) McGraw-Hill Companies 789

14 Answers f 9. a. b 3. e McGraw-Hill Companies 79