Graphs of Equations. A coordinate system is a way to graphically show the relationship between 2 quantities.

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1 Gaphs of Equations CHAT Pe-Calculus A coodinate sstem is a wa to gaphicall show the elationship between quantities. Definition: A solution of an equation in two vaiables and is an odeed pai (a, b) such that when is eplaced b a and is eplaced b b, the esulting equation is a tue statement. The gaph of an equation of this tpe is the collection of all points in the ectangula coodinate sstem that coespond to the solution of the equation. Sketching a Gaph To sketch the gaph of an equation in two vaiables using the point-plotting method, constuct a table of values that consists of seveal solution points of the equation. Plot the solution points on a ectangula coodinate sstem and connect the points with a smooth cuve. Note: A disadvantage of the point-plotting method is that with too few solution points, ou can badl misepesent the gaph of an equation. 1

2 Eample: Sketch the gaph of 1. CHAT Pe-Calculus Solution: Make a table of odeed pais Plot the points and connect them. Eample: Sketch the gaph of 1. Solution:

3 Note: The gaphing utilit does not put aows on the ends of lines and cuves, but the should be thee to show that the gaph goes on infinitel. Intecepts of Gaphs Definition of Intecepts A point at which the gaph of an equation meets the - ais is called an -intecept. It is of the fom (, 0). To find the -intecept: Let = 0 and solve fo. A point at which the gaph of an equation meets the - ais is called an -intecept. It is of the fom (0, ). To find the -intecept: Let = 0 and solve fo.

4 Eample: Find the - and -intecepts of the gaph of Solution: Fo the -intecept, let = 0 and solve fo. (0) Fo the -intecept, let = 0 and solve fo. (0) This means that the points (, 0) and (0, ) ae points on ou gaph. We could find moe points b making a table. 4

5 5 Eample: Find the intecepts of the gaph of Solution: Fo the -intecept, let = 0 and solve fo. 0 0 Fo the -intecept, let = 0 and solve fo. ) )( ( 0 0 o This means that the points (0, -), (-, 0) and (, 0) ae points on ou gaph. We could find moe points b making a table.

6 Using a Gaphing Calculato to Gaph Equations To gaph an equation, do the following: CHAT Pe-Calculus 1. Pess [MODE] and then on the 4 th line down, make sue that [Func] is highlighted.. Pess [ nd ] [QUIT] to get back to the main sceen.. Pess [Y=] to get to the equation edito. Ente ou equation. Note: The equation must be of the fom =. If it is not, solve it fist fo and then ente it in the equation edito. Pess [GRAPH]. To change the window settings, do one of the following: 1. Pess [WINDOW] and manuall set ou anges fo and.. Pess [ZOOM] and then [ZoomFit]. If ou gaph is not appeaing on ou sceen, this should bing at least pat of it into view. Then pess [ZOOM] [Zoom Out] [ENTER] o [ZOOM] [Zoom In] [ENTER] to see ou gaph moe accuatel. (You can move the cuso to the cente of the aea ou want to zoom befoe ou hit [ENTER]. Use the aow kes.). Use the Zoom Bo. To do this, pess [ZOOM] [ZBo] [ENTER]. Use the ight and left aows to move the cuso to the cone of the bo that ou will daw aound the aea ou want to include in ou viewing window. Pess [ENTER]. Then using the aow kes, ceate ou bo and then pess [ENTER] again.

7 *To get the window back to the default standad settings, pess [ZOOM] [ZStandad]. Appoimating Points on a Gaph To appoimate a point on the gaph on ou calculato, fist have the gaph in ou viewing window. Pess [TRACE]. A blinking cuso will appea on ou cuve. Using the left and ight aows, ou can move the cuso along ou cuve. The coodinates will appea at the bottom of the sceen. Finding Intecepts Using a Gaphing Calculato To find the -intecept, do the following: 1. Have the gaph in ou viewing window.. Use the [CALC] featue b pessing [ nd ] [TRACE] and then choose [zeo].. It will ask ou fo a left bounda fist. Use the left and ight aow kes to close in on the aea whee it cosses the -ais. Place the cuso on the cuve to the left of the point whee it cosses. Pess [ENTER]. 4. Use the ight aow ke to move the cuso on the cuve to the ight of the point whee it cosses the - ais. Pess [ENTER]. 5. When ou see Guess?, pess [ENTER] again to find the value. 7

8 To find the -intecept, do the following: 1. Have the gaph in ou viewing window.. Pess [ nd ] [TRACE] [value].. It will ask ou fo the -coodinate. Ente 0 on ou keboad and then pess [ENTER]. The - coodinate will appea on the sceen net to the - coodinate. *Note: This pocess can be used to find an -coodinate if ou know the -coodinate. 8

9 Smmet Smmet is anothe tool that helps us when we gaph. We will look at tpes of smmet: (-, ) (, ) (, ) (, -) -ais smmet -ais smmet (, ) (-, -) oigin smmet 9

10 Gaphical Tests fo Smmet A gaph is smmetic with espect to the -ais if, wheneve (, ) is on the gaph, (-, ) is also on the gaph. A gaph is smmetic with espect to the -ais if, wheneve (, ) is on the gaph, (, -) is also on the gaph. A gaph is smmetic with espect to the oigin if, wheneve (, ) is on the gaph, (-, -) is also on the gaph. Testing fo Smmet The gaph of an equation is smmetic with espect to the -ais if eplacing with - ields an equivalent equation. The gaph of an equation is smmetic with espect to the -ais if eplacing with - ields an equivalent equation. The gaph of an equation is smmetic with espect to the oigin if eplacing with and with ields an equivalent equation. 10

11 Eample: What kind of smmet does the gaph of have? Solution: 1. Test fo -ais smmet: Substitute. ( ) not equivalent, so no -ais smmet. Test fo -ais smmet: Substitute. not equivalent, so no -ais smmet. Test fo oigin smmet: Substitute and -. ( ) This is an equivalent equation, so thee is oigin smmet. 11

12 Eample: What kind of smmet does the gaph of have? Solution: 1. Test fo -ais smmet: Substitute. ( ) equivalent, so thee is -ais smmet. Test fo -ais smmet: Substitute. not equivalent, so no -ais smmet. Test fo oigin smmet: Substitute and -. ( ) not equivalent, so no oigin smmet. 1

13 *We can use smmet to educe the amount of points we need to plot fo a gaph. Cicles A cicle is the set of all points that ae a fied distance fom a fied point. The fied point is the cente, and the given distance is called the adius. (h, k) (, ) Using the distance fomula we get: ( h) ( k) Squaing both sides gives us the standad equation fo a cicle. ( h) ( k) 1

14 Standad Equation of a Cicle The point (, ) lies on the cicle of adius and cente (h, k) if and onl if ( h) ( k) CHAT Pe-Calculus Eample: Find the standad fom of the equation of the cicle with cente at (, -5) and adius 4. ( h) ( k) ( ) ( 5) 4 ( ) ( 5) 1 Eample: Find the standad fom of the equation of the cicle with cente at (, -) and which passes though the point (-1, 1). Solution: Fist find using the distance fomula. ( h) ( k) ( 1 ) (1 )

15 Now put the adius and cente into the standad equation. ( h) ( ) ( ( k) ) 5 15

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