- LET'S SAY YOU HAVE AN EQUATION
AND YOU'RE ASKED TO GRAPH IT.
WELL, THE GOAL OF THIS VIDEO IS TO SHOW YOU
HOW TO GRAPH VARIOUS EQUATIONS
BY PLOTTING POINTS ON THE COORDINATE PLANE.
BEFORE WE ACTUALLY DO SOME GRAPHING OURSELF,
I'D LIKE TO TAKE A LOOK AT SOME GRAPHS THAT CAN BE OBTAINED
BY GRAPHING EQUATIONS.
ALL OF THESE INTERESTING GRAPHS WERE OBTAINED
BY GRAPHING VARIOUS EQUATIONS BY PLOTTING POINTS
ON THE COORDINATE PLANE.
SOME OF THESE ARE BEYOND THE SCOPE OF THIS COURSE,
BUT I DID WANT TO SHOW YOU,
YOU CAN GET SOME VERY INTERESTING GRAPHS.
OKAY, LET'S TAKE A LOOK AT A DEFINITION BEFORE WE START.
THE GRAPH OF AN EQUATION IS A DRAWING THAT REPRESENTS
ALL OF THE ORDERED PAIRS THAT ARE SOLUTIONS OF THE EQUATION.
AND REMEMBER THAT AN ORDERED PAIR
CONSISTS OF TWO COORDINATES,
AN X COORDINATE AND A Y COORDINATE.
SO WE'RE GOING TO FIND VALUES FOR X AND Y
THAT MAKE A GIVEN EQUATION TRUE
AND THEN WE'RE GOING TO PLOT THOSE POINTS
ON THE COORDINATE PLANE.
LET'S TAKE A LOOK AT AN EXAMPLE.
WE WANT TO GRAPH Y = 2X + 1 BY PLOTTING POINTS.
SO REALLY THE MAIN STEP HERE IS WE'RE GOING TO PICK X VALUES,
SUBSTITUTE THOSE VALUES INTO THE EQUATION,
AND FIND THE CORRESPONDING Y VALUE.
WE CAN PICK ANY X VALUE THAT WE WANT.
I'M GOING TO PICK -2, -1, 0, 1, AND 2.
SO I'M GOING TO REPLACE X WITH - 2 IN THE EQUATION.
SO INSTEAD OF WRITING 2X, X IS NOW -2,
AND WHEN I SIMPLIFY THIS, I'M GOING TO GET A -3.
OKAY, SO THIS REPRESENTS ONE SOLUTION
TO THAT EQUATION OF THE ORDERED PAIR OF -2 AND -3
AND WE COULD DO THIS FOREVER.
WE'RE ONLY GOING TO FIND, IN THIS CASE, 5 POINTS
AND THEN DRAW SOME CONCLUSIONS ABOUT THE GRAPH.
WHEN X IS -1, WE'RE GOING TO FOLLOW THIS SAME PROCEDURE
AND YOU CAN CHECK ME, BUT Y WOULD BE -1.
OKAY, AND NOW WE'RE GOING TO LET X = 0,
WOULD RESULT IN Y VALUE OF 1, ORDERED PAIR (0, 1),
WHEN X IS 1, Y IS 3, AND WHEN X = 2, Y = 5.
OKAY. SO EACH OF THESE ORDERED PAIRS
REPRESENTS A SOLUTION TO THAT EQUATION,
AND AS I MENTIONED BEFORE, THERE ARE MANY, MANY MORE.
LET'S PLOT THESE POINTS AND SEE WHAT WE GET.
-2, -3, -1, -1, 0, 1, 1, 3, AND (2, 5).
AS YOU CAN SEE, THE EQUATION DOES RESULT
IN A LINEAR EQUATION, OR A LINEAR GRAPH.
OKAY. THAT'S ESSENTIALLY WHAT WE'RE GOING TO DO
WITH A VARIETY OF DIFFERENT TYPES OF EQUATIONS.
LET'S TAKE A LOOK AT ANOTHER EXAMPLE.
WE WANT TO GRAPH 3X + 5Y = 10.
SINCE WE ARE GOING TO CONTINUE WITH PICKING X VALUES
AND SUBBING THEM IN TO FIND Y,
SOMETIMES IT IS HELPFUL TO SOLVE THIS EQUATION FOR Y.
SO IF I WAS GOING TO DO THAT I WOULD SUBTRACT 3X ON BOTH SIDES.
AND THEN DIVIDE EVERYTHING BY 5.
SO THE EQUIVALENT EQUATION THAT WE ARE GOING TO USE
TO FILL OUT THIS TABLE IS Y = -3/5X + 2.
THE REASON I DID THIS IS BECAUSE NOW IT SHOULD BE EASIER
TO FIND THE CORRESPONDING Y VALUE.
ONE WORD OF CAUTION,
SINCE THE COEFFICIENT OF X IS IN FRACTION FORM
IT'S EASIER TO DO THIS
IF WE PICK MULTIPLES OF THE DENOMINATOR,
IN THIS CASE MULTIPLES OF 5.
SO I'M GOING TO PICK AN X VALUES OF -5, 0, 5.
I COULD PICK 10, BUT 10 WOULD BE
OFF THE COORDINATE PLANE THAT I'VE DRAWN.
OKAY. SO LET'S GO AHEAD AND SEE WHAT THIS WOULD LOOK LIKE.
FIRST I'M GOING TO REPLACE X WITH -5.
NOW, SINCE THIS IS IN FRACTION FORM
I'M GOING TO PUT THE -5 IN FRACTION FORM
AND WHAT HAPPENS IS,
THAT NOTICE HOW WE HAVE A 5 IN THE DENOMINATOR,
A -5 IN THE NUMERATOR, AND THAT SIMPLIFIES NICELY,
AND THE RESULT OF THIS WOULD GIVE US Y = 5.
OKAY, TO SAVE SOME TIME I'M GOING TO GO AHEAD
AND FILL OUT THE REST OF THESE.
WHEN X = 0, Y = 2, WHEN X = 5, Y = -1.
AND I'M GOING TO LEAVE IT AT THESE THREE POINTS.
SO YOU CAN SEE THE ORDERED PAIRS ARE GOING TO FOLLOW.
LET'S GO AHEAD AND JUST PLOT THOSE,
(-5, 5) (0, 2) AND (5, -1),
AND IT IS TRUE WE HAVE ANOTHER LINEAR EQUATION HERE
GOING THROUGH THOSE THREE POINTS AND THE GRAPH IS A LINE.
OKAY, I DO WANT TO TAKE A MOMENT HERE
AND SHOW YOU SOMETHING WITH THE GRAPHING CALCULATOR.
OKAY. SO IF YOU'RE AT THE HOME SCREEN
OF THE GRAPHING CALCULATOR
AND YOU WANT TO ENTER IN AN EQUATION,
BASICALLY IT HAS TO BE SOLVED FOR Y.
SO WE ARE GOING TO USE THE EQUATION IN THIS FORM.
OKAY, LET'S GO BACK TO THE CALCULATOR.
I'M GOING TO HIT Y =, THE TOP LEFT,
AND I'M GOING ENTER IN THIS EQUATION.
ONE WORD OF CAUTION,
SINCE THE COEFFICIENT OF X IS IN FRACTION FORM,
WE SHOULD PUT THAT IN A SET OF PARENTHESIS
TO BE SURE THE CALCULATOR READS THIS PROPERLY, AND +2.
OKAY, NOW, THERE'S TWO WAYS TO CHECK.
IF WE HIT GRAPH, WE DO GET A NICE VISUAL
THAT OUR LINE LOOKS GOOD,
BUT MORE SPECIFICALLY WE WANT TO KNOW IF OUR POINTS ARE CORRECT.
SO IF I HIT SECOND GRAPH, I GET A "T" TABLE
SIMILAR TO THE ONE THAT I HAVE ON THE SCREEN.
HOWEVER, YOU CAN SEE THAT WHEN WE DIDN'T PICK MULTIPLES OF 5,
NOTICE HOW THE Y COORDINATES ARE IN DECIMAL FORM,
WHICH WOULD MAKE THEM MORE DIFFICULT TO GRAPH.
YOU CAN SCROLL UP AND DOWN HERE
AND YOU CAN SEE, THERE'S OUR (-5, 5).
OKAY. I HOPE THAT HELPS.
I'M GOING TO CONCLUDE THIS VIDEO RIGHT NOW.
THIS IS PART ONE OF TWO PARTS.
WE'LL TAKE A LOOK AT A COUPLE OF OTHER EQUATIONS
IN THE NEXT VIDEO THAT ARE NOT LINEAR.