- WELCOME TO LINEAR EQUATIONS AND INTERCEPTS.
THE GOALS OF THIS VIDEO WILL BE TO DETERMINE
IF AN EQUATION IS LINEAR
AND ALSO TO GRAPH A LINEAR EQUATION
BY DETERMINING THE X AND Y-INTERCEPTS.
ANY EQUATION OF THE FORM AX + BY = C,
WHERE "A", B, AND C ARE REAL NUMBERS
AND "A" AND B ARE NOT BOTH 0, IS A LINEAR EQUATION.
AND WE CALL THIS FORM, AX + BY = C,
STANDARD FORM OF A LINEAR EQUATION.
DOWN BELOW HERE YOU CAN SEE,
HERE ARE THREE EXAMPLES OF LINEAR EQUATIONS.
BASED ON THIS DEFINITION, THE REASON THEY'RE LINEAR
IS BECAUSE THEY FIT THIS SPECIAL FORM.
NOTICE THE VALUE OF "A", B, AND C ARE REAL NUMBERS.
THE SAME ON THE SECOND EXAMPLE.
IN THIS THIRD EXAMPLE, IT LOOKS LIKE THE VALUE WOULD BE 0,
WHICH IS OKAY AS LONG "A" AND B ARE NOT EQUAL TO 0.
HERE ARE SOME EXAMPLES OF NON-LINEAR EQUATIONS.
THIS FIRST ONE IS NON-LINEAR
BECAUSE WE HAVE AN X IN THE DENOMINATOR,
WHERE WE HAVE 3 DIVIDED BY X, DOES NOT FIT THIS FORM.
THE SECOND ONE, THE X SQUARED TERM
IS WHAT MAKES THIS NON-LINEAR.
AND THEN IN THIS THIRD EQUATION, THE X x Y,
AGAIN DOES NOT FIT THIS SPECIAL FORM, OR STANDARD FORM,
THEREFORE IT'S NOT A LINEAR EQUATION.
NOW THE MAIN FOCUS OF THIS VIDEO
IS GOING TO BE TO FIND THE INTERCEPTS OF VARIOUS EQUATIONS,
MAINLY LINEAR EQUATIONS.
BUT I WANTED TO SHOW YOU WHAT WE'RE TALKING ABOUT HERE.
ESSENTIALLY WE'RE TRYING TO FIND SPECIFIC POINTS
ON THE GRAPH OF THE EQUATION.
SO FROM THE PREVIOUS VIDEO,
GRAPHING EQUATIONS BY PLOTTING POINTS,
THIS IS JUST A CONTINUATION OF THAT,
BUT WE WANT SPECIFIC POINTS THIS TIME.
NOTICE HERE THE Y-INTERCEPT IS WHERE IT CROSSES THE Y-AXIS.
THE X-INTERCEPT IS WHERE IT CROSSES THE X-AXIS.
IT IS POSSIBLE TO HAVE MORE THAN ONE X-INTERCEPT
OR EVEN MORE THAN ONE Y-INTERCEPT
BASED UPON THE EQUATION.
AND HERE'S JUST A COUPLE OF EXAMPLES
WHERE SOMETIMES YOU MAY NOT HAVE AN X-INTERCEPT.
FOR EXAMPLE, THE CONSTANT FUNCTION, IN THIS CASE Y = -2.
OF COURSE IT CROSSES THE Y-AXIS, BUT NOT THE X-AXIS.
AND IF WE CONSIDER A VERTICAL LINE, X = 3,
THIS HAS AN X-INTERCEPT, BUT NO Y-INTERCEPT.
LET'S START BY LOOKING AT THE X-INTERCEPT.
THE POINT AT WHICH THE GRAPH CROSSES THE X-AXIS
IS CALLED THE X-INTERCEPT.
SO NOTICE IF I PLOT SEVERAL POINTS ON THE X-AXIS,
WHAT YOU'LL NOTICE IS, ALL OF THE Y-COORDINATES
ARE EQUAL TO 0.
AND THIS IS EXACTLY HOW WE ARE GOING TO FIND X-INTERCEPTS.
TO FIND THE X-INTERCEPT OF AN EQUATION'S GRAPH,
REPLACE Y WITH 0 AND SOLVE FOR X.
NOW WHEN WE LOOK AT THE Y-INTERCEPT,
THE POINT AT WHICH THE GRAPH CROSSES THE Y-AXIS,
YOU COULD SEE IF WE PLOT POINTS ALONG THE Y-AXIS,
ALL OF THE X-COORDINATES ARE EQUAL TO 0.
SO TO FIND THE Y-INTERCEPTS OF AN EQUATION'S GRAPH,
WE REPLACE X WITH 0 AND SOLVE FOR Y.
SO LET'S GO AHEAD AND GIVE IT A TRY.
AGAIN IT IS JUST LIKE COMPLETING A T-TABLE,
EXCEPT NOW WE'RE GOING TO HAVE A SPECIALIZED T-TABLE.
WHEN WE SET X EQUAL TO 0 AND SOLVE FOR Y,
WE'RE FINDING THE Y-INTERCEPT.
WHEN WE SET Y EQUAL TO 0 AND SOLVE FOR X,
WE'RE FINDING THE X-INTERCEPT.
SO LET'S TRY IT. LET'S REPLACE X WITH 0.
OF COURSE, 3 x 0 IS 0, SO WE HAVE - 2Y IS EQUAL TO 12.
DIVIDING BY -2, WE HAVE A Y VALUE, OR A Y-INTERCEPT, OF -6.
LET'S GO AHEAD AND PLOT THAT.
BE CAREFUL IF YOU'RE ASKED TO FIND THE Y-INTERCEPT
AND YOU JUST SAY Y = -6.
TECHNICALLY, THIS IS AN EQUATION OF A HORIZONTAL LINE,
SO I LIKE TO LIST IT AS AN ORDERED PAIR.
THIS WOULD BE THE POINT (0,-6).
IN ORDER TO FIND THE X-INTERCEPT,
WE SET Y EQUAL TO 0.
AGAIN, THIS PRODUCT WILL BE 0,
SO WE'RE LEFT WITH 3X EQUAL TO 12.
DIVIDING BY +3, WE HAVE X IS EQUAL TO +4.
OUR X-INTERCEPT IS THE POINT (4,0).
SO BECAUSE THIS IS A LINEAR EQUATION,
IT FITS THE FORM THAT WE JUST DISCUSSED.
THE LINE WE WANT TO GRAPH IS THE LINE THROUGH THESE TWO POINTS.
AND THERE IT IS.
AGAIN, WE HAVE AN X-INTERCEPT OF (4,0)
AND WE HAVE A Y-INTERCEPT OF (0,-6).
LET'S TRY ANOTHER.
LET'S GRAPH Y = -2X + 4 BY DETERMINING THE INTERCEPTS.
THIS IS ANOTHER FORM OF A LINEAR EQUATION
WE HAVEN'T DISCUSSED YET,
SO LET'S STICK TO FINDING THE INTERCEPTS.
SO TO FIND THE Y-INTERCEPT, WE'LL SET X EQUAL TO 0.
THIS PRODUCT IS 0.
WE HAVE Y EQUAL TO 4.
OUR Y- INTERCEPT IS (0,4).
NOW WE'LL REPLACE Y WITH 0 TO FIND THE X-INTERCEPT.
SUBTRACTING 4 ON BOTH SIDES, DIVIDING BY -2,
WE HAVE X IS EQUAL TO 2.
OUR X-INTERCEPT IS THE POINT (2,0).
AGAIN, IF WE WERE TO ADD 2X TO BOTH SIDES OF THIS EQUATION,
WE COULD PUT THIS IN STANDARD FORM,
SO WE KNOW IT'S A LINEAR EQUATION,
SO OUR GRAPH IS A LINE THROUGH THESE TWO POINTS.
AND AGAIN, HERE IT IS.
OUR X-INTERCEPT AND OUR Y-INTERCEPT.
LET'S TAKE A LOOK AT ONE MORE.
WE WANT TO GRAPH Y = X SQUARED - 4 BY DETERMINING THE INTERCEPTS.
RIGHT AWAY I NOTICE BECAUSE OF THE X SQUARED TERM
THIS IS NOT GOING TO BE A LINEAR EQUATION,
BUT WE CAN STILL FIND THE INTERCEPTS.
LET'S START BY FINDING THE Y-INTERCEPT.
WE'LL SET X EQUAL TO 0.
SO OUR Y-INTERCEPT IS -4.
TO FIND THE X-INTERCEPTS, WE'LL SET Y EQUAL TO 0.
NOW HERE WE HAVE A DEGREE 2 EQUATION.
THERE'S A COUPLE OF WAYS OF DOING THIS.
THIS IS FACTORABLE, OR WE COULD JUST ISOLATE THE X SQUARED TERM.
I'M GOING TO GO AHEAD AND ADD 4 TO BOTH SIDES.
SO WHAT NUMBER SQUARED WILL GIVE US +4?
WE CAN TAKE THE SQUARE ROOT OF BOTH SIDES
AND WHAT WE'LL FIND IS X IS EQUAL TO
A POSITIVE OR A NEGATIVE 2.
REMEMBER WE HAD X SQUARED = 4.
-2 SQUARED WOULD ALSO GIVE US 4.
SO IT LOOKS LIKE WE HAVE TWO X- INTERCEPTS.
WE HAVE THE POINT (2,0) AND THE POINT (--2,0).
NOW THESE THREE POINTS MAY NOT BE ENOUGH POINTS
TO MAKE A NICE GRAPH.
IF WE NEEDED TO FIND ADDITIONAL POINTS,
WE WOULD JUST MAKE A LARGER T-TABLE.
BUT WE HAVE FOUND THE X- INTERCEPTS AND THE Y-INTERCEPT.
SO LET'S GO AND LOOK AT THE GRAPH.
AND HERE IT IS. THIS SHAPE IS CALLED A PARABOLA.
WE HAVE TWO X-INTERCEPTS AND ONE Y-INTERCEPT.
OKAY, I HOPE THAT HELPS EXPLAIN
HOW TO FIND THE INTERCEPTS OF AN EQUATION
AND ALSO TO IDENTIFY LINEAR EQUATIONS.
THANK YOU FOR WATCHING AND HAVE A GOOD DAY.