- WE WILL CONTINUE OUR REVIEW OF GRAPHING FUNCTIONS.
WE'LL BE LOOKING AT GRAPHING NONLINEAR FUNCTIONS TODAY.
MORE SPECIFICALLY, OUR GOAL IS TO GRAPH
ABSOLUTE VALUE FUNCTIONS
AND GRAPH SQUARE ROOT FUNCTIONS.
LET'S TAKE A LOOK AT THE MOST BASIC ABSOLUTE VALUE FUNCTION,
F OF X = ABSOLUTE VALUE OF X.
REMEMBER THE ABSOLUTE VALUE OF A NUMBER
IS A NUMBER'S DISTANCE FROM 0,
AND REMEMBER DISTANCE IS ALWAYS POSITIVE.
LET'S GO AHEAD AND SELECT SOME VALUES OF X TO USE.
THE ABSOLUTE VALUE OF X IS ALWAYS POSITIVE.
THEREFORE THE CORRESPONDING Y VALUES WILL BE AS FOLLOWS.
LET'S GO AHEAD AND GRAPH THESE FIVE POINTS, -3,3, -1,1, 0,0,
1,1 AND 3,3,
AND AS YOU CAN SEE, THIS GRAPH HAS A V SHAPE TO IT,
AND ABSOLUTE VALUE FUNCTION GRAPHS
ARE ALWAYS GOING TO BE IN THE SHAPE OF A V.
THEY MAY BE WIDER OR NARROWER, THEY MAY BE SHIFTED UP OR DOWN,
BUT THEY WILL ALWAYS BE IN THE SHAPE OF A V.
GRAPHS OF THE ABSOLUTE VALUE FUNCTION FORM F OF X = "A"
x ABSOLUTE VALUE OF THE QUANTITY BX + C + D
CAN HAVE A VARIETY OF TRANSFORMATIONS
BASED UPON THE VALUES OF "A", B, C, AND D.
LET'S TAKE A LOOK AT THIS.
IN RED WE HAVE THE FUNCTION
DESCRIBED ON THE PREVIOUS SCREEN,
AND BELOW, IN BLUE, WE HAVE THE VALUES OF "A", B, C, AND D.
THIS SLIDER ALLOWS US TO CHANGE THE VALUES OF EACH VARIABLE,
AND SEE HOW IT AFFECTS THE GRAPH.
LET'S TAKE A LOOK AT HOW THE VALUE OF "A" AFFECTS THE GRAPH.
YOU CAN SEE AS WE INCREASE THE VALUE OF "A"
THE GRAPH BECOMES NARROWER, OR STRETCHED, VERTICALLY.
AS "A" BECOMES NEGATIVE, NOW THE V OPENS DOWNWARD.
IF I SWITCH TO THE VALUE OF D, AS IN PREVIOUS EXAMPLES,
THE VALUE OF D DOES SHIFT THE GRAPH UP OR DOWN.
THE VALUE OF C APPEARS TO SHIFT THE GRAPH EITHER LEFT OR RIGHT.
I WOULD ENCOURAGE YOU TO SPEND SOME TIME
EXPLORING HOW THESE VALUES AFFECT THE GRAPH.
LET'S GO BACK TO OUR PRESENTATION,
AND LET'S NOW TAKE A LOOK AT THE MOST BASIC SQUARE ROOT FUNCTION
F OF X = THE SQUARE ROOT OF X.
WHEN I SELECT THE X VALUES THIS TIME I AM GOING TO PICK X VALUES
THAT WILL BE EASY TO TAKE THE SQUARE ROOT OF,
FOR EXAMPLE, 0, 1, 4, 9,
AND I'M GOING TO GO AHEAD AND SELECT -1.
LET'S TAKE THE SQUARE ROOT OF OUR X VALUES.
SQUARE ROOT OF 0 WOULD BE 0.
SQUARE ROOT OF 1, 1. SQUARE ROOT OF 4 IS 2.
SQUARE ROOT OF 9 IS 3.
NOW, THE SQUARE ROOT OF -1 IS ACTUALLY "I",
THE IMAGINARY UNIT,
AND SINCE OUR X AND Y AXES ARE REAL NUMBER LINES
IN THIS GRAPH,
THE Y VALUE WOULD BE UNDEFINED.
LET'S GO AHEAD AND PLOT THE POINTS THAT WE DO HAVE.
0,0, 1,1, 4,2, AND UNFORTUNATELY 9,3 IS OFF THE SCREEN.
BUT WE CAN TELL FROM THE THREE POINTS THAT WE HAVE
THAT THE GRAPH OF THE ABSOLUTE VALUE OF X
IS EQUAL TO SOMETHING THAT LOOKS LIKE THIS.
NOW, THESE GRAPHS CAN GET MUCH MORE COMPLICATED.
FOR EXAMPLE, IF WE HAVE THE SQUARE ROOT FUNCTION
IN THE FORM OF "A" x SQUARE ROOT OF THE QUANTITY BX + C + D,
THE VALUES OF "A", B, C, AND D
CAN HAVE VARIOUS TRANSFORMATIONS ON THE GRAPH.
OF COURSE IN MANY CASES
THE GRAPHING CALCULATOR CAN BE HELPFUL.
I DO HAVE AN EXPLANATION I WOULD LIKE TO SHOW YOU.
AGAIN, IN RED WE HAVE OUR FUNCTION,
AND IN BLUE WE HAVE THE VALUES OF "A", B, C, AND D.
WE'RE ABLE TO CHANGE THESE VALUES
BY USING THE SLIDER BELOW.
IF I CHANGE THE VALUE OF "A"
YOU CAN SEE IT DOES HAVE AN AFFECT ON THE GRAPH,
AND IN FACT IT DOES SEEM TO DETERMINE
WHETHER THE GRAPH POINTS UPWARD OR POINTS DOWNWARD.
IF I CHANGE THE VALUE OF D JUST AS BEFORE,
IT SHIFTS THE FUNCTION EITHER UP OR DOWN.
THE VALUE OF C SHIFTS THE FUNCTION LEFT OR RIGHT,
AND THE VALUE OF B
ALSO SEEMS TO DETERMINE WHETHER THE GRAPH
WILL POINT TO THE RIGHT OR POINT TO THE LEFT.
THERE'S A LOT GOING ON HERE,
BUT THAT'S KIND OF A NICE WAY
TO EXPLORE HOW THESE VALUES AFFECT THE GRAPH.
THIS WAS A QUICK REVIEW
OF HOW TO GRAPH ABSOLUTE VALUE FUNCTIONS
AND SQUARE ROOT FUNCTIONS.
I HOPE IT HELPS, AND I HOPE YOU HAVE A GOOD DAY.