- IN THIS QUESTION, WE'RE ASKED TO MATCH THE GRAPH ON THE LEFT
WITH THE CORRECT FUNCTION ON THE RIGHT.
AND WE'RE TOLD THE BLACK GRAPH IS THE PARENT FUNCTION.
SO THIS BLACK GRAPH HERE IS OUR PARENT FUNCTION,
AND WE SHOULD RECOGNIZE THIS AS A SQUARE ROOT FUNCTION
SO OUR PARENT FUNCTION IS F OF X = THE SQUARE ROOT OF X.
AND THIS QUESTION IS A REVIEW OF REFLECTIONS
ACROSS THE X AND Y AXIS
AS WELL AS HORIZONTAL STRETCHES AND COMPRESSIONS.
SO WE TAKE A LOOK AT THE BLACK GRAPH
AND COMPARE IT TO THE GREEN GRAPH,
NOTICE HOW IT'S A REFLECTION ACROSS THE X AXIS
AND IF WE COMPARE THE BLACK GRAPH TO THE RED GRAPH,
NOTICE HOW IT'S A REFLECTION ACROSS THE Y AXIS.
AND THEN COMPARING THE BLACK GRAPH
TO THE BLUE AND PINK FUNCTION,
IT IS SOMETIMES HARD TO TELL IF THE TRANSFORMATION
IS A VERTICAL STRETCHER COMPRESSION
OR A HORIZONTAL STRETCHER COMPRESSION
BUT IN THIS CASE WE KNOW IT'S GOING TO BE A HORIZONTAL
STRETCHER COMPRESSION.
SO THE BLUE GRAPH IS A HORIZONTAL COMPRESSION
OF THE BLACK GRAPH, OR IT'S COMPRESSED IN THIS DIRECTION
AND THE PINK GRAPH IS A HORIZONTAL STRETCH
OF THE BLACK GRAPH.
SO BEFORE WE MATCH THESE UP,
LET'S DO A QUICK REVIEW ON REFLECTIONS
AND HORIZONTAL STRETCHES AND COMPRESSIONS.
Y = -F OF X WHERE THE OPPOSITE OF F OF X
REFLECTS F OF X ACROSS THE X AXIS.
SO IF OUR PARENT FUNCTION IS F OF X = THE SQUARE ROOT OF X,
AND WE GRAPH THE OPPOSITE OF F OF X OR NEGATIVE SQUARE ROOT X,
NOTICE HOW IT REFLECTS THE PARENT FUNCTION
ACROSS THE X AXIS.
ONE WAY TO REMEMBER THIS IS
BY CHANGING THE SIGN OF THE FUNCTION,
REMEMBER THE FUNCTION VALUES ARE Y VALUES
SO WE'RE CHANGING THE SIGN OF THE Y COORDINATES
THEREFORE REFLECTING THE FUNCTION ACROSS THE X AXIS.
IF WE HAVE Y = F OF -X,
THIS REFLECTS THE PARENT FUNCTION ACROSS THE Y AXIS.
NOTICE IN THIS CASE
WE'RE CHANGING THE SIGN OF THE X COORDINATES.
SO IF WE HAVE F OF -X OR SQUARE ROOT OF -X,
NOTICE HOW IF WE CHANGE THE SIGN OF THE X COORDINATES,
IT WILL REFLECT THE PARENT FUNCTION ACROSS THE Y AXIS.
OKAY, NOW FOR THE HORIZONTAL STRETCHES AND COMPRESSIONS,
THIS IS A LITTLE BIT MORE INVOLVED.
IF Y = F OF B x X, WHERE B IS GREATER THAN 1,
THIS WILL COMPRESS THE GRAPH OF THE PARENT FUNCTION
HORIZONTALLY BY A FACTOR OF 1/B.
SO IF WE HAVE G OF X = F OF 2 x X,
WHICH WOULD BE THE SQUARE ROOT OF 2X,
THIS WILL BE A HORIZONTAL COMPRESSION
OR IT WILL COMPRESS THE BLUE GRAPH HORIZONTALLY
SO IT LOOKS LIKE THE GREEN GRAPH.
AND THEN IF WE HAVE Y = F OF BX WHERE B IS BETWEEN 0 AND 1,
THIS WILL STRETCH THE GRAPH HORIZONTALLY
BY A FACTOR OF 1/B.
SO FOR EXAMPLE IF WE HAVE H OF X = F OF 1/2X,
WHICH WOULD BE THE SQUARE ROOT OF 1/2X,
OR THE SQUARE ROOT OF 0.5X,
THIS WOULD BE HORIZONTAL STRETCH
SO WE WOULD TAKE THE BLUE GRAPH
AND STRETCH IT HORIZONTALLY IN THIS DIRECTION
FORMING THE RED GRAPH.
SO GOING BACK TO OUR EXAMPLE,
LET'S START WITH THE REFLECTIONS ACROSS THE X AND Y AXIS
WHICH WE SHOULD RECOGNIZE
WOULD BE FUNCTION "A" OR FUNCTION B.
LET'S LOOK AT B FIRST.
IF G OF X THE SQUARE ROOT OF -X,
AGAIN WE'RE CHANGING THE SIGN OF THE X COORDINATES
WHICH MEANS IT WOULD REFLECT THE BLACK GRAPH ACROSS THE Y AXIS.
SO B WOULD BE THE RED GRAPH.
THEREFORE FOR G OF X = THE OPPOSITE
OR NEGATIVE SQUARE ROOT X,
WE'D BE CHANGING THE FUNCTION VALUES OR THE Y VALUES
THEREFORE IT'S GOING TO REFLECT THE BLACK GRAPH
ACROSS THE X AXIS
SO THE GRAPH OF THIS FUNCTION WOULD BE THE GREEN GRAPH.
SO NOW WE'RE LEFT WITH C AND D
WHICH WILL BE HORIZONTAL STRETCHES OR COMPRESSIONS.
WELL WHEN THE COEFFICIENT OF THE X HERE IS GREATER THAN 1,
IT'S GOING TO BE A HORIZONTAL COMPRESSION
WHICH MEANS IT WILL TAKE THE BLACK GRAPH HERE
AND COMPRESS IT HORIZONTALLY FORMING THIS BLUE GRAPH.
AND IN COMPARISON WHEN WE HAVE G OF X = THE SQUARE ROOT OF 0.25X,
NOTICE THAT B IS BETWEEN 0 AND 1,
THIS WILL BE A HORIZONTAL STRETCH
WHICH IN THIS CASE IF WE TAKE THE BLACK GRAPH
AND STRETCH IT HORIZONTALLY,
WE WOULD HAVE THE PINK GRAPH.
WELL I DO REALIZE YOU COULD JUST GRAPH THESE
ON THE GRAPHING CALCULATOR AND THEN MATCH THEM,
THE IDEA HERE IS TO BE ABLE TO RECOGNIZE
THE TYPE OF TRANSFORMATION JUST BY LOOKING AT THE FUNCTION.
OKAY, I HOPE YOU FOUND THIS HELPFUL.