- WE WANT TO FIND THE X AND Y INTERCEPTS
OF THE QUADRATIC FUNCTION F OF X = X SQUARED +2X - 9.
TO FIND THE X INTERCEPTS OF ANY FUNCTION,
WE SET Y OR F OF X EQUAL TO ZERO AND SOLVE FOR X.
SO IF WE SUBSTITUTE ZERO FOR F OF X,
WE WOULD HAVE THE EQUATION 0 = X SQUARED + 2X - 9,
BUT THIS DOES NOT FACTOR SO WE CAN'T SOLVE THIS BY FACTORING.
SO WE'LL HAVE TO SOLVE THIS
USING THE QUADRATIC FORMULA GIVEN HERE
WHERE "A" WOULD BE EQUAL TO 1, B WOULD BE EQUAL TO POSITIVE 2
AND C WOULD BE EQUAL TO -9.
SO OUR SOLUTIONS WOULD BE X = -B OR -2
PLUS OR MINUS THE SQUARE ROOT OF B SQUARED,
WHICH WOULD BE 2 SQUARED
- 4 x "A", WHICH IS 1 x C, WHICH IS -9
DIVIDED BY 2 x "A" OR 2 x 1.
SO THE TWO SOLUTIONS HERE WILL BE OUR X INTERCEPTS.
SO WE'D HAVE X EQUALS-- THIS WOULD BE -2 PLUS OR MINUS
THE SQUARE ROOT OF--THIS WOULD BE 4 AND THEN PLUS 36,
THAT'S THE SQUARE ROOT OF 40 DIVIDED BY 2 X 1 or 2.
NEXT WE WANT TO SIMPLIFY THE SQUARE ROOT OF 40.
WE'RE LOOKING FOR THE PERFECT SQUARE FACTORS OF 40,
AND SINCE THE SQUARE ROOT OF 40
IS EQUAL TO THE SQUARE ROOT OF 4 x 10.
WELL, THE SQUARE ROOT OF 4 IS EQUAL TO 2,
THIS SIMPLIFIES TO 2 SQUARE ROOT 10.
SO WE HAVE X = -2 +/- 2 SQUARE ROOT 10 DIVIDED BY 2
WE NEED TO BE CAREFUL WHEN SIMPLIFYING THIS.
WE CANNOT JUST SIMPLIFY THESE 2s
BECAUSE WE CANNOT SIMPLIFY ACROSS ADDITION
OR SUBTRACTION.
SO TO HELP US SIMPLIFY THIS,
LET'S WRITE THIS AS TWO SEPARATE FRACTIONS.
WE HAVE -2 DIVIDED BY 2 +/- 2 SQUARE ROOT 10 DIVIDED BY 2
AND NOW WE CAN SEE IT'S GOING TO SIMPLIFY TO -1
AND THEN JUST PLUS OR MINUS SQUARE ROOT 10.
AGAIN, THIS SIMPLIFIES TO ONE OVER ONE AND SO DOES THIS.
SO WE HAVE TWO SOLUTIONS HERE,
AND THEREFORE WE HAVE TWO X INTERCEPTS.
THE SMALLER X INTERCEPT WOULD BE -1 - SQUARE ROOT 10, 0
AND THE LARGER X INTERCEPT
WOULD BE -1 + SQUARE ROOT 10, 0.
WE ARE GOING TO CHECK THESE POINTS GRAPHICALLY,
SO LET'S GO AHEAD AND GET DECIMAL APPROXIMATIONS
FOR BOTH OF THESE.
-1 - SQUARE ROOT 10 IS APPROXIMATELY -4.16,
AND -1+ SQUARE ROOT 10 IS APPROXIMATELY POSITIVE 2.16.
SO AGAIN, THIS X INTERCEPT IS APPROXIMATELY -4.16, 0
AND THIS X INTERCEPT
IS APPROXIMATELY POSITIVE 2.16, 0.
WE DO WANT TO GIVE THE EXACT VALUES IN THE HOMEWORK,
BUT TO CHECK IT GRAPHICALLY
WE WOULD NEED THE DECIMAL APPROXIMATIONS.
NOW TO FIND THE Y INTERCEPT OF ANY FUNCTION,
WE SET X EQUAL TO ZERO AND SOLVE FOR Y OR F OF X.
SO IF X IS EQUAL TO ZERO, THE Y INTERCEPT WOULD BE F OF 0,
WHICH WOULD BE EQUAL TO 0 SQUARED + 2 x 0 - 9,
WHICH WE CAN SEE WOULD BE EQUAL TO -9.
WHEN WE HAVE A QUADRATIC FUNCTION IN THIS FORM HERE,
THE VALUE OF C OR THE CONSTANT ON THE END
WILL ALWAYS BE THE Y INTERCEPT.
SO THE Y INTERCEPT WOULD BE THE POINT 0, -9.
LET'S LOOK AT THE GRAPH OF OUR FUNCTION
TO VERIFY THESE INTERCEPTS.
HERE IS THE SMALLER X INTERCEPT,
WHICH WE FOUND TO BE -1 - SQUARE ROOT 10, 0,
WHICH WAS APPROXIMATELY -4.16, 0.
HERE IS THE LARGER X INTERCEPT OF -1 + SQUARE ROOT 10, 0
OR APPROXIMATELY 2.16, 0,
AND FINALLY HERE IS THE Y INTERCEPT OF 0, -9.
I HOPE YOU FOUND THIS HELPFUL.