Given: #f(x) = y = -0.5(x+4)(x-6)#
This function is a quadratic function. When #f(x) = 0# you can find the #x#-intercepts (zeros):
#f(x) = -0.5(x+4)(x-6) = 0#
#x + 4 = 0; " "x - 6 = 0#
#x = -4; " "x = 6#
x-intercepts: #" "(-4, 0), (6, 0)#
Put the equation into #y = Ax^2 + Bx + C = 0# form. Use FOIL to distribute:
#f(x) = -0.5 (x^2 -6x +4x -24) = -0.5 (x^2 -2x -24) = 0#
#f(x) = -0.5x^2 +x +12 = 0#
The vertex is #(-B/(2A), f(-B/(2A))), " axis of symmetry is " x = -B/(2A#
#-B/(2A) = -1/(2(-.5)) = 1#
#f(1) = -0.5 (1)^2 + 1 + 12 = 12.5#
The vertex is #(1, 12.5); " axis of symmetry is " x = 1#
You can do point-plotting to find additional points since #x# is the independent variable:
#ul(" "x" "|" "y" ")#
#-3" "|" "4.5#
#-2" "|" "8#
#-1" "|" "10.5#
#" "0" "|" "12#
#" "2" "|" "12#
#" "3" "|" "10.5#
#" "4" "|" "8#
graph{ -0.5(x+4)(x-6) [-10, 10, -5, 15]}
