Factoring and tell whether the parabola opens up or down. Also Identify the minimum and how does the axis of symmetry relate to the x-intercepts?
y=x^2-4x-32
Note that the form is y = x^2 + bx + c
factoring and tell whether the parabola opens up or down. Also Identify the minimum and how does the axis of symmetry relate to the x-intercepts?
Since x^2 is positive and there is no negative in front of the y, term, this parabola opens upward
Factoring we have.... y = ( x - 8) ( x + 4)
Setting y = 0, we can find the x intercepts thusly : 0 = (x - 8) ( x + 4).....setting each of these factors to 0 and solving for x produces the x intercepts of x = 8 and x = -4
The minimum can be found thusly :
The x coordinate of the vertex is given by -b / [ 2a ] ....b = -4 and a = 1 .....so .... -b / [2a ] = -[ -4] / [ 2(1)] =
2
Now....we can find the y coordinate of the vertex by plugging this value into the function...so we have ...
y = (2)^2 - 4(2) - 32 = 4 - 8 - 32 = 4 - 40 = -36 .....and this is the minimum y value of the parabola
The axis of symmetry will be found between the intercepts and is given by adding the intercepts and dividing by 2....so we have ... [ 8 + -4 ] / 2 = 4 / 2 = 2.......so the axis of symmetry is x = 2....this is no coincidence.....the axis of symmetry will occur at the vertex which is ( 2 , - 36)
