What do you need help with here? Graphing? Factoring? Finding the roots? You haven't made it clear.
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?
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)] =
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)
Trouble factoring a quadratic? Easy when you realise that ----
The factors of the constant term add up to the value of the co-efficient of x
Here you have x^2 - 4x - 32. We want the factors of -32 that add up to -4 because that is the
co-efficient of x here.
factors of -32 are plus or minus16 and plus or minus 2
and plus or minus 8 and plus or minus 4. Since there is no way we can get - 4 from the sums of
plus or minus 16 and 2 we look at plus or minus 8 and 4. The only possible sum is -8 + 4.
So the factors are (x-8)(x+4) Try a few more and you'll soon get the technique.