The intercept with the y-axes occurs when x = 0
f(0) = 2*02 + 35*0 + 75 = 0 + 0 + 75 = 75
So intercept with x axes is the point (0,75)
The intercept with the x-axes occurs when y=0
2x2 + 35x + 75 = 0
As there’s no direct way to find x, we can use the Bhaskara formula:x=(-b+- square root (b2-4ac))/(2a)
This formula is for: ax2 +bx + c = 0
So:
a=2
b=35
c=75
x=(-35+- square root (35^2-4*2*75))/(2*2)
So x can be -2.5 or -15
So intercept with y axes are the points (-2.5,0) and (-15,0)
And there is also a formula to find the mínimum. The mínimum is always the vertex:
Xv = -b/(2*a) = -35/(2*2) = -8.75
Yv = f(Xv) = f(-8.75) = 2(-8.75)^2 + 35(-8.75) + 75 = -78.125
So the minimum is the point (-8.75, -78.125)
You can easily verify all these data with a graphic software such as GeoGebra.
Here is the complete list of all the points we need to see:
(0,75); (-2.5,0) ; (-15,0), (-8.75, -78.125)
So the mínimum X has to be -15 and the máximum X has to be 0
and the mínimum Y has to be -78.125 and the máximum Y has to be 75
So this leads to a different scale for X and for Y.
