To graph a parabola without a calculator, we can factor it to find its root values. These are the x values where the graph is equal to 0 (on the x-axis).
x^2 + 11x + 24 can be factored to:
(x+8)(x+3)
Set both factors equal to 0 to find the roots:
x+8=0
x=-8
x+3=0
x=-3
So the graph crosses the x-axis at x = -8 and x = -3.
Next we can find the vertex of the graph. This is the point where the slope changes directions. Since the x^2 term is positive, we know the parabola is facing up (like a smile face). Therefore the vertex is a minimum value.
We can find the x value of the minima by finding the difference between the roots. Half way from -3 to -8 is -5.5.
Next we can find the y value by plugging -5.5 in for x:
y = (-5.5)^2 + 11(-5.5) + 24 = -6.25
So the vertex is at (-5.5, -6.25)
Note: Minima and maxima can be found easier by using calculus.
From these three points we can draw a pretty accurate graph of the equation. See the attached picture.
