Quantitative Reasoning

A lot of higher level GMAT problems can be solved easily if one knows how to draw the curve of a Quadratic Function.

Before we go ahead learning how to draw such scary curves, let's revise a few concepts about Quadratic Functions and Quadratic Equations.

What is a Quadratic Function?
If a function in 'x', has '2' has the highest power of 'x', then that function is called a Quadratic Function.

e.g.
f(x) = ax^2 + bx + c is a Quadratic function
where
a, b and c are real numbers.

A quadratic function, when plotted in Co-ordinate System, has the shape of a PARABOLA.

The values of 'x' at which f(x) = 0, i.e. ax^2 + bx + c = 0 are called the roots of a quadratic equation. A quadratic equation ax^2 + bx + c = 0 may have two real roots, one real root or no real roots at all.

If b^2 - 4ac > 0, two distinct real roots exist.
If b^2 - 4ac = 0, one real roots exists.
If b^2 - 4ac > 0, no real roots exist.

If a > 0, PARABOLA points downwards.
If a < 0, PARABOLA points upwards.



Lets discuss about the shifting of Parabola in vertical and horizontal directions.

Vertical Shifting:

To draw parabola of the type: y = ax^2 + b,
first plot y = ax^2, and then if
b > 0, Parabola shifts upwards, and if
b < 0, Parabola shifts downwards.



Horizontal Shifting:

In parabola of the type: y = a(x - b)^2,
first plot y = ax^2, and if
b > 0, Parabola shifts to the right and if
b < 0, Parabola shifts to the left.



Vertical and Horizontal Shifting Combined:

First plot y = ax^2, and then in two steps,
first shift the parabola in the Horizontal direction, and
then in the vertical direction.
H: Horizontal Shift
V: Vertical Shift



Shapes of Parabola:

The shape of the Parabola y = ax^2 depends on the magnitude of 'a'.



I hope that this will demystify the process of drawing plots of Quadratic Functions. Please let me know if you have any doubts.

Here is a good problem that tests our fundamentals of Quadratic equations:
https://www.beatthegmat.com/inequations- ... tml#477997