Quadratic Function – Example

Explanation

GRAPHING A QUADRATIC FUNCTION

INTRODUCTION

A quadratic function is an important mathematical function. It can be defined as f(x) = ax² + bx + c, where a, b, and c are real numbers, with a ≠ 0. The characteristic feature of a quadratic function is its parabolic graph.

Let's see how to draw the graph of a quadratic function, using the function f(x) = -2x² + 4x + 3 as an example.

ANALYZING THE COEFFICIENTS

COEFFICIENT a (-2): This coefficient determines the shape of the parabola. If 'a' is negative, as in our example (-2), the parabola will open downwards.
COEFFICIENT b (4): Coefficient 'b' influences the location of the vertex of the parabola and its symmetry.
COEFFICIENT c (3): Coefficient 'c' represents the y-intercept, i.e., the point where the graph intersects the y-axis, in our case at c = 3.

GRAPHING PROCEDURE

FIND THE Y-INTERCEPT: Start with the point where the graph intersects the y-axis, which is (0, c). In our example, this is (0, 3).
FIND THE VERTEX OF THE PARABOLA: The vertex of the parabola is found using the formula p = -b/(2a) for the x-coordinate. In our example, this is p = -4 / (2 * (-2)) = -4 / (-4) = 1. When we substitute this value into the function, we get the y-coordinate q = f(p). So, q = -2(1)² + 4(1) + 3 = -2 + 4 + 3 = 5. Therefore, the vertex of the parabola is at (p, q) = (1, 5).
PLOT SYMMETRICAL POINTS: Since the parabola is symmetrical with respect to its axis of symmetry (which passes through the vertex, x=p), we can plot additional points that are symmetrical to points we already know. For example, if (0,3) is a point, and the axis of symmetry is x=1, then another point (2,3) will also be on the parabola.
DRAW THE PARABOLA: Use the identified points – the y-intercept, the vertex, and any symmetrical points – to draw a smooth parabolic curve.

GRAPHING EXAMPLE

For the function f(x) = -2x² + 4x + 3, we have identified:

The y-intercept at (0, 3).
The vertex of the parabola at (p, q) = (1, 5).
Additional points can be chosen as needed for better accuracy in drawing, such as finding the zeros or plotting another point like (2,3) which is symmetrical to (0,3) across the axis of symmetry x=1.