QUADRATIC FUNCTIONS

A function f from to is called a quadratic function if for some a, b, c ∈ f(x) = ax² + bx + c, provided that a ≠ 0. The graph of a quadratic function is a parabola, whose shape is as shown by the blue curve in the figure above.

 

Parabola with the equation y = ax² + bx + c (where a ≠ 0) has the following properties.

 

Property 1

If a > 0 then the parabola opens up (concave up) and if a < 0 then the parabola opens down (concave down).

Figure 1

 

Property 2

The parabola intersects the y-axis at (0,c).

Figure 2

 

Property 3

As for the x-intercept, there are three possibilities. It depends on the value of the discriminant D = b² – 4ac.

D < 0 ⇔ the parabola does not intersect the x-axis

Figure 3

 

D = 0 ⇔ parabola touches the x-axis, intersecting the x-axis at exactly one point

Figure 4

D > 0 ⇔ the parabola intersects the x-axis at two distinct points

Figure 5

 

To find the coordinates of the points where the parabola intersects the x-axis, find the roots of the quadratic equation ax² + bx + c = 0. The roots are the abscissae of the coordinates of the points where the parabola intersects the x-axis. If D = 0 the equation has exactly one root so there is one and only one x-intercept. If D > 0 the equation has two distinct roots, implying that there are two x-intercepts. If D < 0 then the equation has no roots so the parabola does not intersect the x-axis.

 

Property 4

The parabola has an axis of symmetry with the equation .

Figure 6

 

Property 5

The parabola has exactly one turning point. There are two types of turning points. If a > 0 the turning point is the minimum turning point and if a < 0 the turning point is the maximum turning point. The turning points are also called the extreme points. In Figure 7 below, the point P(x_E,y_E) is the extreme point, where and .

 

Figure 7

 

The value y_E is called the extreme value of the quadratic function. As a can be positive or negative, there are two types of extreme values, namely minimum value ​​and maximum value. If a > 0 then y_E is the minimum value, i.e. smallest value that can be achieved by the function. On the other hand, if a < 0 then y_E is the maximum value, i.e. the largest value that can be achieved by the function.

 

Example

Let f be a quadratic function with . Determine a) the coordinates of the points where the graph of f intersects the coordinate axes, b) the coordinates of the turning point and the type of the turning point, c) the extreme value of f, d) the type of the extreme value.

Answer

In this case, a = 0.5, b = -1, and c = -4. Since c = -4, the coordinates of the y-intercept is (0,-4) [Property 2]. To find the coordinates of the x-intercept, solve the quadratic equation . Multiplying both sides of the equation by 2, we get. x² – 2x – 8 = 0, Note that

x² – 2x – 8 = 0  ⇔   (x + 2)(x – 4) = 0  ⇔  x = -2 or x = 4

The equation has two roots, namely x₁ = -2 and x₂ = 4. Thus, the coordinates of the points where the parabola intersects the x-axis are (-2,0) and (4,0). To find the coordinates of the turning point, we can use the formulae in [Property 5] above.

It follows that the coordinates of the turning point of the parabola are (1,-4½). Since a > 0, the parabola opens up. Consequently, it is the minimum turning point. The extreme value of the function is -4½. Since a > 0, it is the minimum value. The parabola in this example is shown in Figure 8.

Figure 8