Quadratic Functions

Quadratic functions are given a specific position in the educational curriculum because of their complexity. They are functions whose values may be readily predicted from their input values; as a result, they are a step up from linear functions and a significant step away from straight-line attachment, respectively. Here, in this article, we’ll learn more about quadratic functions and graphs related to it. The curve outlined by the swing is an illustration of what a quadratic equation looks like when graphed. In mathematics, a quadratic equation is defined as an equation of degree 2, implying that the function’s maximum exponent is 2. A quadratic has the standard form y = ax² + bx + c, where a, b, and c are all numbers and a cannot be 0.

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Quadratic Functions

A quadratic function is defined as f(x) = ax² + bx + c, where a, b, and c are all non-zero values.

Graphing quadratic functions with a parabola is the representation of the graph of that function on a curve. In terms of opening up or down, as well as “width” and “steepness,” parabolas may have a variety of shapes, although they all have the same fundamental “U” form.

All parabolas are symmetric when viewed in reference to a line known as the axis of symmetry. The vertex of a parabola is the point at which the axis of symmetry of the parabola meets with itself. 

You are already aware that a line is produced by the intersection of two points. As a result, if you have two points on the plane, there is only one and only one line that links them, as shown by the symbol. The same may be said with quadratic functions and points, and the results are comparable. 

There is just one quadratic function f whose graph contains all three points given three points in the plane with different first coordinates and do not lie on a line. This is demonstrated in the applet below. There are three spots on the graph, and a parabola connects them all. In the text box below the graph, the associated function is displayed. The function and parabola are updated when you drag any of the points.

Using the techniques of stretching/shrinking and moving the parabola y = x², many quadratic functions may be graphed easily by hand.

What does a quadratic function represent?

If we talk about its representation generally we get a graph from a quadratic equation and it has 3 different forms which are- standard, factor and vertex. Let’s have a look at all the 3 forms.

1. Standard Form

Let’s start with the advantages of a standard form. Formulas and equations are stated with the highest degree first in normal mathematical notation. The exponent is referred to by the degree. Because the maximum exponent in quadratic equations is two, the degree is two. The phrase with an exponent of one follows the x² term, followed by the term with an exponent of zero.

The standard form has the advantage of easily recognising a function’s end behaviour as well as the values of a, b and c.

Standard Form of Quadratic equation

y = ax² + bx + c

The leading coefficient and the degree of a function identify the function’s end behaviour. A quadratic equation’s degree is always two. When expressed in standard form, the leading coefficient of a quadratic equation is always the word a.

The parabola opens up when aa is positive, meaning the function rises to the left and rises to the right. If aa is negative, the parabola opens down, which means the function falls to the left and then to the right.

Example

y = 3x² + 2x – 1 

y = -3x² + 2x + 1

2. Factor Form

Let’s have a look at why the factored form is useful. We factor the equation from standard form to get to factored form, which is exactly what it sounds like.

Factor Form of Quadratic Equation

y = a(x – r₁)(x – r₂)

Additionally, we may utilise the value of a to identify the final behaviour of a quadratic equation in its factored form. Despite the fact that the degree isn’t immediately recognised, we know there are just two components that make the degree two. The final behaviour follows the same guidelines as before.

Identifying the function’s zeros, or x-intercepts, is another advantage of the factored form. 

The value of r₁ and the value of r₂ are both zeroes of the quadratic equation.

Example

y = -(x + 2)(x – 3)

Vertex Form

Finally, we have the quadratic vertex form. It’s important to remember that the vertex of a parabola is the point on the parabola where the axis of symmetry meets.  It’s also the lowest point of an opening parabola or the highest point of an opening parabola.

Vertex Form of Quadratic Equation

y = a(x – h)² + k

The biggest advantage of vertex form, as you might imagine, is the ease with which the vertex may be identified. A parabola’s vertex, or the vertex of a quadratic equation, is expressed as (h,k), where h is the x-coordinate and k is the y-coordinate.

The values of hh and kk are easily distinguishable in this form, as can be shown. The value of a can also be used to determine the final behaviour.

Example 

y = 3(x – 2)² – 1

Elements of Quadratic Function

There are 3 elements of quadratic function namely, graph, domain and vertex.

1) Graph- A quadratic function’s graph is always a parabola that opens upward or downward (end behaviour). 

2) Domain- A quadratic function’s domain is all real numbers.

3) Vertex- When the parabola opens upwards, the vertex is the lowest point.

Conclusion

So, at last we can conclude that a quadratic function is defined as f(x) = ax² + bx + c, where a, b, and c are all non-zero values. A parabola is a curve that represents the graph of a quadratic function. Parabolas can open up or down, and their “width” and “steepness” can vary, but they all share the same basic “U” shape. The zeros of a quadratic equation are the locations on the x-axis where the graph of the equation intersects.