Graphing Functions

The process of graphing functions entails drawing the graph (curve) of the appropriate function. Graphing fundamental functions like linear, quadratic, cubic, and so on is rather simple; however, graphing complex functions like rational, logarithmic, and so on, requires some expertise and knowledge of mathematical concepts.

Let’s look at how to graph functions using some examples.

Definition

Drawing the curve that represents the function on the coordinate plane is known as graphing functions. Every point on a curve (graph) that represents a function fulfills the function equation. The linear function f(x) = -x+ 2 is represented by the graph below.

Graph of Linear function f(x) = -x + 2.

Take any point along this line, for example (-1, 3). Let’s replace (-1, 3) = (x, y) in the function f(x) = -x + 2 (notice that it can also be expressed as y = -x + 2) with (-1, 3) = (x, y). Then

3 = -(-1) + 2

3 = 1+2

3 = 3, hence the function is satisfied by (-1, 3).

Similarly, you can experiment with different points to see if they satisfy the function. Every point on the line (also known as a “curve”) fulfills the function. Graphing functions is the process of drawing such curves to represent functions.

Properties of graph of functions

When does a function increase, decrease, or stay the same?

f(x)

On an open interval I, a function f is increasing. If we have f(x₁)< f(x₂) for any option of x₁ and x₂ in I, then (x₂).

On an open interval, a function f is decreasing. If we have f(x₁)> f(x₂) for any option of x₁ and x₂ in I, then (x₂).

If we have any choice of x₁ and x₂ in I, a function f is constant on an open interval I.

Where f(x₁)=f (x₂). (All x selections in I have the same f(x) value.)

If there is an open interval I containing x=c, a function f has a local minimum at x=c.

C such that f(x) > f for every x c in I (x). A local maximum of f is called f(x).

If there is an open interval I containing c, then f(x)< f(c). A local maximum of f is called f(c).

Because there may be other points outside of the interval I, these are called “local.”

That are less than or larger than the local minimum or maximum.

Examples

Functions of one variable

          Graph of function

          f(x,y) = sinx² . cosx²

The function’s graph

f: {1,2,3}→{a,b,c,d} define by

is a subset of the whole

{1,2,3} *{a,b,c,d}

 G(f)={(1,a),(2,d),(3,c)}

The domain {1,2,3} is recovered from the graph as the set of first components of each pair in the graph.

{1,2,3}={x:there exists y, such that (x,y)∈G(f)}

The range can be regained in the same way.

{a,c,d}={y:there exists x, such that (x,y)∈G(f)}

The codomain {a,b,c,d}, on the other hand, cannot be identified only from the graph.

On the real line, the cubic polynomial graph

 f(x)=x³-9x is

{(x,x³-9x):x is a real number}

The outcome of plotting this set on a Cartesian plane is a curve (see figure)

Functions of two variables

   Plot of the graph of

    f(x,y) = (-cos x² + cos y²)² also 

    showing its gradient projected 

    on the bottom plane.

The trigonometric function’s graph

f(x,y)=sin(x²) cos(y²) is

{(x,y,sin(x²) cos(y²)}:x and y are real numbers

The outcome of plotting this collection on a three-dimensional Cartesian coordinate system is a surface (see figure).

It is frequently useful to display the function’s gradient and numerous level curves on a graph. The level curves can be projected on the bottom plane or mapped on the function surface. The graph of the function is depicted in the second figure:

f(x,y) = (-cos x² + cos y²)² .

Characteristics of functions graph

Positive or negative values for a function indicate whether the graph is above or below the x-axis. It may rise or fall on a given interval, reflecting the graph’s rise or fall. Local maximum and minimum values, which are the y coordinates of the graph’s turning points, may exist.

Graphing Basic Functions

It’s simple to graph basic functions like linear and quadratic functions. The underlying concept behind graphing functions is this:

•  possible, determining the form. For example, if the function is linear and has the form f(x) = ax + b, the graph is a line; if it is quadratic and has the form f(x) = ax² + bx + c, the graph is a parabola.

• By substituting several random x values into the function and determining the associated y values by substituting each value into the function

Here are some illustrations.

Graphing Quadratic Functions

We can also identify some random spots on a quadratic function when graphing it. However, a perfect U-shaped curve may not be achieved. This is because we need to know where the curve is turning to create a perfect U-shaped curve. That is, we must locate its vertex. We can locate two or three random points on either side of the vertices after locating the vertex, which will aid in charting the function.

Graphing Linear Functions

Let’s graph the same linear function (f(x) = -x + 2) as in the previous section. To do so, we make a table of values by selecting some random values for x, such as x = 0 and x = 1. Then, to get y-values, substitute each of these into y = -x + 2.

Thus, (0, 2) and (0, 3) are two locations on the line (1, 1). We can get the graph presented in the previous section by plotting them on a graph and connecting them with a straight line (stretching the line on both sides).

Conclusion

Functions are mathematical building blocks used in the construction of machinery, the prediction of natural disasters, the treatment of diseases, the study of global economies, and the flight of aeroplanes. Functions can accept many different inputs yet always produce the same output, which is unique to that function.