Graphing functions is the way of generating a curve that represents a function on a coordinate plane. If a function is represented by a graph, then every point on the curve satisfies the function equation. A function’s graph is frequently a useful method of representing the relation of the function models, and trying to manipulate a mathematical expression for a function can shed light on the function’s properties. Many essential phenomena can be modelled using functions presented as expressions.
Steps to graph a function
When graphing functions, we take the following steps:

Find the function’s domain and range and keep them in mind as you draw the curve.

Find and plot the xintercept(s) and yintercept(s).

Determine whether or not there are any holes.

Find the asymptotes vertical, horizontal, and slant and draw dotted lines to break the graph along those lines and ensure that the graph does not touch them.

Create a table of values by taking some random x values (on both sides of the xintercept and on both sides of the vertical asymptote) and calculating the corresponding y values.

Plot the points from the table and connect those using asymptotes, domain, and range.
Constructing a basic function graph
Basic function graph
The basic concept of graphing functions is

If possible, identify the shape. If it is a linear function of the form f(x) = ax+ b, its graph is a line; if it is a quadratic function of the form . Its graph is a parabola.

locating some points on it by substituting some random x values and locating the corresponding y values by substituting each value into the function
For example
This section graphs seven fundamental functions that will be used throughout the course. Plotting points is used to graph each function. Keep in mind that f(x)=y, so f(x) and y can be used interchangeably.
A constant function is any function of the form f(x)=c, where c can be any real number. Constant functions are linear and have the form f(x)=0x+c. It is clear from this form that the slope is 0 and the yintercept is (0,c). c is obtained by evaluating any value for x, such as x=2.
The graph of a constant function is a horizontal line. The domain is made up of all real numbers R, and the range is made up of a single value c.
Identifying functions using graphs
We will use two tests to identify functions using graphs. If a horizontal line intersects the curve more than once at some point, the curve lacks an inverse function. In short, if you have a curve, the vertical line test determines whether or not it is a function, and the horizontal line test determines whether or not the inverse of that curve is a function.
Vertical line test
The vertical line test can be used to determine whether a graph represents a function. A vertical line connects all points with the same x value. The y value of a point on a graph where a vertical line intersects it represents an output for that input x value. If any vertical line intersects a graph more than once, the graph does not define a function because that x value has more than one output. For each input value, a function produces only one output value.
If, on the other hand, the vertical line intersects the graph more than once, it indicates that a single xvalue is associated with more than one y value. This condition results in the relation being “disqualified,” or not being considered as a function.
Horizontal Line Test
Once we have determined that a graph defines a function, we can use the horizontal line test to see if it’s a onetoone function. Make horizontal lines across the graph. A horizontal line connects all points with the same y value. The input for that output y value is the x value of a point where a vertical line intersects a function. If any horizontal line intersects a graph more than once, the graph does not represent a function because that y value has more than one input.
Use of Horizontal Line Test
For x R, the graphs of f(x) = x2 + 1 and f(x) = 2x – 1 are shown below.
Conclusion
In this article we conclude that, The process of drawing the graph of the corresponding function is known as graphing functions. The graph of a function is extremely useful. We may not always know the expression for a function, but we do know some of its values maybe from an experiment. The graph can give us a good idea of what function could be used to solve the problem. And for identifying functions using graphs we will perform a vertical line test that determines whether the curve is a function, and the horizontal line test determines whether the curve’s inverse is a function.