### Graphing Functions: Types of Function Graphs

Graphing functions consist of drawing the graph (curve) of the corresponding function. It is fairly easy to graph basic functions like linear, quadratic, cubic, etc. Complicated graphs, such as rational, logarithmic, etc, require some skill and some mathematical concepts to understand.

Let’s look at how to graph functions with examples.

**Graphing Basic Functions**

Basic functions like linear functions and quadratic functions are easy to graph. Function graphing is the process of plotting data in a graph.

- Identifying the shape if possible. Its graph, for example, would have a line if it were a linear function of the form f(x) = ax + b; if it were a quadratic function of the form f(x) = ax
^{2}+ bx + c, then its a parabola. - Finding some points on it by substituting some random values for x and finding the corresponding y values by substituting each value into the function.

Here are some examples.

**Graphing Linear Functions**

Same linear function as mentioned in the previous section (f(x) = -x + 2). We create a table of values for x by taking some random numbers, say x = 0 and x = 1. Substitute each of these in y = -x + 2 to compute the y-values.

There are two points on the line (0, 2) and (1, 1). By plotting them on a graph and joining them by a straight line (extending the line on both sides), we get their graph as shown in the previous section.

**Graphing Quadratic Functions**

For graphing quadratic functions, we can also find some random points on it. But this may not give a perfectly U-shaped curve. This is because, to get a precise U-shaped curve, we need to know where the curve is turning. i.e., we have to find its vertex. After finding the vertex, we can find two or three random points on each side of the vertex that would assist in graphing the function.

Example: Quadratic function f(x) = x^{2} – 2x + 5.

Solution:

Comparing it to f(x) = ax^{2} + bx + c, a = 1, b = -2, and c = 5.

The vertex’s x-coordinate is h = -b/2a = -(-2)/2(1) = 1.

According to its y-coordinate, f(1) = 1^{2} – 2(1) + 5 = 4.

Hence, the vertex is (1, 4).

**Graphing Complex Functions**

Graphing functions are simpler when each of their domains and ranges is a set of real numbers. However, not every function behaves this way. Graphing some complex functions requires that domain, range, asymptotes, and holes be taken into consideration. Common examples include:

- Rational functions – The parent function is f(x) = 1/x (which is called the reciprocal function).
- Exponential functions – Its parent function is of the form f(x) = a
^{x}. - Logarithmic Functions – This function has the form f(x) = log x.

Just have an idea of what the graphs of parent functions of each of these functions look like.

We follow the following steps for graphing functions in each of these cases:

- When drawing the curve, keep the domain and range of the function in mind.
- Find x-intercept(s) and y-intercept(s), and plot them.
- Identify the holes if any.
- We can then draw dotted lines along the asymptotes (vertical, horizontal, and slant) so that we can break the graph along those lines and make sure that it does not touch them.
- Construct a table of values by taking some random numbers for x (on both sides of the x-intercept and/or on both sides of the vertical asymptote), calculate the corresponding values of y.
- Take care of asymptotes, domain, and range when plotting the points from the table.

**Graphing Rational Functions**

A rational function f(x) is defined as (x + 1) / (x – 2). Graph this function according to the above steps.

- Domain = {x ∈ R | x ≠ 2} ; Range = {y ∈ R | y ≠ 1}. To understand how domains and ranges of rational functions are determined.
- Its x-intercept is (-1, 0) and its y-intercept is (0, -0.5).
- There are no holes.
- The vertical asymptote (VA) is x = 2 and the horizontal asymptote (VA) is y = 1.

**Graphing Exponential Functions**

Consider the exponential function f(x) = 2^{-x} + 2. We will graph it in the same way as we did above.

- The domain of this function is the set of all real numbers (R) and its range is y > 2.
- It has no vertical asymptotes. There is, however, a horizontal asymptote at y = 2.
- It has no x-intercepts. Its y-intercept is (0, 3).
- No holes.

**Graphing Logarithmic Functions**

Logarithmic function, such as f(x) = 2 log2 x – 2. We will now graph it by following the steps as explained earlier.

- Its domain is x > 0, and its range is all real numbers (R).
- The x-int is (2, 0) and there is no y-int.
- Its vertical asymptote is y = 0 (x-axis) and there is no horizontal asymptote.
- No holes.
- We have one reference point so far, which is (2, 0). The table will be constructed by taking some random numbers on either side of 0 (we cannot select values of x less than 0 because the domain is x > 0).