Graphing Basics

To create a graph of any given function, we need to plot some points such as intercepts, critical points, and some regular points which can help us trace the graph on the cartesian plane. Let’s further understand these basics in detail as follows:

Plotting Points

We can easily plot various different points of any function on the graph by just using random input and their outputs as the coordinates. This random plot of points helps us connect the final graph after all the necessary calculations are done. For example, we need to graph the function f(x) = e^x, so just putting x = log_e3 we get the output f(log_e3) = 3. Now, we can (log_e3, 3) as a point on the graph. 

Domain and Range

First, analyze the function to check for its domain. We need to find out the points where the value of the function becomes undefined or is discontinuous. For example: 

1/x is not defined at x = 0. Log(x) is defined only at positive values of x. 

Finding Intercepts and Asymptotes

Intercepts are the points where the graph cuts the coordinate axis and to find the x-intercept, we put y = 0 and solve for x. Similarly, to find the y-intercept, we put x = 0 and solve for y.

Asymptotes are lines that the graph approaches but do not intersect. There are three types of asymptotes which are as follows: 

• Horizontal Asymptote
• Vertical Asymptote
• Slant Asymptote

Horizontal asymptotes occur when the function approaches a constant value as x approaches infinity or negative infinity. Vertical asymptotes occur when the function approaches infinity or negative infinity as x approaches a specific value. Slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator.

To calculate Horizontal Asymptote, we need to calculate the limit of a function at infinity, and vertical asymptotes are those points for which functions become not defined i.e., the denominator becomes 0.

Learn more about, Vertical and Horizontal Asymptotes

Example: Find Intercept and Asymptote for f(x) = (2x + 1) / (x – 3).

Solution:

To find the x-intercept, we set f(x) = 0 and solve for x:

⇒ (2x + 1) / (x – 3) = 0

⇒ 2x + 1 = 0 (x ≠ 3)

⇒ x = -1/2

Therefore, the x-intercept of the function is at (-1/2, 0).

To find the y-intercept, we set x = 0 and solve for f(x):

⇒ f(0) = (2(0) + 1) / (0 – 3) = -1/3

Therefore, the y-intercept of the function is at (0, -1/3).

The vertical asymptote occurs at x = 3, since the denominator of the function becomes zero at that point.

To find the horizontal asymptote, we need to examine the behavior of the function as x approaches infinity or negative infinity. We can do this by dividing the numerator and denominator by the highest power of x in the function:

f(x) = (2x + 1) / (x – 3) = (2 + 1/x) / (1 – 3/x)

As x becomes very large or very small, the term 1/x becomes insignificant compared to the other terms in the numerator and denominator, so we can ignore it:

f(x) ≈ 2 / 1 = 2 (as x → ±∞)

Therefore, the horizontal asymptote of the function is y = 2.

Local Extrema and Inflection Points

Local Extrema are those points of the function or graph for which there is no such value of function greater or smaller than the local extrema i.e., no other point in the neighborhood of the local extrema has a more extreme value than it. 

To find out the maxima and minima in any function, we need to find the critical points. Critical points of the function are defined as the points where either slope of the function is not defined or the slope is 0 i.e., f'(x) = 0.

After getting the values of critical points, check the second derivative of the function at those critical points. If f”(x) > 0 for some critical point x=k, then f(k) is the local minima of the function, and if f”(x) < 0 for some critical point x = k, then f(k) is the local maxima of the function. 

If f”(x) = 0 for some critical point x = k then x = k is the Point of Inflection or Inflection Point of the function.

Calculating Slope and Concavity

The slope is the measure of inclination from the positive x-axis and it tells us whether the graph is increasing (slope>0) or decreasing (slope<0). To find the slope of any given function, we differentiate the given function w.r.t to the dependent variable and substitute the value for which we need to calculate the slope.

Concavity is the measure of the curve which tells us whether the graph is concave up or concave down i.e., the direction of curvature of the graph. To calculate the concavity, we use the second derivative w.r.t dependent variable of the function. The second derivative tells us the rate at which the derivative is changing. If the second derivative is positive, then the function is concave up, and if the second derivative is negative, then the function is concave down.

Example: Find the slope and concavity of f(x) = x³ – 3x² + 2x.

Solution:

f(x) = x³ – 3x² + 2x

⇒ f'(x) = 3x² – 6x + 2

To find the slope of the function at a specific point, we substitute the value of x in the derivative:

f'(-1) = 3(-1)^2 – 6(-1) + 2 = 11

Therefore, the slope of the function at x = -1 is 11.

To find the concavity of the function,

f”(x) = 6x – 6

To find the points where the concavity changes, we set f”(x) = 0 and solve for x:

⇒ f”(x) = 6x – 6 = 0

⇒ x = 1

Therefore, the function changes from concave down to concave up at x = 1.

Here, (1, 0) is the inflection point.

Inflection Point: Inflection point is the point where the concavity changes i.e., the second derivative of function = 0. 

Symmetry 

Determine whether the functions are odd, even, or neither of these. Sometimes some functions are periodic in nature. We need to check for their periodicity if they are periodic in nature. Functions satisfying, f(x) = f(-x) are called even functions. While the functions satisfying f(-x) = -f(x) are called odd functions. Some examples of periodic functions are: 

Sin(x), Cos(x), and other trigonometric functions. 

Curve Sketching as its name suggests helps us sketch the approximate graph of any given function which can further help us visualize the shape and behavior of a function graphically. Curve sketching isn’t any sure-shot algorithm that after application spits out the graph of any desired function but it is an active role approach for a visual representation of a function that needs analysis of various features of graphs, such as intercepts, asymptotes, extrema, and concavity, to gain a better understanding of how the function behaves.

In this article, we will explore all the fundamentals of curve sketching and its solved examples. Other than that we will also explore all the aspects in detail which will help us analyze and sketch the function more efficiently.