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.
Curve Sketching is a collection of various techniques which can be used to create the approximate graph of any given function. That can help us analyze different features and behavior of the graph.
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) = ex, so just putting x = loge3 we get the output f(loge3) = 3. Now, we can (loge3, 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.
Read More about Domain and Range.
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:
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.
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.
Let's consider an example for better understanding.
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
Symmetry in curve sketching refers to the property of a curve where one part of the curve mirrors or reflects another. Common types of symmetry includes:
- Symmetry about the y-axis (even functions): If f(−x) = f(x), the curve is symmetric about the y-axis.
- Symmetry about the x-axis: If f(x) = −f(x), the curve exhibits symmetry about the x-axis.
- Symmetry about the origin (odd functions): If f(−x) = −f(x), the curve is symmetric about the origin.
Also read about Odd and Even Functions.
Tracing Different Types of Functions
We can sketch any function using Curve Sketching but some functions can be quite tricky to sketch. Let's consider some examples of various different functions which we will sketch using the techniques of curve sketching.
Linear Functions
Sketching Linear Function is quite an easy task in curve sketching as we just need two points on the graph and the line joining those two points is the graph of a linear function. Let's consider an example:
Example: Sketch the graph for the function, f(x) = 2x + 3.
Solution:
For f(x) = 2x + 3,
Put x = 0 ⇒ f(0) = 3
and Put x = 1 ⇒ f(1) = 5
Now, draw a striaght line passing throught the pionts (0, 3) and (1, 5) which is the graph of the linear function f(x) = 2x+3.
Polynomial Functions
Polynomial functions occur a lot in calculus, and it is essential to know how to sketch their graphs. We will look at a function and use the techniques studied above to infer the graph of the function. The general idea is to look for asymptomatic values, and where they are going, and then find the critical points and draw a graph according to them. Let's see it through examples,
Example: Sketch the graph for the given function,
f(x) = x2 + 4
Solution:
We know that the domain of this function is all real numbers. This functions will tend to infinity as we go towards large positive and negative values of x.
Notice that f(-x) = (-x)2 + 4 = x2 + 4 = f(x). That is this function is even, so its graph must be symmetric about the y-axis.
Now we know that graph goes to infinity and is symmetrical around the y-axis. Now, let's look for critical points.
f'(x) = 2x = 0
⇒ x = 0
Thus, there is only one critical point which is x = 0. Checking the double derivative f''(x) = 2. Since f''(x) > 0 for every x. So, the graph must be convex upward everywhere with minima at x = 0. Now we just need to know the value of the function at minima.
f(0) = 4.
Now we are ready to plot a graph.
Exponential Functions
Exponential functions are an essential part of calculus and are commonly represented as f(x) = ax, where a is any positive constant and x can be any possible real number.
To sketch the graph of Exponential Functions we need to check, the domain, range, and asymptotes. We also need to check whether the function is increasing or decreasing. If the base of the exponential function lies between 0 and 1 then it decreases in its domain otherwise it is an increasing function.
Let's consider an example of sketching the exponential function.
Example: Sketch the graph for the given function, f(x) = 2x - 1
Solution:
The domain of this function is all real numbers. As x goes to negative infinity, the function approaches zero, and as x goes to infinity, the function approaches infinity.
The base of the exponential function is 2, which is greater than one, so the function increases as x increases.
To find critical points, we need to find where the derivative is zero. f'(x) = 2^x ln(2). This derivative is zero only when x = 0.
Thus, the critical point occurs at x = 0. To determine the concavity, we can find the second derivative of the function. f''(x) = 2^x ln^2(2). Since the second derivative is always positive, the graph must be convex upward everywhere.
Now we just need to find the value of the function at the critical point. f(0) = 2^0 - 1 = 0.
We can now use this information to sketch the graph of the exponential function.
Logarithmic Functions
We know that logarithmic functions are inverse of exponential functions. The function y = logbx is the inverse of y = bx. The graph of the exponential function is given below. We also know that the graph of an inverse of a function is basically a mirror image of the graph in y = x. So we can derive the shape of the graph of log function from the given graph of the exponential function.
The mirror image of the Logarithmic function is the exponential function both of them are shown in the image below,
Let's see an example of graphic logarithmic functions.
Example: Plot the graph for log10x + 5.
Solution:
We can see that the function is f(x) = log10x + 5.
The graph of this equation will be shifted 5 units in the upwards direction.
Read More,
Sample Problems on Curve Sketching
Problem 1: Sketch the graph for the given function,
f(x) = x + 8
Solution:
We know that the domain of this function is all real numbers. This functions will tend to infinity as we go towards large positive and negative values of x.
Now we know that graph goes to positive infinity for larger positive values of x and negative infinity for larger negative values of x.Now, let's look for critical points.
f'(x) = 1
There is no critical point, that means derivatives change sign remains same and constant throughout.
Let's see where the equation cuts the x-axis.
x+ 8 = 0
⇒x = -8
Now we are ready to plot a graph.
Problem 2: Sketch the graph for the given function,
f(x) = x2 - 6x + 8
Solution:
We know that the domain of this function is all real numbers. This functions will tend to infinity as we go towards large positive and negative values of x.
Now we know that graph goes to positive infinity for larger positive values of x and negative infinity for larger negative values of x.Now, let's look for critical points.
f'(x) = 2x -6 = 0
⇒x = 3
There is one critical point, that means derivatives change sign at that, but we don't know which sign changes to what. So, we will check the sign.
From x ∈ (-∞,3] f'(x) < 0. That is in this interval, the graph is decreasing.
From x ∈ (3,∞) f'(x) > 0. That is in this interval, the graph is increasing.
That means the critical point is a minimum.
Let's see where the equation cuts the x-axis.
x2 -6x + 8 = 0
⇒x2 -4x -2x + 8 = 0
⇒x(x - 4) -2(x - 4) = 0
⇒(x - 2)(x - 4) = 0
Now we are ready to plot a graph.
Problem 3: Sketch the graph for the given function,
f(x) = x3 - 3x + 4
Solution:
We know that the domain of this function is all real numbers. This functions will tend to infinity as we go towards large positive and negative values of x.
Now we know that graph goes to positive infinity for larger positive values of x and negative infinity for larger negative values of x.Now, let's look for critical points.
f'(x) = 3x2 -3 = 0
⇒x2 = 1
⇒x = -1 or 1
There are two critical points, that means derivatives change sign at them, but we don't know which sign changes to what. So, we will check the sign.
From x ∈ (-∞,-1] f'(x) > 0. That is in this interval, the graph is increasing.
From x ∈ (-1,1] f'(x) < 0. That is in this interval, the graph is decreasing.
From x ∈ (1,∞) f'(x) > 0. That is in this interval, the graph is increasing.
f(0) = 4.
Now we are ready to plot a graph.
Problem 4: Plot the graph for the equation f(x) = ex + 2.
Solution:
We know that f(x) = ex + 2 is an exponential function, it increases with increasing value of x.
f'(x) = ex
This will never become zero, so there are no critical points. The graph is continuously increasing.
f''(x) > 0 thus it's shape is always convex upward. Due to the addition of 2 to the exponential function. The whole graph will be shifted two units upwards.
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Finding Derivative with Fundamental Theorem of CalculusIntegrals are the reverse process of differentiation. They are also called anti-derivatives and are used to find the areas and volumes of the arbitrary shapes for which there are no formulas available to us. Indefinite integrals simply calculate the anti-derivative of the function, while the definit
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Evaluating Definite IntegralsIntegration, as the name suggests is used to integrate something. In mathematics, integration is the method used to integrate functions. The other word for integration can be summation as it is used, to sum up, the entire function or in a graphical way, used to find the area under the curve function
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Properties of Definite IntegralsProperties of Definite Integrals: An integral that has a limit is known as a definite integral. It has an upper limit and a lower limit. It is represented as \int_{a}^{b}f(x) = F(b) â F(a)There are many properties regarding definite integral. We will discuss each property one by one with proof.Defin
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Definite Integrals of Piecewise FunctionsImagine a graph with a function drawn on it, it can be a straight line or a curve, or anything as long as it is a function. Now, this is just one function on the graph. Can 2 functions simultaneously occur on the graph? Imagine two functions simultaneously occurring on the graph, say, a straight lin
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Improper IntegralsImproper integrals are definite integrals where one or both of the boundaries are at infinity or where the Integrand has a vertical asymptote in the interval of integration. Computing the area up to infinity seems like an intractable problem, but through some clever manipulation, such problems can b
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Riemann SumsRiemann Sum is a certain kind of approximation of an integral by a finite sum. A Riemann sum is the sum of rectangles or trapezoids that approximate vertical slices of the area in question. German mathematician Bernhard Riemann developed the concept of Riemann Sums. In this article, we will look int
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Riemann Sums in Summation NotationRiemann sums allow us to calculate the area under the curve for any arbitrary function. These formulations help us define the definite integral. The basic idea behind these sums is to divide the area that is supposed to be calculated into small rectangles and calculate the sum of their areas. These
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Trapezoidal RuleThe Trapezoidal Rule is a fundamental method in numerical integration used to approximate the value of a definite integral of the form bâ«a f(x) dx. It estimates the area under the curve y = f(x) by dividing the interval [a, b] into smaller subintervals and approximating the region under the curve as
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Definite Integral as the Limit of a Riemann SumDefinite integrals are an important part of calculus. They are used to calculate the areas, volumes, etc of arbitrary shapes for which formulas are not defined. Analytically they are just indefinite integrals with limits on top of them, but graphically they represent the area under the curve. The li
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Antiderivative: Integration as Inverse Process of DifferentiationAn antiderivative is a function that reverses the process of differentiation. It is also known as the indefinite integral. If F(x) is the antiderivative of f(x), it means that:d/dx[F(x)] = f(x)In other words, F(x) is a function whose derivative is f(x).Antiderivatives include a family of functions t
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Indefinite IntegralsIntegrals are also known as anti-derivatives as integration is the inverse process of differentiation. Instead of differentiating a function, we are given the derivative of a function and are required to calculate the function from the derivative. This process is called integration or anti-different
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Particular Solutions to Differential EquationsIndefinite integrals are the reverse of the differentiation process. Given a function f(x) and it's derivative f'(x), they help us in calculating the function f(x) from f'(x). These are used almost everywhere in calculus and are thus called the backbone of the field of calculus. Geometrically speaki
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Integration by U-substitutionFinding integrals is basically a reverse differentiation process. That is why integrals are also called anti-derivatives. Often the functions are straightforward and standard functions that can be integrated easily. It is easier to solve the combination of these functions using the properties of ind
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Reverse Chain RuleIntegrals are an important part of the theory of calculus. They are very useful in calculating the areas and volumes for arbitrarily complex functions, which otherwise are very hard to compute and are often bad approximations of the area or the volume enclosed by the function. Integrals are the reve
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Partial Fraction ExpansionIf f(x) is a function that is required to be integrated, f(x) is called the Integrand, and the integration of the function without any limits or boundaries is known as the Indefinite Integration. Indefinite integration has its own formulae to make the process of integration easier. However, sometime
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Trigonometric Substitution: Method, Formula and Solved ExamplesTrigonometric substitution is a process in which the substitution of a trigonometric function into another expression takes place. It is used to evaluate integrals or it is a method for finding antiderivatives of functions that contain square roots of quadratic expressions or rational powers of the
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Chapter 8: Applications of Integrals
Area under Simple CurvesWe know how to calculate the areas of some standard curves like rectangles, squares, trapezium, etc. There are formulas for areas of each of these figures, but in real life, these figures are not always perfect. Sometimes it may happen that we have a figure that looks like a square but is not actual
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Area Between Two Curves: Formula, Definition and ExamplesArea Between Two Curves in Calculus is one of the applications of Integration. It helps us calculate the area bounded between two or more curves using the integration. As we know Integration in calculus is defined as the continuous summation of very small units. The topic "Area Between Two Curves" h
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Area between Polar CurvesCoordinate systems allow the mathematical formulation of the position and behavior of a body in space. These systems are used almost everywhere in real life. Usually, the rectangular Cartesian coordinate system is seen, but there is another type of coordinate system which is useful for certain kinds
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Area as Definite IntegralIntegrals are an integral part of calculus. They represent summation, for functions which are not as straightforward as standard functions, integrals help us to calculate the sum and their areas and give us the flexibility to work with any type of function we want to work with. The areas for the sta
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Chapter 9: Differential Equations
Differential EquationsA differential equation is a mathematical equation that relates a function with its derivatives. Differential Equations come into play in a variety of applications such as Physics, Chemistry, Biology, Economics, etc. Differential equations allow us to predict the future behavior of systems by captur
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Particular Solutions to Differential EquationsIndefinite integrals are the reverse of the differentiation process. Given a function f(x) and it's derivative f'(x), they help us in calculating the function f(x) from f'(x). These are used almost everywhere in calculus and are thus called the backbone of the field of calculus. Geometrically speaki
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Homogeneous Differential EquationsHomogeneous Differential Equations are differential equations with homogenous functions. They are equations containing a differentiation operator, a function, and a set of variables. The general form of the homogeneous differential equation is f(x, y).dy + g(x, y).dx = 0, where f(x, y) and h(x, y) i
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Separable Differential EquationsSeparable differential equations are a special type of ordinary differential equation (ODE) that can be solved by separating the variables and integrating each side separately. Any differential equation that can be written in form of y' = f(x).g(y), is called a separable differential equation. Separ
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Exact Equations and Integrating FactorsDifferential Equations are used to describe a lot of physical phenomena. They help us to observe something happening in real life and put it in a mathematical form. At this level, we are mostly concerned with linear and first-order differential equations. A differential equation in âyâ is linear if
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Implicit DifferentiationImplicit Differentiation is the process of differentiation in which we differentiate the implicit function without converting it into an explicit function. For example, we need to find the slope of a circle with an origin at 0 and a radius r. Its equation is given as x2 + y2 = r2. Now, to find the s
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Implicit differentiation - Advanced ExamplesIn the previous article, we have discussed the introduction part and some basic examples of Implicit differentiation. So in this article, we will discuss some advanced examples of implicit differentiation. Table of Content Implicit DifferentiationMethod to solveImplicit differentiation Formula Solve
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Advanced DifferentiationDerivatives are used to measure the rate of change of any quantity. This process is called differentiation. It can be considered as a building block of the theory of calculus. Geometrically speaking, the derivative of any function at a particular point gives the slope of the tangent at that point of
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Disguised Derivatives - Advanced differentiation | Class 12 MathsThe dictionary meaning of âdisguiseâ is âunrecognizableâ. Disguised derivative means âunrecognized derivativeâ. In this type of problem, the definition of derivative is hidden in the form of a limit. At a glance, the problem seems to be solvable using limit properties but it is much easier to solve
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Derivative of Inverse Trigonometric FunctionsDerivative of Inverse Trigonometric Function refers to the rate of change in Inverse Trigonometric Functions. We know that the derivative of a function is the rate of change in a function with respect to the independent variable. Before learning this, one should know the formulas of differentiation
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Logarithmic DifferentiationMethod of finding a function's derivative by first taking the logarithm and then differentiating is called logarithmic differentiation. This method is specially used when the function is type y = f(x)g(x). In this type of problem where y is a composite function, we first need to take a logarithm, ma
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