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.
Curve Sketching
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.
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