Find the domain and sketch the graph of the function \(f(x) = x^2 – 6x\).

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Solution:

Given

y = x2 – 6x  = x(x – 6)  ——- (i)

A. Domain:  Clearly y is defined for x ∈ (-∞, ∞)

So, domain of y = (-∞ ∞).

B. Intercept: At x = 0 we get y = 0

At y = 0, we get x = 0, x = 6

This means y-intercept is x= 0 and x = 6 and x-intercept is y = 0. That is the crve meets the x-axes only at (0, 0) and (6, 0).

C. Symmetry: Her

f(-x) = (-x)2 – 6(-x) = x2 + 6x  f(x) and ≠ -f(x)

That is y is neither symmetrical about axes

D. Asymptotes:

\(\lim_{x \to ± ∞}(y)\)

\( = \lim_{x \to ± ∞}(x^2 – 6x) = ± ∞\)

So, the curve has no horizontal asymptotes.

And there is finite value of x such that y approaches to ∞. So, the curve has no vertical asymptotes.

Thus, the curve has no horizontal and vertical asymptotes.

E. Extremity:

f(x) = 6x2 – 6x

So, f'(x) = 2x – 6 = 2(x – 3)  ——- (ii)

Since the critical point of f(x) is x = a if f'(a) = 0 or f'(a) = ∞

By (ii), the critical points of f(x) are x = 3.

So,

Interval (-∞, 3) (3, ∞)
Sign of f'(x) -ve +ve
Nature of curve Decrease Increase

F. Extreme:  Here, f(3) = 0

So, the curve f(x) has maxima at (3, 0)

G. Concavity: Here

f”(x) = 2 ≠ 0 and nor undefined for all x

So, there is not point of inflection and f”(x) > 0 for all x, so the curve of f(x) is concave up for all x.

H. Summary above table

Interval (-∞, 3) (3, ∞)
Sign of f(x) Decrease Increase
Nature of curve Concave up Concave up

With these information, the graph of the equation is as:

User Loaded Image | CSIT Guide

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