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

Question

Find the domain and sketch the graph of the function f(x) = x^2 &#8211; 6x

Suresh Chand · Accepted Answer

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:

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

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

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