Sketch the graph of the function f(x) = x^2 - 9.
Solution. To graph f(x), we need to follow the steps below:
A. Domain:
f is defined on the set Df = (-∞, +∞).
B. Intersection with the axes:
The curve intersects the x-axis at x = -3 and x = 3.
The curve intersects the y-axis at y = -9.
C. Symmetry:
For each x ∈ Df, we have -x ∈ Df.
Comparing f(-x) with f(x), we deduce that f is an even function. Thus, the graph is symmetric with respect to the y-axis.
D. Asymptotes:
lim(x→+∞) f(x) = +∞
lim(x→-∞) f(x) = +∞
E. Monotony:
By computing the derivative of f(x), we obtain:
f'(x) = 2x
The critical point of f is x = 0.
- f is decreasing for x < 0
- f is increasing for x > 0
G. Concavity and inflection points:
The second derivative is given by f''(x) = 2
- f is convex for all x in Df.
H. Graph of f: Sketch the graph by yourself, then check with the solution.