Calculus AB and Calculus BC

CHAPTER 4 Applications of Differential Calculus

G. FURTHER AIDS IN SKETCHING

It is often very helpful to investigate one or more of the following before sketching the graph of a function or of an equation:

(1) Intercepts. Set x = 0 and y = 0 to find any y- and x-intercepts respectively.

(2) Symmetry. Let the point (x, y) satisfy an equation. Then its graph is symmetric about

the x-axis if (x, −y) also satisfies the equation;

the y-axis if (−x, y) also satisfies the equation;

the origin if (−x, −y) also satisfies the equation.

(3) Asymptotes. The line y = b is a horizontal asymptote of the graph of a function f if either Image inspect the degrees of P(x) and Q(x), then use the Rational Function Theorem. The line x = c is a vertical asymptote of the rational function Image if Q(c) = 0 but P(c) ≠ 0.

(4) Points of discontinuity. Identify points not in the domain of a function, particularly where the denominator equals zero.

EXAMPLE 19

Sketch the graph of Image

SOLUTION: If x = 0, then y = −1. Also, y = 0 when the numerator equals zero, which is when Image A check shows that the graph does not possess any of the symmetries described above. Since y → 2 as x → ±∞, y = 2 is a horizontal asymptote; also, x = 1 is a vertical asymptote. The function is defined for all reals except x = 1 ; the latter is the only point of discontinuity.

We find derivatives: Image

From y we see that the function decreases everywhere (except at x = 1), and from y that the curve is concave down if x < 1, up if x > 1. See Figure N4–8.

Image

FIGURE N4–8

Verify the preceding on your calculator, using [−4,4] × [−4, 8].

EXAMPLE 20

Describe any symmetries of the graphs of

(a) 3y2 + x = 2; (b) y = x + Image (c) x2 − 3y2 = 27.

SOLUTIONS:

(a) Suppose point (x, y) is on this graph. Then so is point (x, −y), since 3(−y)2 + x = 2 is equivalent to 3y2 + x = 2. Then (a) is symmetric about the x-axis.

(b) Note that point (−x, −y) satisfies the equation if point (x, y) does:

Image

Therefore the graph of this function is symmetric about the origin.

(c) This graph is symmetric about the x-axis, the y-axis, and the origin. It is easy to see that, if point (x, y) satisfies the equation, so do points (x, −y), (−x, y), and (−x, −y).