## 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 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 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

**SOLUTION:** If *x* = 0, then *y* = −1. Also, *y* = 0 when the numerator equals zero, which is when 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:

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.

**FIGURE N4–8**

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

**EXAMPLE 20**

Describe any symmetries of the graphs of

(a) 3*y*^{2} + *x* = 2; (b) *y* = *x* + (c) *x*^{2} − 3*y*^{2} = 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 3*y*^{2} + *x* = 2. Then (a) is symmetric about the *x*-axis.

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

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*).