7.5 Graphs I. Polynomial, Rational, & Root Functions

7.5.1  Polynomial Functions

 

Example 1. Given the function f together with its first and second derivatives,

f (x) = x²(x² − 4),     f ^‘(x) = 4x(x² − 2),    f ^”(x) = 4(3x² − 2),

sketch its graph by completing steps (I)–(VI).

(I)   Domain and Intercepts

Since f is a polynomial, D_f = \( \mathbb{R} \).

f (x) = 0\( \Longleftrightarrow x^{2} (x^{2} – 4) = 0 \Longleftrightarrow x \in {0, \pm 2}. \Longrightarrow \text{x-intercepts:} {(0,0), (\pm 2, 0)} \)

(II)  Symmetry

 

f (−x) = (−x)²((−x)² − 4) = x²(x² − 4) = f (x). \(\Longrightarrow\) f is even.

\( \Longrightarrow \)the graph is symmetric with respect to the y-axis.

(III)  Asymptotic Behavior

Given that f is a polynomial function, it is continuous everywhere, thus, it does not have any veritcal asymptotes.

To understand how the graph looks like when x \( \rightarrow \pm \infty \), we rewrite f as follows:

f (x) = x²(x² − 4) = x⁴ − 4x² = x⁴ 1 − \( \frac{4}{x^{4}} \)

And note that

\( x \rightarrow \pm \infty \Longrightarrow x^{4} \rightarrow \infty \text{and} ( 1 – \frac{4}{x^{4}} ) \rightarrow \text{since} \frac{4}{x^{4}} \rightarrow 0 \)

Thus

\( \lim _{x \rightarrow \infty} f(x)=\infty \quad \text { and } \quad \lim _{x \rightarrow-\infty} f(x)=\infty \)

 

(IV) Monotonicity and Local Extrema

Since f\(\prime\)(x) = 4x(x² − 2), the critical numbers of f  are  \( \{ 0, \pm \sqrt{2} \} \)  .  Thus, the signs of f\(\prime\) are as in Figure 7.26(a) (verify this).

Thus, f is increasing on ( – \(\sqrt{2}, 0 \) ) \(\cup (\sqrt{2}, \infty)\), and decreasing on \( ( – \infty, – \sqrt{2} ) \cup (0,\sqrt{2}) \)
It follows that f has a local maximum value f (0) = 0 and local minimum values f ( – \(\sqrt{2})\) = -4 and f ( \(\sqrt{2})\) = -4.

 

 

7.5.2  Rational Functions and Asymptotes

Degree of the numerator ≤ degree of the denominator, Horizontal asymptotes

Example 1. Given the function f together with its first and second derivatives,

\( f(x) = –   \frac{2x^{2} + 10x + 13}{(x + 2)^{2}},   f\prime (x) = \frac{2(x + 3)}{(x + 2)^{3}},  f\prime\prime = – \frac{2(2x + 7)}{(x + 2)^{4}} \)

 

sketch its graph by completing steps (I)–(VI).

 

(I)   Domain and Intercepts

\( D_{f} = ( – \infty, -1 ) \cup ( -1, \infty ) \) x-Intercepts: (0,0); y-intercept: (0,0)

(II)  Symmetry

The function is neither even nor odd. Thus, no symmetry with respect to the y-axis nor the origin.

 

(III)  Asymptotic Behavior Vertical Asymptotes

Vertical Asymptotes

Since

\( \lim\limits_{x \to -2^{+}} \frac{2x^{2} + 10x + 13}{(x+2)^{2}} = – \infty, \lim\limits_{x \to -2^{-}}  –  \frac{2x^{2} + 10x + 13}{(x+2)^{2}} = – \infty \)

 

Horizontal Asymptotes

A rational function \( f = \frac{P}{Q} \) for which degree of P \( \le \) degree of Q has a horizontal asymptote.

Definition 1. A line y = L is said to be a horizontal asymptote of the graph of y = f (x) if

\( \lim\limits_{x \to \infty} f(x) = L or \lim\limits_{x \to – \infty} f(x) = L. \)

Method to find asymptotes of rational functions such that \( f = \frac{P}{Q} \) for which degree of P \( \le \) degree of Q.

Divide numerator and denominator by the highest power of x in the denominator and let \( x \rightarrow \pm \infty \), and use the fact that
lim \( \frac{a}{x^{n}} = 0 \), a contstant n \( \in N, n \ge 1 \).

Thus, in our present example, we have

\( \lim\limits_{x \to\infty} -\frac{2x^{2} + 10x + 13}{(x+2)^{2}} = \lim\limits_{x \to\infty} -\frac{\frac{2x^{2}+10x+13}{x^{2}}}{\frac{(x+2)^{2}}{x^{2}}} = \lim\limits_{x \to\infty} -\frac{2 + \frac{10}{x} + \frac{13}{x^{2}}}{(1 + \frac{2}{x^{2}})^{2}} = -\frac{2}{1} = -2 \)

A similar calculation shows that \( \lim\limits_{x \to\infty} – \frac{2x^{2} + 10 + 13}{(x+2)^{2}} = -2 \)

Hence, the graph has a horizontal asymptote given by y = −2.

 

(IV)  Monotonicity and Local Extrema

Since \( f\prime (x) = \frac{2(x+3)}{(x+2)^{3}} \), f has only one critical number: {-3}. Thus, the signs of f \(\prime \) are as in Figure 7.28(a) (verify this).

Thus, f is increasing on (−∞, −3) ∪ (−2, ∞), and decreasing on (−3, −2).

It follows that f has a local maximum value f (−3) = −1 and no local minimum values.

 

 

Since \( f \prime\prime (x) = – \frac{2(2x+7)}{(x+2)^{4}}, f \prime\prime vanishes at \{ – \frac{7}{2} \). Thus, the signs of \( f\prime\prime \) are as in Figure 7.28(b) (verify this).

Thus, f is concave upward on \( ( – \infty, -\frac{7}{2}), \) and downward on \( ( -\frac{7}{2}, -2 ) \cup (-2, \infty) \).

The inflection point: \( (-\frac{7}{2}, -\frac{10}{9}) \).

(VI) Graph

Based on all the above information, the graph is as shown in Figure 7.27.

 

 

 

Degree of the numerator > degree of the denominator + 1. Other asymptotes.

Example 3. [Other Asymptotic Behavior] Given the function f together with its first and second derivatives,

\( f (x) = \frac{x^{4}}{(x-1)^{2}}, f\prime(x) = \frac{2x^{3}(x-2)}{(x-1)^{3}}, f\prime\prime(x) = \frac{2x^{2}(x^{2} -4x + 6)}{(x-1)^{4}}, \)

sketch its graph by completing steps (I)–(VI).

 

(I) Domain and Intercepts

D_f = (−∞, −1) ∪ (−1, ∞). x-intercepts: (0, 0); y-intercept: (0, 0).

 

(II) Symmetry

The function is neither even nor odd. Thus, no symmetry with respect to the y-axis nor the origin.

 

(III) Asymptotic Behavior

Vertical Asymptotes

Since

\( \lim _{x \rightarrow 1^{+}} \frac{x^{4}}{(x-1)^{2}}=\infty, \quad \lim _{x \rightarrow 1^{-}} \frac{x^{4}}{(x-1)^{2}}=\infty, \Longrightarrow \) vertical asymptote: x=1.

Other Asymptotic Behavior

A rational function \( f = \frac{P}{Q} \) for which degree of P > degree of Q + 1 has asymptotic behavior that does not correspond to a vertical or slant asymptote.

Method to study asymptotc behavior of rational functions such that f = \(\frac{P}{Q} \) for which degree of P > degree of Q + 1.

 

Use polynomial division to divide the numerator by the denominator. Since the quotient q = q(x) is not a linear expression, then the curve y = q(x) is an an asympotote to the graph of f but it is not a line. In our present example, we have (verify this)

\( f(x) = \frac{x^{4}}{(x-1)^{2}} = x^{2} + 2x + 3 + \frac{4x-3}{(x-1)^{2}}. \Longrightarrow \)

\( \Longrightarrow \lim\limits_{x \to\pm\infty} ( \frac{x^{4}}{(x-1)^{2}} – (x^{2} + 2x + 3) ) = \lim\limits_{x \to\pm\infty} \frac{4x-3}{(x-1)^{2}} = 0. \Longrightarrow \)

\( \Longrightarrow \) the graph of f approaches the parabola y = x² + 2x + 3 asymptotically as \( x\rightarrow\pm\infty \)

(see Figure 7.31).

 

(IV)  Monotonicity and Local Extrema

Since \( f\prime(x) = \frac{2x^{3}(x-2)}{(x-1)^{3}} \), f has critical numbers: {0, 2}. Verify that the following is true:

f is increasing on (0, 1) ∪ (2, ∞), and decreasing on (−∞, 0) ∪ (1, 2).

It follows that f has local minimum values f (0) = 0 and f (2) = 16 and no local maximum values.

(V)  Concavity and Inflection Points

Since \( f\prime\prime (x) \frac{2x^{2}(x^{2} – 4x + 6)}{(x-1)^{4}}, f\prime\prime \) vanishes at {0} (explain why). Thus

f is concave upward on (−∞, 1) ∪ (1, ∞), and downward nowhere (verify this). There are no inflection points.

 

(VI)  Graph

Based on all the above information, the graph is as shown in Figure 7.31.

 

 

 

7.5.3 Functions Involving Radicals

Example 1. Given the function f together with its first and second derivatives,

\( f(x) = \frac{x}{\sqrt{4x^{2} – 9}}, f\prime (x) = \frac{9}{(4x^{2} – 9)\sqrt{4x^{2} – 9}}, f\prime\prime (x) = \frac{108x}{(4x^{2} – 9)^{2}\sqrt{4x^{2} – 9}} \)

sketch its graph by completing steps (I)–(VI).

 

(I) Domain and Intercepts

\( D_{f} = ( -\infty, – \frac{3}{2} ) \cup (\frac{3}{2},\infty,). \) no x- nor y- intercepts

 

(II) Symmetry

Given that f (−x) = −f (x) (verify this), f is an odd function, thus its graph is symmetric with respect to the origin.

 

(III) Asymptotic Behavior

Vertical Asymptotes

Since

\( \lim\limits_{x \to(3/2)^{+}} \frac{x}{\sqrt{4x^{2} – 9}} = \infty, \lim\limits_{x \to(-3/2)^{-}} \frac{x}{\sqrt{4x^{2} – 9}} = -\infty, \Longrightarrow \)

Vertical asymptotes: \( x = \pm \frac{3}{2} \)

 

Horizontal Asymptotes

Here, we apply the same method used to find horizontal asymptotes of rational functions (see Example 1 in 7.5.2), and note that the highest power of x in the denominator is x, thus, we compute

\( \lim\limits_{x \to\infty} \frac{x}{\sqrt{4x^{2} – 9}} = \lim\limits_{x \to\infty} \frac{\frac{x}{x}}{\frac{\sqrt{4x^{2} – 9}}{x}} = \lim\limits_{x \to\infty} \frac{1}{ – \sqrt{\frac{4x^{2} – 9}{x^{2}}}} = \lim\limits_{x \to\infty} -\frac{1}{\sqrt{4 – \frac{9}{x^{2}}}} = \frac{1}{2} \)

and

\( \lim\limits_{x \to -\infty} \frac{x}{\sqrt{4x^{2} – 9}} = \lim\limits_{x \to -\infty} \frac{\frac{x}{x}}{\frac{\sqrt{4x^{2} – 9}}{x}} = \lim\limits_{x \to -\infty} \frac{1}{ – \sqrt{\frac{4x^{2} – 9}{x^{2}}}} = \lim\limits_{x \to -\infty} -\frac{1}{\sqrt{4 – \frac{9}{x^{2}}}} = \frac{1}{2} \)

 

 

(^∗) Recall that \( \sqrt{x^{2}} = | x | \), thus when \( x \rightarrow – \infty, \sqrt{x^{2}} = -x. \) Hence the presence of the negative sign.

It follows that the graph has two horizontal asymptotes: \(  x = \pm\frac{1}{2} \)

 

(IV)  Monotonicity and Local Extrema

Since \( f \prime (x) = – \frac{9}{(4x^{2} – 9)\sqrt{4x^{2} – 9}}, f\prime (x) < 0  \text{for all}  x \in D_{f} \) thus f is decreasing on \( D_{f} \) and, therefore, has no local extrema.

(V)  Concavity and Inflection Points

Since \( f^{\prime \prime}(x)=\frac{6 x}{(x-1)^{4}}, f^{\prime \prime} \)

\( f\prime\prime (x) \lt 0 \text{for} x \in (-\infty,-\frac{3}{2} ) \text{and} f\prime\prime (x) \gt 0 \text{for} x \in (\frac{3}{2},\infty) \)

Thus,

f is concave upward on \( \frac{3}{2}, \infty ), and downward on (-\infty, -\frac{3}{2}. \) There are no inflection points.

 

(VI)  Graph

Based on all the above information, the graph is as shown in Figure 7.32.

 

7.5.4   Exercises

In problems 1–10, given the function f together with its first and second derivatives, sketch its graph by completing steps (I)–(VI) as in the examples above.

1.  \( f(x) = x^{3}(5-6x^{2}), f\prime (x) = -15x^{2}(2x^{2} – 1 ), f\prime\prime (x) = 30x(1 – 4x^{2} ) \).
2. \( f (x) = \frac{x}{x^{2} – 4}, f\prime (x) = \frac{x^{2} + 4}{(x^{2} – 4)^{2}}, f\prime\prime (x) = -\frac{2x(x^{2} + 12)}{(x^{2} – 4)^{3}} \)
3. \(  f(x) = \frac{x^{3}}{x^{2}-4}, f\prime (x) = \frac{x^{2}(x^{2} – 12)}{(x^{2} – 4)^{2}}, f\prime\prime (x) = \frac{8x(x^{2} + 12 )}{(x^{2} – 4)^{3}} \)
4. \( f(x) = \frac{x^{2}}{x^{2} – 4}, f\prime (x) = – \frac{8x}{(x^{2} – 4 )^{2}}, f\prime\prime (x) = \frac{8(3x^{2} + 4)}{(x^{2} – 4)^{3}} \)
5. \( f(x) = \frac{2x + 4}{\sqrt{x^{2} + 2}}, f\prime (x) = -\frac{4(x-1)}{(x^{2} + 2)\sqrt{x^{2} + 2}}, f\prime\prime (x) = \frac{4(2x^{2} -3x – 2)}{(x^{2} + 2)^{2}\sqrt{x^{2} + 2}}. \)
6. \( f(x) = \frac{x^{2}(x + 2)}{(x-1)^{2}}, f\prime (x) = \frac{x(x^{2} -3x -4)}{(x-1)^{3}}, f\prime\prime (x) = \frac{2(7x + 2)}{(x-1)^{3}} \)
7. \( f(x) = \frac{x^{4}}{x^{2} -2}, f\prime (x) = \frac{2x^{3}(x^{2} – 4)}{(x^{2} – 2)^{2}}, f\prime\prime (x) = \frac{2x^{2}(x^{4} -6x^{2} + 24)}{(x^{2} – 2)^{3}} \)
8. \( f(x) = \frac{\sqrt{x^{2} – 2}}{x-2}, f\prime (x) = – \frac{2(x-1)}{(x-2)^{2}\sqrt{x^{2} – 2}}, f\prime\prime (x) = \frac{2x^{2}(2x – 3)}{(x – 2)^{3}(x^{2} – 2)\sqrt{x^{2} – 2}} \)
9. \( f (x) = \frac{x^{4}}{(2x-1)^{2}}, f\prime (x) = \frac{4x^{3}(x-1)}{(2x – 1)^{3}}, f\prime\prime (x) = \frac{4x^{2}(2x^{2} – 4x + 3)}{(2x – 1)^{4}} \)
10. \( f (x) = \frac{x^{4}}{8 – x^{3}}, f\prime (x) = -\frac{x^{3}(x^{3} – 32)}{(x^{3} – 8)^{2}}, f \prime\prime (x) = -\frac{48x^{2}(x^{3} + 16)}{(x^{3} – 8)^{3}} \)

 

In problems 11–14, sketch the graph of a function that matches the given information.

11. (I) \(D_{f}=(-\infty,-\sqrt{2}) \cup(-\sqrt{2}, \sqrt{2}) \cup(\sqrt{2}, \infty)\); x-int: (0,0); (II) symmetric with respect to the origin; (III) \(\lim\limits_{x \to \sqrt{2}} f(x)=\infty ; \lim\limits_{x \to-\infty} f(x)=0 ;(\mathrm{IV}) f^{\prime}(x)>0 \text { if } x \in(-\sqrt{2}, \sqrt{2}))\); \(f^{\prime}(x)<0 \text { if } x \in(-\infty,-\sqrt{2}) \cup(\sqrt{2}, \infty) ;(\mathrm{V}) f^{\prime \prime}(x)>0 \text { if } x \in D_{f}, x>0 ; f^{\prime \prime}(x)<0 \text { if } x \in D_{f}, x< 0 \)
     
12. (I) \( D_{f}=(-\infty, 1) \cup(1, \infty)\); x-int: (0,0); (II) No symmetry; (III) \( \lim\limits_{x \to 1} f(x)=-\infty\); slant asymptote: y = -x – 2; (IV) \( f^{\prime}(x)>0 \text { if } x \in(1,3) ; f^{\prime}(x)<0\) else where in its domain; (V) \(f^{\prime \prime}(x)>0 \text { if } x \in D_{f}, x<0 ; f^{\prime \prime}(x)<0 \text { if } x x \in D_{f}, x>0\)
     
13. (I) \( D_{f}=\left(-\infty,-\frac{3}{2}\right) \cup\left(-\frac{3}{2}, 1\right) \cup(1, \infty)\); x-int: (\(\pm\frac{3}{2}, 0)\); (II) No symmetry (III) \(\lim\limits_{x \to-\infty} f(x)=2 ; \lim\limits_{x \to \infty} f(x)=-2 \); (IV) \(f^{\prime}(x)>0 \text { if } x \in\left(\frac{9}{4}, \infty\right) ; f^{\prime}(x)<0\) elsewhere in \(D_{f}\); local min. at f(2.25) = -2.7; (V) \(f^{\prime \prime}(x)>0 \text { if } x \in\left(\frac{3}{2}, 2.9\right) ; f^{\prime \prime}(x)<0 \) elsewhere in \(D_{f}\);.
     
14. (I) \(D_{f}=(-\infty .-3) \cup(-3, \infty)\); x-int: (-5.4, 0), (-2.6,0); (II) No symmetry; (III) \(\lim\limits_{x \to-3} f(x)=-\infty ; \lim\limits_{x \to \pm \infty} f(x)=1\); (IV) \(f^{\prime}(x)>0 \text { if } x \in(-3,-2) ; f^{\prime}(x)<0\) elsewhere in \(D_{f}\); local max. f(-2) = 2; (V)  \(f^{\prime \prime}(x)>0 \text { if } x \in\left(-\frac{3}{2}, \infty\right) ; f^{\prime \prime}(x)<0\) elsewhere in \(D_{f}\), infl. pt. -1.5, -1.9).