# Rational functions can have any polynomial in the numerator and denominator. Analyse the key features of each function and sketch its graph. Describe the common features of the graphs. a) f(x)=frac{x}{x^{2}-1} b)g(x)=frac{x-2}{x^{2}+3x+2} c)h(x)=frac{x+5}{x^{2}-x-12}

Question
Polynomial graphs
Rational functions can have any polynomial in the numerator and denominator. Analyse the key features of each function and sketch its graph. Describe the common features of the graphs. $$\displaystyle{a}{)}{f{{\left({x}\right)}}}={\frac{{{x}}}{{{x}^{{{2}}}-{1}}}}\ {b}{)}{g{{\left({x}\right)}}}={\frac{{{x}-{2}}}{{{x}^{{{2}}}+{3}{x}+{2}}}}\ {c}{)}{h}{\left({x}\right)}={\frac{{{x}+{5}}}{{{x}^{{{2}}}-{x}-{12}}}}$$

2020-12-04

Step 1
a. Function: $$\displaystyle{f{{\left({x}\right)}}}={\frac{{{x}}}{{{x}^{{{2}}}-{1}}}}$$
Step 2 For vertical asymptote, equate the expression in the denominator with 0 and solve for x, therefore: $$\displaystyle{x}^{{{2}}}-{1}={0}$$
Solve for x: $$\displaystyle{x}^{{{2}}}={1}$$
Apply square root on both sides of the equation: $$\displaystyle{x}=\sqrt{{{1}}}=\pm{1}$$
Step 3
The equation of vertical asymptote also gives information about the domain of this function, therefore the domain of $$\displaystyle{f{{\left({x}\right)}}}\ {i}{s}\ {\left(-\infty,-{1}\right)}\cup{\left(-{1},{1}\right)}\cup{\left({1},\infty\right)}$$
Step 4
SInce the degree of the expression in the denominator is 2, greater than the degree of the expression in the numerator, horizontal asymptote of this function is $$\displaystyle{y}={0}$$
Step 5
The range of this function $$\displaystyle{f{{\left({x}\right)}}}\ {i}{s}\ {\left(-\infty,\infty\right)}.$$ Step 6
Graph of $$\displaystyle{f{{\left({x}\right)}}}={\frac{{{x}}}{{{x}^{{{2}}}-{1}}}}$$

Step 7
b. Function: $$\displaystyle{g{{\left({x}\right)}}}={\frac{{{x}-{2}}}{{{x}^{{{2}}}+{3}{x}+{2}}}}$$
Step 8
For vertical asymptote, equate the expression in the denominator with 0 and solve for x, therefore: $$\displaystyle{x}^{{{2}}}+{3}{x}+{2}={0}$$
Factor using ac method: $$\displaystyle{x}^{{{2}}}+{x}+{2}{x}+{2}={0}$$
Factor: $$\displaystyle{\left({x}+{2}\right)}{\left({x}+{1}\right)}={0}$$
Solve for x: $$\displaystyle{x}=-{2}$$
And:$$\displaystyle{x}=-{1}$$
The equation of vertical asymptotes also gives information about the domain of this function, therefore the domain of $$\displaystyle{g{{\left({x}\right)}}}\ {i}{s}\ {\left(-\infty,-{2}\right)}\cup{\left(-{2},-{1}\right)}\cup{\left(-{1},\infty\right)}.$$
Step 9
Since the degree of the expression in the denominator is 2, greater than the degree of the expression in the numerator, horizontal asymptote of this function is $$\displaystyle{y}={0}$$
Step 10
The range of this function PSKg(x)\ is\ (-\infty, 0) \cup (13,928, \infty) (from graph)
Step 11
Graph of $$\displaystyle{g{{\left({x}\right)}}}={\frac{{{x}-{2}}}{{{x}^{{{2}}}+{3}{x}+{2}}}}$$

Step 12
c.Function: $$\displaystyle{h}{\left({x}\right)}={\frac{{{x}+{5}}}{{{x}^{{{2}}}-{x}-{12}}}}$$
Step 13
For vertical asymptote, equate the expression in the denominator with 0 and solve for x, therefore: $$\displaystyle{x}^{{{2}}}-{x}-{12}={0}$$
Factor using ac method: $$\displaystyle{x}^{{{2}}}-{4}{x}+{3}{x}-{12}={0}$$
Factor: $$\displaystyle{\left({x}-{4}\right)}{\left({x}+{3}\right)}={0}$$
Solve for x: $$\displaystyle{x}=-{3}$$
And: $$\displaystyle{x}={4}$$
Step 14
The equation of vertical asymptote also gives information about the domain of this function, therefore the domain of $$\displaystyle{h}{\left({x}\right)}\ {i}{s}\ {\left(-\infty,-{3}\right)}\cup\ {\left(-{3},{4}\right)}\cup{\left({4},\infty\right)}.$$
Step 15
SInce the degree of the expression in the denominator is 2, greater than the degree of the expression in the numerator, horizontal asymptote of this function is $$\displaystyle{y}={0}$$.
Step 16
The range of this function h(x)\ is\ (-\infty, -0.398) \cup (-0.398, \infty) (from graph).
Step 17
$$\displaystyle{h}{\left({x}\right)}={\frac{{{x}+{5}}}{{{x}^{{{2}}}-{x}-{12}}}}$$

### Relevant Questions

Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.
a) Identify the parameters a, k, d, and c in the polynomial function $$\displaystyle{y}={\frac{{{1}}}{{{3}}}}{\left[-{2}{\left({x}+{3}\right)}\right]}^{{{4}}}-{1}$$. Describe how each parameter transforms the base function $$\displaystyle{y}={x}^{{{4}}}$$. b) State the domain and range, the vertex, and the equation of the axis of symmetry of the transformed function. c) Describe two possible orders in which the transformations can be applied to the graph of $$\displaystyle{y}={x}^{{{4}}}$$ to produce the graph of $$\displaystyle{y}={\frac{{{1}}}{{{3}}}}{\left[-{2}{\left({x}+{3}\right)}\right]}^{{{4}}}-{1}$$. d) Sketch graphs of the base function and the transformed function on the same set of axes.
a) Identify the parameters a, b, h, and k in the polynomial $$\displaystyle{y}={\frac{{{1}}}{{{3}}}}{\left({x}+{3}\right)}^{{{3}}}-{2}$$ Describe how each parameter transforms the base function $$\displaystyle{y}={x}^{{{3}}}$$.
b) State the domain and range of the transformed function.
c) Sketch graphs of the base function and the transformed function on the same set of axes.
For each of the following functions f (x) and g(x), express g(x) in the form a: f (x + b) + c for some values a,b and c, and hence describe a sequence of horizontal and vertical transformations which map f(x) to g(x).
$$\displaystyle{f{{\left({x}\right)}}}={x}^{{2}}-{2},{g{{\left({x}\right)}}}={2}+{8}{x}-{4}{x}^{{2}}$$
Using calculus, it can be shown that the arctangent function can be approximated by the polynomial
$$\displaystyle{\arctan{\ }}{x}\ \approx\ {x}\ -\ {\frac{{{x}^{{{3}}}}}{{{3}}}}\ +\ {\frac{{{x}^{{{5}}}}}{{{5}}}}\ -\ {\frac{{{x}^{{{7}}}}}{{{7}}}}$$
where x is in radians.
a) Use a graphing utility to graph the arctangent function and its polynomial approximation in the same viewing window. How do the graphs compare?
b) Study the pattern in the polynomial approximation of the arctangent function and predict the next term. Then repeat part (a). How does the accuracy of the approximation change when an additional term is added?
The bulk density of soil is defined as the mass of dry solidsper unit bulk volume. A high bulk density implies a compact soilwith few pores. Bulk density is an important factor in influencing root development, seedling emergence, and aeration. Let X denotethe bulk density of Pima clay loam. Studies show that X is normally distributed with $$\displaystyle\mu={1.5}$$ and $$\displaystyle\sigma={0.2}\frac{{g}}{{c}}{m}^{{3}}$$.
(a) What is thedensity for X? Sketch a graph of the density function. Indicate onthis graph the probability that X lies between 1.1 and 1.9. Findthis probability.
(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than $$\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}$$.
(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of $$\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}$$? Explain, based on theprobability of this occurring.
(d) What point has the property that only 10% of the soil samples have bulk density this high orhigher?
(e) What is the moment generating function for X?
Part II
29.[Poles] (a) For each of the pole diagrams below:
(i) Describe common features of all functions f(t) whose Laplace transforms have the given pole diagram.
(ii) Write down two examples of such f(t) and F(s).
The diagrams are: $$(1) {1,i,-i}. (2) {-1+4i,-1-4i}. (3) {-1}. (4)$$ The empty diagram.
(b) A mechanical system is discovered during an archaeological dig in Ethiopia. Rather than break it open, the investigators subjected it to a unit impulse. It was found that the motion of the system in response to the unit impulse is given by $$w(t) = u(t)e^{-\frac{t}{2}} \sin(\frac{3t}{2})$$
(i) What is the characteristic polynomial of the system? What is the transfer function W(s)?
(ii) Sketch the pole diagram of the system.
(ii) The team wants to transport this artifact to a museum. They know that vibrations from the truck that moves it result in vibrations of the system. They hope to avoid circular frequencies to which the system response has the greatest amplitude. What frequency should they avoid?
Sketch graphs of each of the following polynomial functions. Be sure to label the x- and they-intercepts of each graph. $$\displaystyle{a}.{y}={x}{\left({2}{x}+{3}\right)}{\left({2}{x}-{5}\right)}{b}.{y}={\left({11}-{2}{x}\right)}^{{{2}}}{\left({x}-{2}\right)}$$

A random sample of $$n_1 = 14$$ winter days in Denver gave a sample mean pollution index $$x_1 = 43$$.
Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
Previous studies show that $$\sigma_2 = 13$$.
Assume the pollution index is normally distributed in both Englewood and Denver.
(a) State the null and alternate hypotheses.
$$H_0:\mu_1=\mu_2.\mu_1>\mu_2$$
$$H_0:\mu_1<\mu_2.\mu_1=\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1<\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1\neq\mu_2$$
(b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations.
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
(c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate.
(Test the difference $$\mu_1 - \mu_2$$. Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)
(e) Based on your answers in parts (i)−(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \alpha?
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are not statistically significant.
(f) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for
$$\mu_1 - \mu_2$$.
(Round your answers to two decimal places.)
lower limit
upper limit
(h) Explain the meaning of the confidence interval in the context of the problem.
Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.
Using calculus, it can be shown that the secant function can be approximated by the polynomial $$\displaystyle{\sec{{x}}}\approx{1}+{\frac{{{x}^{{{2}}}}}{{{2}!}}}+{\frac{{{5}{x}^{{{4}}}}}{{{4}!}}}$$ where x is in radians. Use a graphing utility to graph the secant function and its polynomial approximation in the same viewing window. How do the graphs compare?