1.2.2Did you look far left and far right?

End Behavior and Asymptotes

1-29.

Eunsun spilled coffee on her math book, making some of the graphs difficult to read!

Coordinate plane, large random edged splotch, covering the plane, between negative 4 & positive 4 horizontally, & between negative 5 & positive 5 vertically, with upper right & lower left ends of a line, passing through the approximate points (5 comma 3), & (negative 5, comma negative 5).

Coordinate plane, with skinny random edged splotch, covering the plane between negative 1 & positive 4 horizontally, & between negative 5 & positive 5 vertically, with upper right & lower left ends of a line passing through the approximate points (5 comma 3), & (negative 5, comma negative 5).

Coordinate plane, large random edged splotch, covering the plane between negative 1 & positive 4 horizontally, & between negative 5 & positive 1 vertically, with additional splotch in first quadrant, with upper right & lower left ends of a line, passing through the approximate points (5 comma 3), & (negative 5, comma negative 5)

Help Eunsun determine the equation of each function graphed above.

When Eunsun arrived at school, she looked at Rudy’s book and discovered that the graphs looked like this:

Compare the graphs in part (a) with the graphs in part (b). Use the words asymptote and hole in your explanation.
The actual equations are:
$y = \frac { x ^ { 2 } - 3 x + 2 } { x - 1 }$
$y = \frac { x ^ { 2 } - 3 x + 3 } { x - 1 }$
$y = \frac { x ^ { 3 } - 2 x ^ { 2 } + x - 1 } { x ^ { 2 } + 1 }$
Use algebra to simplify each expression. What do you notice?

1-30.

Even though Eunsun’s equations were incorrect, to her surprise, she received partial credit on the assignment. The next day, Eunsun’s teacher used her homework to teach a new concept, end behavior!

Graph $f(x) = -x + 2 +\frac { 2 x } { x ^ { 2 } + 1 }$ , on your calculator.

Use your calculator to “zoom out” so you can look far left and far right. Write an equation for the end-behavior function of the graph.
Describe the connection between the equation of the end-behavior function and the equation of the original function.
The linear equation you wrote in part (a) is a slant asymptote for this function. Explain why the name is appropriate.

Digital Graphing Calculator Click in the lower right corner of the graph to view it in full-screen mode.

1-31.

Sketch a graph of $y=\frac { 1 } { x ^ { 2 } + 1 }+3$ . Then look far to the left and far to the right. Determine the equation for the end behavior of this graph. How can this end behavior be predicted from the equation of the original function?

Test your ideas with the 1-31 Student eTool. Click in the lower right corner of the graph to view it in full-screen mode.

1-32.

The tables below give select values of continuous functions $f$ and $g$ . Based on the patterns, describe the general shape of the end behavior of each function. If you can name the end-behavior function, do so.

$x$	$f \left(x\right)$
$–1000$	$499$
$–950$	$470$
$–900$	$440$
$–100$	$300$
$–50$	$200$
$–25$	$300$
$–10$	$500$
$0$	$600$
$10$	$500$
$25$	$300$
$50$	$200$
$100$	$300$
$900$	$440$
$950$	$470$
$1000$	$499$

$x$	$g\left(x\right)$
$–1000$	$–6.99$
$–950$	$–6.97$
$–900$	$–6.92$
$–100$	$–6.42$
$–50$	$–6.20$
$–25$	$–4.80$
$–10$	$200$
$0$	DNE
$10$	$–200$
$25$	$4.80$
$50$	$6.20$
$100$	$6.42$
$900$	$6.92$
$950$	$6.97$
$1000$	$6.99$

Use the 1-32 Student eTool to input the data and make an estimate for the end behavior. Click in the lower right corner of the graph to view it in full-screen mode.

1-33.

Let $f(x)=\frac{6x^2-x+3}{3x+1}$ .

Demonstrate that $f(x) = 2x - 1 +\frac { 4 } { 3 x + 1 }$ .
Using your graphing calculator and a suitable window, graph $y = f (x)$ . Then describe the end behavior. If you can, write an equation for the end behavior.
How can the equation of the end-behavior function be predicted from the equation you wrote in part (a)? How can it be predicted by analyzing the graph of $y = f(x)$ ?

Digital Graphic Caluclator Click in the lower right corner of the graph to view it in full-screen mode.

1-34.

Use algebra to find the end behavior of each equation below.

$f(x) =\frac { - 5 x ^ { 2 } + 3 } { x - 2 }$

$f(x) =\frac { 2 x + 4 } { 3 x - 2 }$

$f(x) =\frac { x ^ { 2 } - 4 x + 4 } { x - 2 }$

$f(x) =\frac { 2 x ^ { 3 } + 2 x } { x }$

Which of the functions above has a horizontal asymptote? Which has a slant asymptote? Which has neither? Explain how knowing the end-behavior function can help answer these questions.
Write the equation of a function that has a horizontal asymptote of $y =\frac { 5 } { 4 }$ .

1-35.

Sketch an example of each the following functions. If it is impossible to sketch, explain why.

A function that has a horizontal asymptote $y=2$ .
A function that has horizontal asymptotes $y=2$ and $y=−3$ .
A function that has horizontal asymptotes $y=2$ , $y=−3$ , and $y=7$ .
A function that passes through the origin and has a horizontal asymptote $y=0$ .
A function that has a horizontal asymptote $y=2$ and a slant asymptote $y=x$ .

Review and Preview problems below

1-36.

The graph of $j(x)=\frac{2}{5}x^2+1$ is shown at right. 1-36 HW eTool. Homework Help ✎

Approximate the area under the curve for $−3\le x\le3$ by calculating the sum of the areas of the six left endpoint rectangles as shown. (The height of a left endpoint rectangle is determined by the function’s value at the left $x$ -value.)
Is the approximation in part (a) too high or too low? How can you tell?
Now, sketch this function with six right endpoint rectangles and compute the approximate area.
You should have obtained the same answers using right and left endpoint rectangles. Will this be true for all functions? If so, explain why. If not, explain what was special about this case that made the area estimates equal. Give an example of a case where the area estimates will be different.

Upward parabola, vertex at (0, comma 1), & 7 vertical shaded bars, bottom edges on x axis, each with width of 1, starting at x = negative 3, with the top left vertex of each bar, on the parabola.

1-37.

A car travels at a rate of $v(t) = 20t + 30$ miles per hour for $0 ≤ t ≤ T$ . 1-37 HW eTool Homework Help ✎

Sketch a velocity graph and label the axes with the correct units.
Shade the area under the curve for $0 ≤ t ≤ T$ . What does this area represent?
What are the units of the area? Explain how you know.
Compute the area under the curve for $0 ≤ t ≤ 2$ . What does your answer represent?

1-38.

If $f(x)=\frac{3}{x^2}+1$ . 1-38 HW eTool Homework Help ✎

State the domain and range of $f$ .
Write expressions for $f(−x)$ , $f(\sqrt { x })$ , and $f(x + h)$ .

1-39.

Graph the following functions on your graphing calculator and zoom out until you can clearly see its end behavior. Then, write an equation for the end-behavior function. 1-39 HW eTool Homework Help ✎

$y = 1 −\frac { 1 } { x }$

$y =\frac { 3 x ^ { 2 } } { 6 x + 1 }$

1-40.

State the domain for each of the functions below. Homework Help ✎

$f(x) =\frac { x } { x ^ { 2 } + 1 }$

$g(x) =\frac { 1 } { x } - \frac { x } { x + 1 }$

$h(x)=\sqrt{x^2-9}$

$k(x)=\frac{\log\left(x-3\right)}{\sqrt{x+4}}$

1-41.

Wei Kit knows that radical expressions can be rewritten using rational exponents. Study his examples below.

Examples: $\sqrt{x}=x^{1/2 }$ $( \sqrt [ 5 ] { z } ) ^ { 2 }= z^{2/5}$ $\sqrt [ 3 ] { m ^ { 2 } } = m^{2/3}$

Rewrite the following radicals expressions with rational exponents. Homework Help ✎

$\sqrt { k ^ { 7 } }$

$\sqrt [ 3 ] { t ^ { 4 } }$

$( \sqrt { n } ) ^ { 4 }$

$\sqrt [ 5 ] { b ^ { 31 } }$

1-42.

Imagine rotating the flag at right about its pole. Homework Help ✎

Describe the resulting three-dimensional figure. Draw a picture of this figure on your paper. To help you visualize this, use the 1-42 eTool. Click in the lower right corner of the graph to view it in full-screen mode.
Calculate the volume of the rotated flag.

Vertical segment, labeled pole, with right triangle, labeled flag, whose left vertical leg, shares most of the upper part of the segment, vertical leg labeled 4, horizontal leg labeled, 5.

1-43.

Copy the graph at right. Then complete it so it will have each type of symmetry described below. 1-43 HW eTool. Homework Help ✎

Reflection symmetry across the $y$ -axis.
Reflection symmetry across the $x$ -axis.
Point symmetry about the origin. (This means a $180^\circ$ rotation about the origin leaves the graph unchanged.)
Recall the definitions of even and odd functions. For each part above, state if the graph is even, odd, or neither.

Increasing curve, starting at the origin, opening down, passing through the points (1, comma 1), & (4, comma 2).

Calculus Third Edition

1.2.2Did you look far left and far right?

Horizontal Asymptotes (informal)