2. Graph g(x)-4x2%3DList and identify on the graph:x2-x-6vertical asymptotesa.b. horizontal asymptotec. y-interceptd. x-interceptsIn your graph you should label all intercepts and asymptotes along with at least one extra point inbetween any vertical asymptotes and/or x-intercepts.2--8-6-4-246.-2--4-6- 9. Sketch the graphs of the following logarithmic functions. Be sure to clearly indicate at least twopoints and any asymptotes. If a point does not lie on a grid point, label with the ordered pair.a. f(x) = log4 xb. g(x) = log4(x + 3)5

Answered: 2. Graph g(x) -4x2 %3D x2-x-6 List and…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

100%

2. Graph g(x)
-4x2
%3D
List and identify on the graph:
x2-x-6
vertical asymptotes
a.
b. horizontal asymptote
c. y-intercept
d. x-intercepts
In your graph you should label all intercepts and asymptotes along with at least one extra point in
between any vertical asymptotes and/or x-intercepts.
2-
-8
-6
-4
-2
4
6.
-2-
-4
-6-

9. Sketch the graphs of the following logarithmic functions. Be sure to clearly indicate at least two
points and any asymptotes. If a point does not lie on a grid point, label with the ordered pair.
a. f(x) = log4 x
b. g(x) = log4(x + 3)
5

Expert Solution

Step 1

"Hey, since there are multiple questions posted, we will answer first question. If you want any specific question to be answered then specify the question number in your message".

2. $g (x) = \frac{- 4 x^{2}}{x^{2} - x - 6}$

a) Vertical Asymptotes:

To find the vertical asymptotes, solve the equation $x^{2} - x - 6 = 0$

$x^{2} - x - 6 = 0 x^{2} + 2 x - 3 x - 6 = 0 x (x + 2) - 3 (x + 2) = 0 (x - 3) (x + 2) = 0 x = - 2, 3$

Therefore, Vertical asymptotes are $x = - 2 a n d x = 3$

Step 2

b) Horizontal asymptotes:

Line y = L is a horizontal asymptote of the function $y = g (x),$ if either $\lim_{x \to \infty} g (x) = L o r \lim_{x \to - \infty} g (x) = L$ and L is finite.

$\lim_{x \to \infty} \frac{- 4 x^{2}}{x^{2} - x - 6} = - 4 \cdot \lim_{x \to \infty} \frac{x^{2}}{x^{2} - x - 6} (\lim_{x \to \infty} c f (x) = c \lim_{x \to \infty} f (x)) = - 4 \cdot \lim_{x \to \infty} \frac{1}{1 - \frac{1}{x} - \frac{6}{x^{2}}} (D i v i d e b y x^{2}) = - 4 \cdot \frac{1}{1 - 0 - 0} = - 4$

Similarly $\lim_{x \to - \infty} \frac{- 4 x^{2}}{x^{2} - x - 6} = - 4$

Therefore, Horizontal asymptote = -4

Step 3

c) y-intercept:

To find y-intercept, set x = 0

Substitute x = 0 in $g (x) = \frac{- 4 x^{2}}{x^{2} - x - 6}$

$y = \frac{- 4 x^{2}}{x^{2} - x - 6} y = \frac{- 4 (0)}{0 - 0 - 6} y = 0$

The y-intercept (0,0)

Step by step

Solved in 5 steps with 1 images

Knowledge Booster

$Hyperbolas$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions