Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Experience the ease of finding quick and accurate answers to your questions from professionals on our platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To graph the function [tex]\( q(x) = \frac{-4}{2x - 3} \)[/tex] and identify its asymptotes, we need to follow a systematic approach. Let's break it down step-by-step:
### Step 1: Identify the Vertical Asymptotes
Vertical asymptotes occur where the denominator of the function is equal to zero because the function value tends towards infinity at these points.
For [tex]\( q(x) = \frac{-4}{2x - 3} \)[/tex]:
Set the denominator equal to zero:
[tex]\[ 2x - 3 = 0 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ 2x = 3 \][/tex]
[tex]\[ x = \frac{3}{2} \][/tex]
So, there is a vertical asymptote at [tex]\( x = \frac{3}{2} \)[/tex].
### Step 2: Identify the Horizontal Asymptote
Horizontal asymptotes are identified by analyzing the behavior of the function as [tex]\( x \)[/tex] approaches positive or negative infinity.
For the function [tex]\( q(x) = \frac{-4}{2x - 3} \)[/tex]:
Consider the limit as [tex]\( x \)[/tex] approaches infinity:
[tex]\[ \lim_{x \to \infty} \frac{-4}{2x - 3} \][/tex]
As [tex]\( x \)[/tex] becomes very large, the term [tex]\( -3 \)[/tex] in the denominator becomes negligible. Thus, the function approaches:
[tex]\[ \lim_{x \to \infty} \frac{-4}{2x - 3} \approx \frac{-4}{2x} \][/tex]
[tex]\[ = \lim_{x \to \infty} \frac{-4}{2} \cdot \frac{1}{x} \][/tex]
[tex]\[ = \frac{-4}{2} \cdot 0 \][/tex]
[tex]\[ = 0 \][/tex]
So, the horizontal asymptote is at [tex]\( y = 0 \)[/tex].
### Step 3: Graph the Function
To plot the function, we follow these steps:
1. Plot the Asymptotes:
- Draw a vertical line at [tex]\( x = \frac{3}{2} \)[/tex].
- Draw a horizontal line at [tex]\( y = 0 \)[/tex].
2. Plot the Function:
- Choose several points on either side of the vertical asymptote to get the shape of the curve.
- As [tex]\( x \)[/tex] approaches the vertical asymptote from either side, the function value tends towards [tex]\( \pm \infty \)[/tex].
- As [tex]\( x \)[/tex] approaches [tex]\( \pm \infty \)[/tex], the function approaches the horizontal asymptote [tex]\( y = 0 \)[/tex].
Here is a sketch of the graph based on these observations:
### Graph of [tex]\( q(x) = \frac{-4}{2x - 3} \)[/tex]:
1. Vertical Asymptote: [tex]\( x = \frac{3}{2} \)[/tex]
2. Horizontal Asymptote: [tex]\( y = 0 \)[/tex]
[tex]\[ \begin{array}{c|c} x & q(x) \\ \hline -2 & \frac{-4}{-4 - 3} = \frac{-4}{-7} \approx 0.571 \\ 0 & \frac{-4}{-3} \approx 1.333 \\ 1 & \frac{-4}{-1} = 4 \\ \end{array} \][/tex]
As [tex]\( x \to \frac{3}{2}^- \)[/tex], [tex]\( q(x) \to -\infty \)[/tex].
As [tex]\( x \to \frac{3}{2}^+ \)[/tex], [tex]\( q(x) \to \infty \)[/tex].
### Drawing:
On a Cartesian plane, draw the vertical line [tex]\( x = \frac{3}{2} \)[/tex] and the horizontal line [tex]\( y = 0 \)[/tex]. Plot points such as [tex]\( (-2, 0.571) \)[/tex], [tex]\( (0, 1.333) \)[/tex], and [tex]\( (1, 4) \)[/tex]. Draw the curve approaching the asymptotes:
- For [tex]\( x < \frac{3}{2} \)[/tex], the curve falls from positive values approaching the vertical asymptote.
- For [tex]\( x > \frac{3}{2} \)[/tex], the curve rises from negative values approaching the vertical asymptote.
In this manner, the graph will display the behavior defined by the asymptotes and calculated points.
### Step 1: Identify the Vertical Asymptotes
Vertical asymptotes occur where the denominator of the function is equal to zero because the function value tends towards infinity at these points.
For [tex]\( q(x) = \frac{-4}{2x - 3} \)[/tex]:
Set the denominator equal to zero:
[tex]\[ 2x - 3 = 0 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ 2x = 3 \][/tex]
[tex]\[ x = \frac{3}{2} \][/tex]
So, there is a vertical asymptote at [tex]\( x = \frac{3}{2} \)[/tex].
### Step 2: Identify the Horizontal Asymptote
Horizontal asymptotes are identified by analyzing the behavior of the function as [tex]\( x \)[/tex] approaches positive or negative infinity.
For the function [tex]\( q(x) = \frac{-4}{2x - 3} \)[/tex]:
Consider the limit as [tex]\( x \)[/tex] approaches infinity:
[tex]\[ \lim_{x \to \infty} \frac{-4}{2x - 3} \][/tex]
As [tex]\( x \)[/tex] becomes very large, the term [tex]\( -3 \)[/tex] in the denominator becomes negligible. Thus, the function approaches:
[tex]\[ \lim_{x \to \infty} \frac{-4}{2x - 3} \approx \frac{-4}{2x} \][/tex]
[tex]\[ = \lim_{x \to \infty} \frac{-4}{2} \cdot \frac{1}{x} \][/tex]
[tex]\[ = \frac{-4}{2} \cdot 0 \][/tex]
[tex]\[ = 0 \][/tex]
So, the horizontal asymptote is at [tex]\( y = 0 \)[/tex].
### Step 3: Graph the Function
To plot the function, we follow these steps:
1. Plot the Asymptotes:
- Draw a vertical line at [tex]\( x = \frac{3}{2} \)[/tex].
- Draw a horizontal line at [tex]\( y = 0 \)[/tex].
2. Plot the Function:
- Choose several points on either side of the vertical asymptote to get the shape of the curve.
- As [tex]\( x \)[/tex] approaches the vertical asymptote from either side, the function value tends towards [tex]\( \pm \infty \)[/tex].
- As [tex]\( x \)[/tex] approaches [tex]\( \pm \infty \)[/tex], the function approaches the horizontal asymptote [tex]\( y = 0 \)[/tex].
Here is a sketch of the graph based on these observations:
### Graph of [tex]\( q(x) = \frac{-4}{2x - 3} \)[/tex]:
1. Vertical Asymptote: [tex]\( x = \frac{3}{2} \)[/tex]
2. Horizontal Asymptote: [tex]\( y = 0 \)[/tex]
[tex]\[ \begin{array}{c|c} x & q(x) \\ \hline -2 & \frac{-4}{-4 - 3} = \frac{-4}{-7} \approx 0.571 \\ 0 & \frac{-4}{-3} \approx 1.333 \\ 1 & \frac{-4}{-1} = 4 \\ \end{array} \][/tex]
As [tex]\( x \to \frac{3}{2}^- \)[/tex], [tex]\( q(x) \to -\infty \)[/tex].
As [tex]\( x \to \frac{3}{2}^+ \)[/tex], [tex]\( q(x) \to \infty \)[/tex].
### Drawing:
On a Cartesian plane, draw the vertical line [tex]\( x = \frac{3}{2} \)[/tex] and the horizontal line [tex]\( y = 0 \)[/tex]. Plot points such as [tex]\( (-2, 0.571) \)[/tex], [tex]\( (0, 1.333) \)[/tex], and [tex]\( (1, 4) \)[/tex]. Draw the curve approaching the asymptotes:
- For [tex]\( x < \frac{3}{2} \)[/tex], the curve falls from positive values approaching the vertical asymptote.
- For [tex]\( x > \frac{3}{2} \)[/tex], the curve rises from negative values approaching the vertical asymptote.
In this manner, the graph will display the behavior defined by the asymptotes and calculated points.
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.
