Answer :
To sketch the graph of the equation [tex]\( y = \frac{-3}{x} + 6 \)[/tex], we need to gather and analyze critical information about the function. This includes identifying asymptotes, intercepts, and the overall behavior of the function. Let’s break this down step by step:
### Step 1: Identify the Vertical Asymptote
The function has a vertical asymptote where the denominator equals zero, causing the function to approach infinity. For the equation [tex]\( y = \frac{-3}{x} + 6 \)[/tex]:
- Set the denominator equal to zero: [tex]\( x = 0 \)[/tex].
- Therefore, there is a vertical asymptote at [tex]\( x = 0 \)[/tex].
### Step 2: Identify the Horizontal Asymptote
As [tex]\( x \)[/tex] approaches infinity or negative infinity, the term [tex]\( \frac{-3}{x} \)[/tex] approaches zero. Thus, the equation approaches the value of the constant term added to it:
- As [tex]\( x \to \infty \)[/tex] or [tex]\( x \to -\infty \)[/tex], [tex]\( y \to 6 \)[/tex].
- Therefore, the horizontal asymptote is [tex]\( y = 6 \)[/tex].
### Step 3: Find the Y-Intercept
The y-intercept occurs where the graph intersects the y-axis (i.e., where [tex]\( x = 0 \)[/tex]). However, since [tex]\( x = 0 \)[/tex] leads to an undefined value, we choose a nearby point, say [tex]\( x = 1 \)[/tex]:
- Substitute [tex]\( x = 1 \)[/tex] into the equation: [tex]\( y = \frac{-3}{1} + 6 = -3 + 6 = 3 \)[/tex].
- Therefore, the y-intercept is at [tex]\( (1, 3) \)[/tex].
### Step 4: Find the X-Intercept
The x-intercept occurs where the graph intersects the x-axis (i.e., where [tex]\( y = 0 \)[/tex]). To find the x-intercept, solve for [tex]\( x \)[/tex] when [tex]\( y = 0 \)[/tex]:
- Set [tex]\( y = 0 \)[/tex]: [tex]\( \frac{-3}{x} + 6 = 0 \)[/tex].
- Solve for [tex]\( x \)[/tex]: [tex]\( \frac{-3}{x} = -6 \)[/tex], so [tex]\( x = \frac{1}{2} \)[/tex].
- Therefore, the x-intercept is at [tex]\( (0.5, 0) \)[/tex].
### Step 5: Sketch the Graph
Using the information gathered:
1. Asymptotes:
- Vertical asymptote at [tex]\( x = 0 \)[/tex].
- Horizontal asymptote at [tex]\( y = 6 \)[/tex].
2. Intercepts:
- x-intercept at [tex]\( (0.5, 0) \)[/tex].
- y-intercept at [tex]\( (1, 3) \)[/tex].
### Graph Behavior
- For [tex]\( x > 0 \)[/tex], the function [tex]\( y = \frac{-3}{x} + 6 \)[/tex] decreases from [tex]\( y = 6 \)[/tex] and approaches the vertical asymptote at [tex]\( x = 0 \)[/tex] as [tex]\( x \)[/tex] gets smaller.
- For [tex]\( x < 0 \)[/tex], the function increases from [tex]\( y = 6 \)[/tex] and also approaches the vertical asymptote at [tex]\( x = 0 \)[/tex] as [tex]\( x \)[/tex] gets larger.
### Final Sketch
1. Draw the vertical asymptote [tex]\( x = 0 \)[/tex].
2. Draw the horizontal asymptote [tex]\( y = 6 \)[/tex].
3. Plot the x-intercept [tex]\( (0.5, 0) \)[/tex] and the y-intercept [tex]\( (1, 3) \)[/tex].
4. Draw a smooth curve indicating the behavior:
- Approaching the vertical asymptote from both sides.
- Approaching the horizontal asymptote as [tex]\( x \)[/tex] moves towards positive and negative infinity.
This sketch provides a clear visualization of the function [tex]\( y = \frac{-3}{x} + 6 \)[/tex].
### Step 1: Identify the Vertical Asymptote
The function has a vertical asymptote where the denominator equals zero, causing the function to approach infinity. For the equation [tex]\( y = \frac{-3}{x} + 6 \)[/tex]:
- Set the denominator equal to zero: [tex]\( x = 0 \)[/tex].
- Therefore, there is a vertical asymptote at [tex]\( x = 0 \)[/tex].
### Step 2: Identify the Horizontal Asymptote
As [tex]\( x \)[/tex] approaches infinity or negative infinity, the term [tex]\( \frac{-3}{x} \)[/tex] approaches zero. Thus, the equation approaches the value of the constant term added to it:
- As [tex]\( x \to \infty \)[/tex] or [tex]\( x \to -\infty \)[/tex], [tex]\( y \to 6 \)[/tex].
- Therefore, the horizontal asymptote is [tex]\( y = 6 \)[/tex].
### Step 3: Find the Y-Intercept
The y-intercept occurs where the graph intersects the y-axis (i.e., where [tex]\( x = 0 \)[/tex]). However, since [tex]\( x = 0 \)[/tex] leads to an undefined value, we choose a nearby point, say [tex]\( x = 1 \)[/tex]:
- Substitute [tex]\( x = 1 \)[/tex] into the equation: [tex]\( y = \frac{-3}{1} + 6 = -3 + 6 = 3 \)[/tex].
- Therefore, the y-intercept is at [tex]\( (1, 3) \)[/tex].
### Step 4: Find the X-Intercept
The x-intercept occurs where the graph intersects the x-axis (i.e., where [tex]\( y = 0 \)[/tex]). To find the x-intercept, solve for [tex]\( x \)[/tex] when [tex]\( y = 0 \)[/tex]:
- Set [tex]\( y = 0 \)[/tex]: [tex]\( \frac{-3}{x} + 6 = 0 \)[/tex].
- Solve for [tex]\( x \)[/tex]: [tex]\( \frac{-3}{x} = -6 \)[/tex], so [tex]\( x = \frac{1}{2} \)[/tex].
- Therefore, the x-intercept is at [tex]\( (0.5, 0) \)[/tex].
### Step 5: Sketch the Graph
Using the information gathered:
1. Asymptotes:
- Vertical asymptote at [tex]\( x = 0 \)[/tex].
- Horizontal asymptote at [tex]\( y = 6 \)[/tex].
2. Intercepts:
- x-intercept at [tex]\( (0.5, 0) \)[/tex].
- y-intercept at [tex]\( (1, 3) \)[/tex].
### Graph Behavior
- For [tex]\( x > 0 \)[/tex], the function [tex]\( y = \frac{-3}{x} + 6 \)[/tex] decreases from [tex]\( y = 6 \)[/tex] and approaches the vertical asymptote at [tex]\( x = 0 \)[/tex] as [tex]\( x \)[/tex] gets smaller.
- For [tex]\( x < 0 \)[/tex], the function increases from [tex]\( y = 6 \)[/tex] and also approaches the vertical asymptote at [tex]\( x = 0 \)[/tex] as [tex]\( x \)[/tex] gets larger.
### Final Sketch
1. Draw the vertical asymptote [tex]\( x = 0 \)[/tex].
2. Draw the horizontal asymptote [tex]\( y = 6 \)[/tex].
3. Plot the x-intercept [tex]\( (0.5, 0) \)[/tex] and the y-intercept [tex]\( (1, 3) \)[/tex].
4. Draw a smooth curve indicating the behavior:
- Approaching the vertical asymptote from both sides.
- Approaching the horizontal asymptote as [tex]\( x \)[/tex] moves towards positive and negative infinity.
This sketch provides a clear visualization of the function [tex]\( y = \frac{-3}{x} + 6 \)[/tex].
