Graph the function by using transformations of the graph of . Plot all necessary asymptotes. For vertical
asymptotes, make sure there are at least two points on each side.

Ask by Mckenzie Mcfarlane. in the United States

Mar 22,2025

Question

Graph the function by using transformations of the graph of . Plot all necessary asymptotes. For vertical
asymptotes, make sure there are at least two points on each side.

Ask by Mckenzie Mcfarlane. in the United States

Mar 22,2025

UpStudy AI · Accepted Answer

To graph the function $$ h(x) = -\frac{1}{x} $$ using transformations of the graph of $$ y = \frac{1}{x} $$, we can follow these steps:

1. **Identify the basic function**: The graph of $$ y = \frac{1}{x} $$ has a vertical asymptote at $$ x = 0 $$ and a horizontal asymptote at $$ y = 0 $$.

2. **Apply transformations**:
   - The negative sign in front of the function indicates a reflection across the x-axis. Therefore, the graph of $$ h(x) = -\frac{1}{x} $$ will be a reflection of $$ y = \frac{1}{x} $$ in the x-axis.

3. **Determine asymptotes**:
   - The vertical asymptote remains at $$ x = 0 $$.
   - The horizontal asymptote also remains at $$ y = 0 $$.

4. **Plot points**:
   - For $$ h(x) = -\frac{1}{x} $$, we can choose a few values of $$ x $$ to find corresponding $$ y $$ values:
     - If $$ x = 1 $$, then $$ h(1) = -1 $$.
     - If $$ x = 2 $$, then $$ h(2) = -0.5 $$.
     - If $$ x = -1 $$, then $$ h(-1) = 1 $$.
     - If $$ x = -2 $$, then $$ h(-2) = 0.5 $$.

5. **Graph the function**:
   - We will plot the points and draw the asymptotes.

Now, let's proceed to graph the function $$ h(x) = -\frac{1}{x} $$ and include the necessary asymptotes.
Function by following steps:
- step0: Find the range:
  $$h\left(xight)=-\frac{1}{x}$$
- step1: Evaluate:
  $$h\left(xight)
eq 0$$
Analyze the extrema of the function $$ h(x)=-\frac{1}{x} $$
Function by following steps:
- step0: Find the local extrema:
  $$h\left(xight)=-\frac{1}{x}$$
- step1: Find the domain:
  $$h\left(xight)=-\frac{1}{x},x
eq 0$$
- step2: Find the derivative:
  $$h^{\prime}\left(xight)=\frac{1}{x^{2}}$$
- step3: Find the domain:
  $$h^{\prime}\left(xight)=\frac{1}{x^{2}},x
eq 0$$
- step4: Substitute $$h^{\prime}\left(xight)=0:$$
  $$0=\frac{1}{x^{2}}$$
- step5: Swap the sides:
  $$\frac{1}{x^{2}}=0$$
- step6: Cross multiply:
  $$1=x^{2}	imes 0$$
- step7: Simplify the equation:
  $$1=0$$
- step8: The statement is false:
  $$x \in \varnothing $$
- step9: The function has no local extrema:
  $$	extrm{No local extrema}$$
Analyze the critical points of the function $$ h(x)=-\frac{1}{x} $$
Function by following steps:
- step0: Find the critical numbers:
  $$h\left(xight)=-\frac{1}{x}$$
- step1: Find the domain:
  $$h\left(xight)=-\frac{1}{x},x
eq 0$$
- step2: Find the derivative:
  $$h^{\prime}\left(xight)=\frac{1}{x^{2}}$$
- step3: Find the domain:
  $$h^{\prime}\left(xight)=\frac{1}{x^{2}},x
eq 0$$
- step4: Substitute $$h^{\prime}\left(xight)=0:$$
  $$0=\frac{1}{x^{2}}$$
- step5: Swap the sides:
  $$\frac{1}{x^{2}}=0$$
- step6: Cross multiply:
  $$1=x^{2}	imes 0$$
- step7: Simplify the equation:
  $$1=0$$
- step8: The statement is false:
  $$x \in \varnothing $$
- step9: Evaluate:
  $$	extrm{Critical numbers does not exist}$$
It seems there was an issue retrieving the graph directly. However, I can summarize the characteristics of the function $$ h(x) = -\frac{1}{x} $$ based on transformations and asymptotes:

1. **Domain**: The function is defined for all real numbers except $$ x = 0 $$. Thus, the domain is $$ (-\infty, 0) \cup (0, \infty) $$.

2. **Range**: The range of the function is also all real numbers except $$ y = 0 $$. Therefore, the range is $$ (-\infty, 0) \cup (0, \infty) $$.

3. **Asymptotes**:
   - **Vertical Asymptote**: $$ x = 0 $$
   - **Horizontal Asymptote**: $$ y = 0 $$

4. **Critical Points and Extrema**: There are no critical points or local extrema for this function.

### Points to Plot:
- $$ (1, -1) $$
- $$ (2, -0.5) $$
- $$ (-1, 1) $$
- $$ (-2, 0.5) $$

### Graphing Instructions:
- Plot the points mentioned above.
- Draw the vertical asymptote at $$ x = 0 $$ (dashed line).
- Draw the horizontal asymptote at $$ y = 0 $$ (dashed line).
- The graph will approach these asymptotes but never touch them.

If you have graphing software or a graphing calculator, you can input the function $$ h(x) = -\frac{1}{x} $$ to visualize it. The graph will show two branches: one in the second quadrant and one in the fourth quadrant, reflecting the behavior of the function across the x-axis.
To graph $$ h(x) = -\frac{1}{x} $$, reflect the graph of $$ y = \frac{1}{x} $$ across the x-axis. The vertical asymptote is at $$ x = 0 $$ and the horizontal asymptote is at $$ y = 0 $$. Plot points like $$ (1, -1) $$, $$ (2, -0.5) $$, $$ (-1, 1) $$, and $$ (-2, 0.5) $$, and draw the asymptotes.

Graph the function by using transformations of the graph of . Plot all necessary asymptotes. For vertical
asymptotes, make sure there are at least two points on each side.

Upstudy AI Solution

Answer

Solution

Points to Plot:

Graphing Instructions:

Extra Insights

Related Questions

Latest Pre Calculus Questions

Graph the function by using transformations of the graph of . Plot all necessary asymptotes. For vertical asymptotes, make sure there are at least two points on each side.

Upstudy AI Solution

Answer

Solution

Points to Plot:

Graphing Instructions:

Extra Insights

Related Questions

Latest Pre Calculus Questions

Graph the function by using transformations of the graph of . Plot all necessary asymptotes. For vertical
asymptotes, make sure there are at least two points on each side.