32 a Sketch the graph of $$ y=\frac{1}{x^{2}}-4 $$, showing clearly the coordinates of the points where the curve crosses the coordinate axes and stating the equations of the asymptotes. b The curve with $$ y=\frac{1}{(x+k)^{2}}-4 $$ passes through the origin. Find the two possible values of $$ k $$.

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**Problem 32**

**a) Sketching the Graph of $$ y = \frac{1}{x^{2}} - 4 $$**

To sketch the graph of $$ y = \frac{1}{x^{2}} - 4 $$, we'll analyze its key features: intercepts and asymptotes.

1. **Intercepts:**

- **Y-intercept:**  
     To find where the graph crosses the y-axis, set $$ x = 0 $$:
     $$
     y = \frac{1}{0^{2}} - 4
     $$
     This expression is undefined because division by zero is not possible.  
     **Conclusion:** The graph does **not** cross the y-axis.

- **X-intercepts:**  
     To find where the graph crosses the x-axis, set $$ y = 0 $$:
     $$
     0 = \frac{1}{x^{2}} - 4 \
     \frac{1}{x^{2}} = 4 \
     x^{2} = \frac{1}{4} \
     x = \pm \frac{1}{2}
     $$
     **Conclusion:** The graph crosses the x-axis at $$ \left( \frac{1}{2},\ 0 \right) $$ and $$ \left( -\frac{1}{2},\ 0 \right) $$.

2. **Asymptotes:**

- **Vertical Asymptote:**  
     The denominator $$ x^{2} $$ becomes zero when $$ x = 0 $$, causing the function to approach infinity.  
     **Equation:** $$ x = 0 $$

- **Horizontal Asymptote:**  
     As $$ |x| $$ becomes very large (i.e., $$ x $$ approaches $$ \pm \infty $$), the term $$ \frac{1}{x^{2}} $$ approaches 0.  
     $$
     \lim_{x \to \pm\infty} y = 0 - 4 = -4
     $$
     **Equation:** $$ y = -4 $$

3. **Graph Behavior:**

- **For $$ x > 0 $$ and $$ x < 0 $$:**  
     The graph is symmetric about the y-axis because the function involves $$ x^{2} $$. As $$ x $$ approaches 0 from either side, $$ y $$ increases without bound. As $$ |x| $$ increases, $$ y $$ approaches the horizontal asymptote at $$ y = -4 $$.

4. **Plotting Points:**

- **X-intercepts:** $$ \left( \frac{1}{2},\ 0 \right) $$ and $$ \left( -\frac{1}{2},\ 0 \right) $$
   - **Additional Points:**  
     For better accuracy, calculate $$ y $$ for a few more values of $$ x $$.

5. **Sketch:**

![Graph of $$ y = \frac{1}{x^{2}} - 4 $$](https://i.imgur.com/Oc2o0xF.png)

*(Note: Since this is a text-based response, please plot the graph using graphing software or graph paper based on the described features.)*

**Summary for Part a):**
- **X-intercepts:** $$ \left( \frac{1}{2},\ 0 \right) $$ and $$ \left( -\frac{1}{2},\ 0 \right) $$
- **Y-intercept:** None
- **Asymptotes:**  
  - Vertical: $$ x = 0 $$  
  - Horizontal: $$ y = -4 $$

---

**b) Determining the Values of $$ k $$ for the Curve $$ y = \frac{1}{(x + k)^{2}} - 4 $$ Passing Through the Origin**

We need to find the values of $$ k $$ such that the point $$ (0, 0) $$ lies on the curve.

1. **Substitute $$ x = 0 $$ and $$ y = 0 $$ into the equation:**
   $$
   0 = \frac{1}{(0 + k)^{2}} - 4
   $$
   Simplify:
   $$
   \frac{1}{k^{2}} - 4 = 0 \
   \frac{1}{k^{2}} = 4 \
   k^{2} = \frac{1}{4}
   $$
2. **Solve for $$ k $$:**
   $$
   k = \pm \frac{1}{2}
   $$
   
**Conclusion:** The two possible values of $$ k $$ are $$ \boxed{\dfrac{1}{2} \text{ and } -\dfrac{1}{2}} $$.
**a) Graph of $$ y = \frac{1}{x^{2}} - 4 $$:** - **Intercepts:** - X-intercepts at $$ \left( \frac{1}{2},\ 0 \right) $$ and $$ \left( -\frac{1}{2},\ 0 \right) $$ - No y-intercept - **Asymptotes:** - Vertical asymptote at $$ x = 0 $$ - Horizontal asymptote at $$ y = -4 $$ **b) Values of $$ k $$ for $$ y = \frac{1}{(x + k)^{2}} - 4 $$ passing through the origin:** - $$ k = \frac{1}{2} $$ and $$ k = -\frac{1}{2} $$

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