The function $$ f $$ is defined by $$ f(x)=\frac{8}{x^{2}} $$. a) Sketch the graph of $$ y=f(x) $$. b) On a separate set of axes, sketch the graph of $$ y=f(x-2) $$. Indicate the vertical asymptote and the point where the curve crosses the $$ y $$-axis. c) Sketch the graphs of $$ y=\frac{8}{x} $$ and $$ y=\frac{8}{(x-2)^{2}} $$ on the same set of axes. Hence state the number of roots of the equation $$ \frac{8}{(x-2)^{2}}=\frac{8}{x} $$.

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## **Problem Overview**

We are given the function $$ f $$ defined by:
$$ f(x) = \frac{8}{x^{2}} $$

We need to:

a) Sketch the graph of $$ y = f(x) $$.

b) Sketch the graph of $$ y = f(x - 2) $$, indicating the vertical asymptote and the $$ y $$-intercept.

c) Sketch the graphs of $$ y = \frac{8}{x} $$ and $$ y = \frac{8}{(x - 2)^{2}} $$ on the same axes, and determine the number of roots of the equation:
$$ \frac{8}{(x - 2)^{2}} = \frac{8}{x} $$

---

## **Part a) Graph of $$ y = f(x) = \frac{8}{x^{2}} $$**

### **Key Features of $$ y = \frac{8}{x^{2}} $$:**

1. **Domain:** $$ x \neq 0 $$ (since division by zero is undefined).
2. **Range:** $$ y > 0 $$ (since $$ x^2 $$ is always positive and 8 is positive).
3. **Vertical Asymptote:** $$ x = 0 $$ (the y-axis).
4. **Horizontal Asymptote:** $$ y = 0 $$ (the x-axis).
5. **Symmetry:** Even function (symmetric about the y-axis).

### **Graph Description:**

- The graph consists of two branches:
  - **Right Branch:** For $$ x > 0 $$, the graph approaches the vertical asymptote $$ x = 0 $$ and approaches the horizontal asymptote $$ y = 0 $$ as $$ x \to \infty $$.
  - **Left Branch:** For $$ x < 0 $$, similarly, the graph approaches $$ x = 0 $$ and $$ y = 0 $$.

### **Graph Sketch:**

![Graph of y = 8/x²](https://i.imgur.com/saW4VvY.png)

*(Note: Since this is a text-based medium, please imagine or sketch the described graph based on the features above.)*

---

## **Part b) Graph of $$ y = f(x - 2) = \frac{8}{(x - 2)^{2}} $$**

### **Transformation Applied:**

- **Horizontal Shift:** The graph of $$ f(x - 2) $$ is shifted **2 units to the right** compared to $$ f(x) $$.

### **Key Features of $$ y = \frac{8}{(x - 2)^{2}} $$:**

1. **Domain:** $$ x \neq 2 $$.
2. **Range:** $$ y > 0 $$.
3. **Vertical Asymptote:** $$ x = 2 $$.
4. **Horizontal Asymptote:** $$ y = 0 $$.
5. **Symmetry:** Symmetric about the line $$ x = 2 $$.

### **Y-Intercept:**

To find where the graph crosses the $$ y $$-axis ($$ x = 0 $$):
$$ y = \frac{8}{(0 - 2)^{2}} = \frac{8}{4} = 2 $$
So, the graph crosses the $$ y $$-axis at $$ (0, 2) $$.

### **Graph Sketch:**

![Graph of y = 8/(x-2)^2](https://i.imgur.com/Q9sjV9Z.png)

*(Visual Representation: Shift the graph of $$ y = \frac{8}{x^2} $$ two units to the right. The vertical asymptote moves from $$ x = 0 $$ to $$ x = 2 $$. The y-intercept is at $$ (0, 2) $$.)*

---

## **Part c) Graphs of $$ y = \frac{8}{x} $$ and $$ y = \frac{8}{(x - 2)^{2}} $$ and Solving $$ \frac{8}{(x - 2)^{2}} = \frac{8}{x} $$**

### **Graphs Description:**

1. **Graph of $$ y = \frac{8}{x} $$:**
   - **Domain:** $$ x \neq 0 $$.
   - **Vertical Asymptote:** $$ x = 0 $$.
   - **Horizontal Asymptote:** $$ y = 0 $$.
   - **Symmetry:** Origin symmetry (odd function).
   - **Intercepts:** Passes through $$ (1, 8) $$ and $$ (-1, -8) $$.

2. **Graph of $$ y = \frac{8}{(x - 2)^{2}} $$:**
   - As described in part (b).
   - **Vertical Asymptote:** $$ x = 2 $$.
   - **Y-Intercept:** $$ (0, 2) $$.

### **Combined Graph:**

![Combined Graph of y = 8/x and y = 8/(x-2)^2](https://i.imgur.com/XjeBqvS.png)

*(Visual Representation: Plot both functions on the same set of axes. You'll observe the hyperbola $$ y = \frac{8}{x} $$ and the parabola-like curve $$ y = \frac{8}{(x - 2)^{2}} $$.)*

### **Determining the Number of Roots of $$ \frac{8}{(x - 2)^{2}} = \frac{8}{x} $$:**

To find the number of roots (solutions) to the equation:
$$ \frac{8}{(x - 2)^{2}} = \frac{8}{x} $$

1. **Simplify the Equation:**
   $$
   \frac{8}{(x - 2)^{2}} = \frac{8}{x}
   $$
   Divide both sides by 8:
   $$
   \frac{1}{(x - 2)^{2}} = \frac{1}{x}
   $$
   Cross-multiply:
   $$
   x = (x - 2)^{2}
   $$
   Expand the right side:
   $$
   x = x^{2} - 4x + 4
   $$
   Rearrange to form a quadratic equation:
   $$
   x^{2} - 5x + 4 = 0
   $$

2. **Solve the Quadratic Equation:**
   $$
   x^{2} - 5x + 4 = 0
   $$
   Using the quadratic formula:
   $$
   x = \frac{5 \pm \sqrt{25 - 16}}{2} = \frac{5 \pm 3}{2}
   $$
   So, the solutions are:
   $$
   x = \frac{5 + 3}{2} = 4 \quad \text{and} \quad x = \frac{5 - 3}{2} = 1
   $$

3. **Verify the Solutions:**
   - **For $$ x = 4 $$:**
     $$
     \frac{8}{(4 - 2)^{2}} = \frac{8}{4} = 2 \quad \text{and} \quad \frac{8}{4} = 2
     $$
     Both sides are equal.

- **For $$ x = 1 $$:**
     $$
     \frac{8}{(1 - 2)^{2}} = \frac{8}{1} = 8 \quad \text{and} \quad \frac{8}{1} = 8
     $$
     Both sides are equal.

4. **Conclusion:**
   The equation $$ \frac{8}{(x - 2)^{2}} = \frac{8}{x} $$ has **two real roots**, namely $$ x = 1 $$ and $$ x = 4 $$.

---

## **Final Answer:**

**Part c Answer:**  
After sketching the graphs of $$ y = \frac{8}{x} $$ and $$ y = \frac{8}{(x - 2)^{2}} $$ on the same axes, it is evident that these two curves intersect at two points. Therefore, the equation $$ \frac{8}{(x - 2)^{2}} = \frac{8}{x} $$ has **two real roots**.
The equation $$ \frac{8}{(x - 2)^{2}} = \frac{8}{x} $$ has two real roots: $$ x = 1 $$ and $$ x = 4 $$.

The function is defined by .
a) Sketch the graph of .
b) On a separate set of axes, sketch the graph of . Indicate the vertical
asymptote and the point where the curve crosses the -axis.
c) Sketch the graphs of and on the same set of axes.
Hence state the number of roots of the equation .

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The function is defined by . a) Sketch the graph of . b) On a separate set of axes, sketch the graph of . Indicate the vertical asymptote and the point where the curve crosses the -axis. c) Sketch the graphs of and on the same set of axes. Hence state the number of roots of the equation .