Question Three $$ y=\frac{2}{4} x^{3}-3 x^{2}-6 x+9 $$ - For the function $$ y-2=2^{-x} $$, sketch its graph clearly showing the intercepts ar asymptote $$ [5 \mathrm{mks}] $$

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Simplify the expression by following steps:
- step0: Solution:
  $$\frac{2}{4}x^{3}-3x^{2}-6x+9$$
- step1: Reduce the fraction:
  $$\frac{1}{2}x^{3}-3x^{2}-6x+9$$
Solve the equation $$ y-2=2^{-x} $$.
Solve the equation by following steps:
- step0: Solve for $$y$$:
  $$y-2=2^{-x}$$
- step1: Move the constant to the right side:
  $$y=2^{-x}+2$$
Let's break down the problem step by step.

### Step 1: Simplify the Polynomial Function
The given polynomial function is:

$$
y = \frac{2}{4} x^{3} - 3 x^{2} - 6 x + 9
$$

Simplifying this, we get:

$$
y = \frac{1}{2} x^{3} - 3 x^{2} - 6 x + 9
$$

### Step 2: Analyze the Exponential Function
The second function given is:

$$
y - 2 = 2^{-x}
$$

Rearranging this, we find:

$$
y = 2^{-x} + 2
$$

### Step 3: Find Intercepts
To find the intercepts of the function $$ y = 2^{-x} + 2 $$:

1. **Y-Intercept**: Set $$ x = 0 $$:
   $$
   y = 2^{-0} + 2 = 1 + 2 = 3
   $$
   So, the y-intercept is $$ (0, 3) $$.

2. **X-Intercept**: Set $$ y = 0 $$:
   $$
   0 = 2^{-x} + 2 \implies 2^{-x} = -2
   $$
   This equation has no solution since $$ 2^{-x} $$ is always positive. Therefore, there are no x-intercepts.

### Step 4: Identify Asymptotes
The function $$ y = 2^{-x} + 2 $$ has a horizontal asymptote. As $$ x \to \infty $$, $$ 2^{-x} \to 0 $$, so:

$$
y \to 2
$$

Thus, the horizontal asymptote is $$ y = 2 $$.

### Step 5: Sketch the Graph
1. **Plot the y-intercept** at $$ (0, 3) $$.
2. **Draw the horizontal asymptote** at $$ y = 2 $$.
3. The function $$ y = 2^{-x} + 2 $$ will approach the asymptote as $$ x $$ increases and will be above the asymptote for all $$ x $$.

### Summary
- **Y-Intercept**: $$ (0, 3) $$
- **X-Intercept**: None
- **Horizontal Asymptote**: $$ y = 2 $$

This information can be used to sketch the graph of the function clearly showing the intercepts and the asymptote.
To sketch the graph of $$ y = 2^{-x} + 2 $$: 1. **Y-Intercept**: Plot at $$ (0, 3) $$. 2. **Horizontal Asymptote**: Draw $$ y = 2 $$. 3. The graph approaches $$ y = 2 $$ as $$ x $$ increases and is always above $$ y = 2 $$. No x-intercept exists for this function.

Question Three -
For the function , sketch its graph clearly showing the intercepts ar
asymptote

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Question Three - For the function , sketch its graph clearly showing the intercepts ar asymptote