Draw the graph of the equation $$ y=6-x-x^{2} $$ and find the gradient of the curve at $$ x=2 $$.

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To graph the equation $$ y = 6 - x - x^{2} $$ and determine the gradient of the curve at $$ x = 2 $$, follow these steps:

---

## 1. Understanding the Equation

The given equation is:
$$
y = 6 - x - x^{2}
$$
This is a **quadratic equation** of the form $$ y = ax^{2} + bx + c $$, where:
- $$ a = -1 $$
- $$ b = -1 $$
- $$ c = 6 $$

Since the coefficient of $$ x^{2} $$ is negative ($$ a = -1 $$), the parabola **opens downward**.

---

## 2. Key Features of the Graph

### a. **Vertex of the Parabola**

The vertex form of a quadratic equation helps identify the peak (vertex) of the parabola. The vertex $$(h, k)$$ can be found using:
$$
h = -\frac{b}{2a}, \quad k = y(h)
$$

**Calculations:**
$$
h = -\frac{-1}{2(-1)} = \frac{1}{-2} = -\frac{1}{2}
$$
$$
k = 6 - \left(-\frac{1}{2}\right) - \left(-\frac{1}{2}\right)^{2} = 6 + \frac{1}{2} - \frac{1}{4} = 6.25
$$

**Vertex:** $$\left( -\frac{1}{2}, \, 6.25 \right)$$

### b. **X-Intercepts (Roots)**

X-intercepts occur where $$ y = 0 $$:
$$
0 = -x^{2} - x + 6 \
x^{2} + x - 6 = 0
$$

**Solving the Quadratic Equation:**
$$
x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \
a = 1, \quad b = 1, \quad c = -6 \
x = \frac{-1 \pm \sqrt{1 + 24}}{2} = \frac{-1 \pm 5}{2}
$$

**Solutions:**
$$
x = \frac{-1 + 5}{2} = 2 \
x = \frac{-1 - 5}{2} = -3
$$

**X-Intercepts:** $$(2, 0)$$ and $$(-3, 0)$$

### c. **Y-Intercept**

Y-intercept occurs where $$ x = 0 $$:
$$
y = 6 - 0 - 0 = 6
$$

**Y-Intercept:** $$(0, 6)$$

### d. **Additional Points for Accuracy**

Calculating $$ y $$ for other $$ x $$-values helps in sketching a more accurate graph.

| $$ x $$ | $$ y = 6 - x - x^{2} $$ |
|---------|------------------------|
| -2      | $$ 6 - (-2) - (-2)^2 = 6 + 2 - 4 = 4 $$ |
| -1      | $$ 6 - (-1) - (-1)^2 = 6 + 1 - 1 = 6 $$ |
| 1       | $$ 6 - 1 - 1 = 4 $$ |
| 3       | $$ 6 - 3 - 9 = -6 $$ |

---

## 3. Sketching the Graph

Using the key points identified, we can sketch the parabola.

**Key Points:**
- Vertex: $$\left( -\frac{1}{2}, \, 6.25 \right)$$
- X-Intercepts: $$(2, 0)$$ and $$(-3, 0)$$
- Y-Intercept: $$(0, 6)$$
- Additional Points: $$(-2, 4)$$, $$(-1, 6)$$, $$(1, 4)$$, $$(3, -6)$$

**ASCII Graph Representation:**

```
y
|
8 |              
7 |              
6 |        *      *
5 |              
4 |     *     *    
3 |              
2 |              
1 |              
0 |*------------*------ x
   -3  -2  -1 0 1 2 3
```

*Note:* ASCII art is a simplified representation. For a precise graph, it's recommended to use graphing software or plot on graph paper.

---

## 4. Determining the Gradient at $$ x = 2 $$

The **gradient** of the curve at a particular point is the slope of the tangent to the curve at that point. It can be found by taking the derivative of $$ y $$ with respect to $$ x $$.

### a. **Finding the Derivative $$ \frac{dy}{dx} $$**

Given:
$$
y = 6 - x - x^{2}
$$

Differentiate with respect to $$ x $$:
$$
\frac{dy}{dx} = \frac{d}{dx}(6) - \frac{d}{dx}(x) - \frac{d}{dx}(x^{2}) = 0 - 1 - 2x = -1 - 2x
$$

### b. **Evaluating the Derivative at $$ x = 2 $$**

$$
\left.\frac{dy}{dx}\right|_{x=2} = -1 - 2(2) = -1 - 4 = -5
$$

**Gradient at $$ x = 2 $$:** $$-5$$

---

## 5. Summary

- **Graph:** The parabola $$ y = 6 - x - x^{2} $$ opens downward with its vertex at $$( -\frac{1}{2}, 6.25 )$$, x-intercepts at $$(-3, 0)$$ and $$(2, 0)$$, and y-intercept at $$(0, 6)$$.
- **Gradient at $$ x = 2 $$:** The slope of the tangent to the curve at $$ x = 2 $$ is $$-5$$.

---

If you have access to graphing tools like Desmos, GeoGebra, or even a graphing calculator, you can visualize the parabola more accurately using the points and features discussed above.
To graph the equation $$ y = 6 - x - x^{2} $$, plot the vertex at $$\left( -\frac{1}{2}, 6.25 \right)$$, x-intercepts at $$(-3, 0)$$ and $$(2, 0)$$, and y-intercept at $$(0, 6)$$. The parabola opens downward. The gradient of the curve at $$ x = 2 $$ is $$-5$$.

Draw the graph of the equation \( y=6-x-x^{2} \) and find the gradient of the curve at \( x=2 \).

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