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

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.