4. \( y=-4 x^{2}-4 x-1 \)

Certainly! Let's analyze the quadratic function you've provided: \[ y = -4x^{2} - 4x - 1 \] This is a quadratic equation in the form \( y = ax^{2} + bx + c \), where: - \( a = -4 \) - \( b = -4 \) - \( c = -1 \) ### 1. **Graph Overview** Since \( a = -4 \) is negative, the parabola opens **downwards**. ### 2. **Vertex** The vertex of a parabola given by \( y = ax^{2} + bx + c \) can be found using the formula: \[ h = -\frac{b}{2a} \] \[ k = y(h) \] **Calculating \( h \):** \[ h = -\frac{-4}{2 \times -4} = \frac{4}{-8} = -0.5 \] **Calculating \( k \):** \[ y(-0.5) = -4(-0.5)^2 - 4(-0.5) - 1 \] \[ y(-0.5) = -4(0.25) + 2 - 1 \] \[ y(-0.5) = -1 + 2 - 1 = 0 \] **Vertex:** \( (-0.5, 0) \) ### 3. **Axis of Symmetry** The axis of symmetry is the vertical line that passes through the vertex. \[ x = h = -0.5 \] ### 4. **Y-intercept** The y-intercept occurs where \( x = 0 \): \[ y(0) = -4(0)^2 - 4(0) - 1 = -1 \] **Y-intercept:** \( (0, -1) \) ### 5. **X-intercepts (Roots)** To find the x-intercepts, set \( y = 0 \) and solve for \( x \): \[ -4x^{2} - 4x - 1 = 0 \] Using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] \[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(-4)(-1)}}{2(-4)} \] \[ x = \frac{4 \pm \sqrt{16 - 16}}{-8} \] \[ x = \frac{4 \pm \sqrt{0}}{-8} \] \[ x = \frac{4}{-8} = -0.5 \] **X-intercept:** \( (-0.5, 0) \) *Note: There is only one real x-intercept because the discriminant is zero, meaning the parabola touches the x-axis at the vertex.* ### 6. **Summary of Key Features** - **Vertex:** \( (-0.5, 0) \) - **Axis of Symmetry:** \( x = -0.5 \) - **Parabola Opens Downwards** - **Y-intercept:** \( (0, -1) \) - **X-intercept:** \( (-0.5, 0) \) (vertex) ### 7. **Graph Sketch** Here's a rough sketch of the parabola based on the information: ``` | 1 | * | 0 |----------*---------- | -1 | * | |____________________ -1 -0.5 0 1 ``` - The vertex is at \((-0.5, 0)\). - The y-intercept is at \((0, -1)\). - The parabola opens downward. ### 8. **Additional Insights** - **Maximum Value:** Since the parabola opens downward, the vertex represents the **maximum point** of the function. The maximum y-value is **0**. - **Symmetry:** The function is symmetric around the line \( x = -0.5 \). ### 9. **Function Behavior** - **For \( x < -0.5 \):** As \( x \) decreases, \( y \) decreases. - **For \( x > -0.5 \):** As \( x \) increases, \( y \) decreases. ### 10. **Graphing Tips** - Plot the vertex \((-0.5, 0)\). - Plot the y-intercept \((0, -1)\). - Since there's only one x-intercept at the vertex, Ensure the parabola touches the x-axis at \((-0.5, 0)\) and opens downward. - Optionally, plot another point for better accuracy, for example, when \( x = 1 \): \[ y = -4(1)^2 - 4(1) - 1 = -4 - 4 - 1 = -9 \] So, another point is \( (1, -9) \). ### 11. **Conclusion** The quadratic function \( y = -4x^{2} - 4x - 1 \) represents a downward-opening parabola with its vertex at \((-0.5, 0)\), intersecting the y-axis at \((0, -1)\), and having a single x-intercept at the vertex. This information should help in graphing the function and understanding its key features.