h. \( y=4 x^{2}+8 x+2 \)

Certainly! Let's analyze the quadratic function \( y = 4x^{2} + 8x + 2 \). We'll explore its key features, including its vertex, intercepts, axis of symmetry, and graph characteristics. ## 1. **Standard Form** The given quadratic function is in the standard form: \[ y = ax^{2} + bx + c \] where: - \( a = 4 \) - \( b = 8 \) - \( c = 2 \) ## 2. **Vertex** The vertex of a parabola described by a quadratic function can be found using the formulas: \[ x_{\text{vertex}} = -\frac{b}{2a} \] \[ y_{\text{vertex}} = f(x_{\text{vertex}}) \] Calculating the \( x \)-coordinate: \[ x_{\text{vertex}} = -\frac{8}{2 \times 4} = -\frac{8}{8} = -1 \] Substituting \( x = -1 \) into the original equation to find \( y \): \[ y_{\text{vertex}} = 4(-1)^2 + 8(-1) + 2 = 4(1) - 8 + 2 = 4 - 8 + 2 = -2 \] **Vertex:** \( (-1, -2) \) ## 3. **Axis of Symmetry** The axis of symmetry is a vertical line that passes through the vertex: \[ x = -1 \] ## 4. **Y-Intercept** The y-intercept occurs when \( x = 0 \): \[ y = 4(0)^2 + 8(0) + 2 = 0 + 0 + 2 = 2 \] **Y-Intercept:** \( (0, 2) \) ## 5. **X-Intercepts (Roots)** To find the x-intercepts, set \( y = 0 \) and solve for \( x \): \[ 0 = 4x^{2} + 8x + 2 \] Using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] \[ x = \frac{-8 \pm \sqrt{8^2 - 4 \times 4 \times 2}}{2 \times 4} \] \[ x = \frac{-8 \pm \sqrt{64 - 32}}{8} \] \[ x = \frac{-8 \pm \sqrt{32}}{8} \] \[ x = \frac{-8 \pm 4\sqrt{2}}{8} \] \[ x = \frac{-2 \pm \sqrt{2}}{2} \] **Approximate X-Intercepts:** - \( x \approx \frac{-2 + 1.414}{2} \approx -0.293 \) - \( x \approx \frac{-2 - 1.414}{2} \approx -1.707 \) So, the x-intercepts are approximately \( (-0.293, 0) \) and \( (-1.707, 0) \). ## 6. **Direction of Opening** Since the coefficient of \( x^2 \) (which is 4) is positive, the parabola opens **upwards**. ## 7. **Graph Sketch** Here's a summary of the key features to plot the graph: - **Vertex:** \( (-1, -2) \) - **Axis of Symmetry:** \( x = -1 \) - **Y-Intercept:** \( (0, 2) \) - **X-Intercepts:** Approximately \( (-0.293, 0) \) and \( (-1.707, 0) \) - **Opening:** Upwards **Steps to Sketch:** 1. Plot the vertex at \( (-1, -2) \). 2. Draw the axis of symmetry as a dashed vertical line through \( x = -1 \). 3. Plot the y-intercept at \( (0, 2) \). 4. Plot the x-intercepts at approximately \( (-0.293, 0) \) and \( (-1.707, 0) \). 5. Since the parabola is symmetric, plot another point on the opposite side of the axis of symmetry for each x-intercept (e.g., reflect the y-intercept across the axis). 6. Draw a smooth curve through these points, opening upwards. **Graph Illustration:** While I can't provide a visual graph here, you can use graphing tools like Desmos or GeoGebra to input the function \( y = 4x^{2} + 8x + 2 \) and visualize the parabola with the features described above. ## 8. **Additional Features** - **Minimum Value:** Since the parabola opens upwards, the vertex represents the minimum point of the function. Therefore, the minimum value of \( y \) is **-2**, occurring at \( x = -1 \). - **Range:** \( y \geq -2 \) - **Domain:** All real numbers (\( -\infty < x < \infty \)) ## 9. **Summary** - **Vertex:** \( (-1, -2) \) - **Axis of Symmetry:** \( x = -1 \) - **Y-Intercept:** \( (0, 2) \) - **X-Intercepts:** Approximately \( (-0.293, 0) \) and \( (-1.707, 0) \) - **Direction:** Opens upwards - **Minimum Value:** \( y = -2 \) at \( x = -1 \) - **Range:** \( y \geq -2 \) - **Domain:** All real numbers Feel free to ask if you need further clarification or additional information about this quadratic function!