5. \( y=-2 x^{2}+3 x-5 \)

Certainly! Let's analyze the quadratic function provided: \[ y = -2x^{2} + 3x - 5 \] This is a quadratic equation of the form \( y = ax^{2} + bx + c \), where: - \( a = -2 \) - \( b = 3 \) - \( c = -5 \) ### 1. **Direction of the Parabola** Since \( a = -2 \) is negative, the parabola opens **downward**. ### 2. **Vertex of the Parabola** The vertex form of a quadratic equation is: \[ y = a(x - h)^2 + k \] Where \((h, k)\) is the vertex. To find the vertex, we can use the formula: \[ h = -\frac{b}{2a} \] \[ k = y(h) \] **Calculating \( h \):** \[ h = -\frac{b}{2a} = -\frac{3}{2(-2)} = -\frac{3}{-4} = \frac{3}{4} \] **Calculating \( k \):** \[ k = -2\left(\frac{3}{4}\right)^2 + 3\left(\frac{3}{4}\right) - 5 \] \[ k = -2\left(\frac{9}{16}\right) + \frac{9}{4} - 5 \] \[ k = -\frac{18}{16} + \frac{36}{16} - \frac{80}{16} \] \[ k = \left(-18 + 36 - 80\right) \div 16 \] \[ k = -62 \div 16 \] \[ k = -\frac{31}{8} \] **Vertex Coordinates:** \[ \left( \frac{3}{4}, -\frac{31}{8} \right) \] ### 3. **Axis of Symmetry** The axis of symmetry is the vertical line that passes through the vertex: \[ x = h = \frac{3}{4} \] ### 4. **Y-intercept** The y-intercept occurs where \( x = 0 \): \[ y = -2(0)^2 + 3(0) - 5 = -5 \] **Y-intercept: \( (0, -5) \)** ### 5. **X-intercepts (Roots)** To find the x-intercepts, set \( y = 0 \) and solve for \( x \): \[ 0 = -2x^{2} + 3x - 5 \] This is a quadratic equation of the form \( ax^2 + bx + c = 0 \). We can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] **Calculating the discriminant (\( \Delta \)):** \[ \Delta = b^2 - 4ac = 3^2 - 4(-2)(-5) = 9 - 40 = -31 \] Since the discriminant is negative (\( \Delta < 0 \)), there are **no real x-intercepts**. Instead, the roots are complex. **Complex Roots:** \[ x = \frac{-3 \pm \sqrt{-31}}{-4} = \frac{-3 \pm i\sqrt{31}}{-4} = \frac{3}{4} \mp \frac{i\sqrt{31}}{4} \] ### 6. **Plotting the Parabola** Given the above information: - **Vertex:** \( \left( \frac{3}{4}, -\frac{31}{8} \right) \) - **Axis of Symmetry:** \( x = \frac{3}{4} \) - **Y-intercept:** \( (0, -5) \) - **No Real X-intercepts** - **Opens Downward** The parabola will have its highest point at the vertex and will not cross the x-axis. ### 7. **Summary of Key Features** - **Standard Form:** \( y = -2x^{2} + 3x - 5 \) - **Vertex:** \( \left( \frac{3}{4}, -\frac{31}{8} \right) \) - **Axis of Symmetry:** \( x = \frac{3}{4} \) - **Direction:** Opens downward - **Y-intercept:** \( (0, -5) \) - **X-intercepts:** None (Complex roots) If you have a specific question or need further analysis related to this quadratic function, feel free to ask!