Questions: f(x)=-2 x^2-5 x+8

Solution Steps

To analyze the quadratic function \( f(x) = -2x^2 - 5x + 8 \), we can perform the following steps:

1. Find the vertex: The vertex of a quadratic function \( ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). Substitute this \( x \) value back into the function to get the \( y \)-coordinate of the vertex.
2. Find the roots: Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the roots of the function.
3. Determine the y-intercept: The y-intercept is the value of the function when \( x = 0 \).

The vertex of the quadratic function \( f(x) = -2x^2 - 5x + 8 \) can be found using the formula for the x-coordinate of the vertex: \[ x = -\frac{b}{2a} \] Substituting \( a = -2 \) and \( b = -5 \): \[ x = -\frac{-5}{2(-2)} = -\frac{5}{-4} = 1.25 \]

To find the y-coordinate of the vertex, substitute \( x = 1.25 \) back into the function: \[ y = -2(1.25)^2 - 5(1.25) + 8 \] \[ y = -2(1.5625) - 6.25 + 8 \] \[ y = -3.125 - 6.25 + 8 \] \[ y = 11.125 \]

Thus, the vertex is: \[ \left( -1.25, 11.125 \right) \]

The roots of the quadratic function can be found using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substituting \( a = -2 \), \( b = -5 \), and \( c = 8 \): \[ \text{Discriminant} = b^2 - 4ac = (-5)^2 - 4(-2)(8) = 25 + 64 = 89 \]

Since the discriminant is positive, there are two real roots: \[ x = \frac{-(-5) \pm \sqrt{89}}{2(-2)} \] \[ x = \frac{5 \pm \sqrt{89}}{-4} \]

Calculating the roots: \[ x_1 = \frac{5 + \sqrt{89}}{-4} \approx -3.6085 \] \[ x_2 = \frac{5 - \sqrt{89}}{-4} \approx 1.1085 \]

Thus, the roots are: \[ x_1 \approx -3.6085 \] \[ x_2 \approx 1.1085 \]

The y-intercept is the value of the function when \( x = 0 \): \[ f(0) = -2(0)^2 - 5(0) + 8 = 8 \]

Thus, the y-intercept is: \[ 8 \]

Final Answer