Questions: Graph the parabola and give its vertex, axis of symmetry, x-intercepts, and y-intercept. y=x^2-11 x+24 The vertex is (Type an ordered pair.)

Solution Steps

The vertex form of a parabola \( y = ax^2 + bx + c \) is given by: \[ x = -\frac{b}{2a} \] For the given equation \( y = x^2 - 11x + 24 \): \[ a = 1, \quad b = -11, \quad c = 24 \] \[ x = -\frac{-11}{2 \cdot 1} = \frac{11}{2} = 5.5 \] Substitute \( x = 5.5 \) back into the equation to find \( y \): \[ y = (5.5)^2 - 11(5.5) + 24 = 30.25 - 60.5 + 24 = -6.25 \] So, the vertex is: \[ (5.5, -6.25) \]

The axis of symmetry is the vertical line that passes through the vertex: \[ x = 5.5 \]

To find the x-intercepts, set \( y = 0 \) and solve for \( x \): \[ x^2 - 11x + 24 = 0 \] Factor the quadratic equation: \[ (x - 3)(x - 8) = 0 \] So, the x-intercepts are: \[ x = 3 \quad \text{and} \quad x = 8 \]

Final Answer