Consider the parabola given by the equation: \( f(x)=-4 x^{2}-14 x+3 \) Find the following for this parabola: A) The vertex: B) The vertical intercept is the point C) Find the coordinates of the two \( x \)-intercepts of the parabola and write them as a list, separated by commas: It is OK to round your value(s) to to two decimal places.

To analyze the parabola given by the equation \( f(x) = -4x^2 - 14x + 3 \), we will find the vertex, the vertical intercept, and the \( x \)-intercepts. ### A) The Vertex The vertex of a parabola in the form \( f(x) = ax^2 + bx + c \) can be found using the formula for the \( x \)-coordinate of the vertex: \[ x = -\frac{b}{2a} \] Here, \( a = -4 \) and \( b = -14 \). Calculating the \( x \)-coordinate of the vertex: \[ x = -\frac{-14}{2 \cdot -4} = \frac{14}{-8} = -\frac{7}{4} = -1.75 \] Next, we substitute \( x = -1.75 \) back into the function to find the \( y \)-coordinate of the vertex: \[ f(-1.75) = -4(-1.75)^2 - 14(-1.75) + 3 \] \[ = -4(3.0625) + 24.5 + 3 \] \[ = -12.25 + 24.5 + 3 \] \[ = 15.25 \] Thus, the vertex is: \[ \text{Vertex} = \left(-\frac{7}{4}, 15.25\right) \quad \text{or} \quad (-1.75, 15.25) \] ### B) The Vertical Intercept The vertical intercept occurs when \( x = 0 \). We can find it by evaluating \( f(0) \): \[ f(0) = -4(0)^2 - 14(0) + 3 = 3 \] Thus, the vertical intercept is the point: \[ (0, 3) \] ### C) The \( x \)-Intercepts The \( x \)-intercepts occur when \( f(x) = 0 \). We need to solve the equation: \[ -4x^2 - 14x + 3 = 0 \] We can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = -4 \), \( b = -14 \), and \( c = 3 \). Calculating the discriminant: \[ b^2 - 4ac = (-14)^2 - 4(-4)(3) = 196 + 48 = 244 \] Now substituting into the quadratic formula: \[ x = \frac{-(-14) \pm \sqrt{244}}{2 \cdot -4} = \frac{14 \pm \sqrt{244}}{-8} \] Calculating \( \sqrt{244} \): \[ \sqrt{244} \approx 15.62 \] Now substituting back: \[ x = \frac{14 \pm 15.62}{-8} \] Calculating the two possible values for \( x \): 1. For \( x = \frac{14 + 15.62}{-8} \): \[ x_1 = \frac{29.62}{-8} \approx -3.70 \] 2. For \( x = \frac{14 - 15.62}{-8} \): \[ x_2 = \frac{-1.62}{-8} \approx 0.20 \] Thus, the coordinates of the two \( x \)-intercepts are approximately: \[ \text{Intercepts} = -3.70, 0.20 \] ### Summary of Results A) Vertex: \((-1.75, 15.25)\) B) Vertical intercept: \((0, 3)\) C) \( x \)-intercepts: \(-3.70, 0.20\)