\( f(x)=x^{2}-14 x+24 \) 2. Which statement below is true for all parabolas and will prove useful in finding the vertex? Parabolas are symmetrical. The vertex lies on the line of symmetry. The vertex is equidistant from each of \( x \)-intercepts. the 6

To analyze the function \( f(x) = x^2 - 14x + 24 \) and determine which statement is true for all parabolas, let's break down the properties of parabolas and their vertices. 1. **Parabolas are symmetrical**: This is true. Parabolas are symmetric about their axis of symmetry. 2. **The vertex lies on the line of symmetry**: This is also true. The vertex of a parabola is located on its axis of symmetry. 3. **The vertex is equidistant from each of \( x \)-intercepts**: This is true as well. The vertex is the midpoint of the \( x \)-intercepts. Now, let's find the vertex of the given parabola to illustrate these properties. The vertex \( (h, k) \) of a parabola given in the standard form \( f(x) = ax^2 + bx + c \) can be found using the formula: \[ h = -\frac{b}{2a} \] where \( a = 1 \) and \( b = -14 \). Let's calculate \( h \): \[ h = -\frac{-14}{2 \cdot 1} = \frac{14}{2} = 7 \] Next, we can find \( k \) by substituting \( h \) back into the function: \[ k = f(7) = 7^2 - 14 \cdot 7 + 24 \] Calculating \( k \): \[ k = 49 - 98 + 24 = -25 \] Thus, the vertex of the parabola is \( (7, -25) \). ### Conclusion All three statements are true for parabolas, but the most fundamental and universally applicable statement is: **The vertex lies on the line of symmetry.** This statement is essential for finding the vertex of any parabola.