Question

find the y-intercept, the axis of symmetry, and the vertex of the graph of the function.\\( f(x) = 3x^2 + 24x + 7 \\)\\( \\)\\( \\)the y-intercept is \\( (0,7) \\) (type an ordered pair)\\( \\)the axis of symmetry is \\( x = -4 \\) (simplify your answer. type an equation)\\( \\)the vertex is \\( \square \\) (type an ordered pair)

Explanation:

Step1: Recall the vertex formula

For a quadratic function \( f(x) = ax^2 + bx + c \), the x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \), and we already know from the axis of symmetry that \( x = -4 \). To find the y-coordinate, we substitute \( x = -4 \) into the function \( f(x) = 3x^2 + 24x + 7 \).

Step2: Substitute \( x = -4 \) into the function

\[

$$\begin{align*} f(-4)&= 3(-4)^2 + 24(-4) + 7\\ &= 3(16) - 96 + 7\\ &= 48 - 96 + 7\\ &= -48 + 7\\ &= -41 \end{align*}$$

\]

Answer:

\((-4, -41)\)