Question

1. find the vertex of $y = -3x^2 - 24x - 46$

Explanation:

Step1: Recall the vertex formula for a quadratic function

For a quadratic function in the form \( y = ax^2+bx + c \), the x - coordinate of the vertex is given by \( x=-\frac{b}{2a} \).
In the function \( y=-3x^{2}-24x - 46 \), we have \( a=-3 \), \( b = - 24 \), and \( c=-46 \).

Step2: Calculate the x - coordinate of the vertex

Substitute \( a=-3 \) and \( b=-24 \) into the formula \( x =-\frac{b}{2a} \).
\( x=-\frac{-24}{2\times(-3)}=-\frac{24}{6}=- 4 \)

Step3: Calculate the y - coordinate of the vertex

Substitute \( x = - 4 \) into the original function \( y=-3x^{2}-24x - 46 \).
\( y=-3\times(-4)^{2}-24\times(-4)-46 \)
First, calculate \( (-4)^{2}=16 \), then \( -3\times16=-48 \), \( -24\times(-4) = 96 \)
\( y=-48 + 96-46=2 \)

Answer:

The vertex of the parabola \( y=-3x^{2}-24x - 46 \) is \( (-4,2) \)