To find the axis of symmetry for the given parabola, we need to use the standard formula for the axis of symmetry of a quadratic function in the form [tex]\( y = ax^2 + bx + c \)[/tex].
Given the quadratic equation:
[tex]\[ y = -3x^2 - 24x - 36 \][/tex]
1. Identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] from the equation. Here:
[tex]\[
a = -3, \quad b = -24, \quad c = -36
\][/tex]
2. The formula for the axis of symmetry is:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
3. Substitute the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the formula:
[tex]\[
x = -\frac{-24}{2 \cdot -3}
\][/tex]
4. Simplify the expression step by step:
[tex]\[
x = \frac{24}{-6}
\][/tex]
[tex]\[
x = -4
\][/tex]
Therefore, the axis of symmetry for the parabola [tex]\( y = -3x^2 - 24x - 36 \)[/tex] is:
[tex]\[ x = -4 \][/tex]
This means that the axis of symmetry is the vertical line [tex]\( x = -4 \)[/tex].
Note: When you're plotting or analyzing the graph using graphing technology, you should see that the parabola is symmetric around this line [tex]\( x = -4 \)[/tex].
