Question

write the equation of the parabola that has its vertex at $(-4, -2)$ and passes through the point $(-3, -9)$ in $f(x)=a(x - h)^2 + k$ form.
the equation of the parabola in $f(x)=a(x - h)^2 + k$ form is $f(x)=\square$.

Explanation:

Step1: Identify vertex \((h,k)\)

Given vertex \((-4, -2)\), so \(h = -4\), \(k = -2\). The vertex form is \(f(x)=a(x - h)^2 + k\), substitute \(h\) and \(k\): \(f(x)=a(x - (-4))^2 + (-2)=a(x + 4)^2 - 2\).

Step2: Substitute point \((-3, -9)\)

The parabola passes through \((-3, -9)\), so substitute \(x = -3\), \(f(x) = -9\) into the equation: \(-9 = a(-3 + 4)^2 - 2\).

Step3: Solve for \(a\)

Simplify the equation: \(-9 = a(1)^2 - 2\) → \(-9 = a - 2\). Add 2 to both sides: \(a = -9 + 2 = -7\).

Step4: Write the equation

Substitute \(a = -7\), \(h = -4\), \(k = -2\) into vertex form: \(f(x)= -7(x + 4)^2 - 2\).

Answer:

\(f(x)= -7(x + 4)^2 - 2\)