Question

write the equation of the following parabola in vertex form.
the vertex is $(-4, -3)$ and the graph passes through the point $(2,9)$.
choose the correct equation below.
\\(\bigcirc\\) a. $f(x)=\frac{1}{3}(x + 4)^2 - 3$
\\(\bigcirc\\) b. $f(x)= -\frac{4}{3}(x - 4)^2 + 3$
\\(\bigcirc\\) c. $f(x)=\frac{1}{3}(x + 4)^2 + 3$
\\(\bigcirc\\) d. $f(x)=(x - 4)^2 - 3$

Explanation:

Step1: Recall vertex form of parabola

The vertex form of a parabola is \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. Given the vertex \((-4, -3)\), so \( h = -4 \) and \( k = -3 \). Substituting these into the vertex form, we get \( f(x) = a(x - (-4))^2 + (-3) = a(x + 4)^2 - 3 \).

Step2: Substitute the point \((2, 9)\) into the equation

We know the graph passes through \((2, 9)\), so substitute \( x = 2 \) and \( f(x) = 9 \) into \( f(x) = a(x + 4)^2 - 3 \). This gives \( 9 = a(2 + 4)^2 - 3 \).

Step3: Solve for \( a \)

First, simplify the equation: \( 9 = a(6)^2 - 3 \), which is \( 9 = 36a - 3 \). Add 3 to both sides: \( 9 + 3 = 36a \), so \( 12 = 36a \). Then divide both sides by 36: \( a = \frac{12}{36} = \frac{1}{3} \).

Step4: Write the final equation

Substitute \( a = \frac{1}{3} \), \( h = -4 \), and \( k = -3 \) back into the vertex form: \( f(x) = \frac{1}{3}(x + 4)^2 - 3 \).

Answer:

A. \( f(x)=\frac{1}{3}(x + 4)^2 - 3 \)