Question

question 38 · 1 point
determine the equation of the parabola whose graph is given below. write the equation in
general form $f(x) = ax^2 + bx + c$.

graph of a parabola with vertex at (-4, 3) and a point at (-3, 5) on a coordinate grid

provide your answer below:

$f(x) = \square$

Explanation:

Step1: Identify the vertex

The vertex of the parabola is at \((-4, 3)\). The vertex form of a parabola is \(f(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex. So substituting \(h = -4\) and \(k = 3\), we get \(f(x) = a(x + 4)^2 + 3\).

Step2: Find the value of \(a\)

We can use another point on the parabola. From the graph, the point \((-3, 5)\) lies on the parabola. Substitute \(x = -3\) and \(f(x) = 5\) into the vertex form:

$$ 5 = a(-3 + 4)^2 + 3 $$

$$ 5 = a(1)^2 + 3 $$

$$ 5 = a + 3 $$

Subtract 3 from both sides: \(a = 5 - 3 = 2\).

Step3: Convert to general form

Now that we have \(a = 2\), the vertex form is \(f(x) = 2(x + 4)^2 + 3\). Expand this:

$$ f(x) = 2(x^2 + 8x + 16) + 3 $$

$$ f(x) = 2x^2 + 16x + 32 + 3 $$

$$ f(x) = 2x^2 + 16x + 35 $$

Answer:

\(f(x) = 2x^2 + 16x + 35\)