Question

What is the function in Vertex form: f(x) = 15x^2 + 60x - 19

Answer 1

To convert the quadratic function \( f(x) = 15x^2 + 60x - 19 \) into vertex form, we need to complete the square. The vertex form of a quadratic function is given by:

\[
f(x) = a(x - h)^2 + k
\]

where \((h, k)\) is the vertex of the parabola.

### Step 1: Factor out the coefficient of \(x^2\)

First, factor out the coefficient of \(x^2\) (which is 15) from the terms involving \(x\):

\[
f(x) = 15(x^2 + 4x) - 19
\]

### Step 2: Complete the square

Next, we need to complete the square for the expression in parentheses, \(x^2 + 4x\). To do this, take half of the coefficient of \(x\) (which is 4), square it, and add/subtract that value inside the parentheses:

1. Half of 4 is 2.
2. Squaring it gives \(2^2 = 4\).

Now we can rewrite the expression:

\[
x^2 + 4x = (x^2 + 4x + 4) - 4 = (x + 2)^2 - 4
\]

### Step 3: Rewrite the function

Substituting this back into the equation gives:

\[
f(x) = 15((x + 2)^2 - 4) - 19
\]

Distributing the 15:

\[
f(x) = 15(x + 2)^2 - 60 - 19
\]

Combining the constant terms:

\[
f(x) = 15(x + 2)^2 - 79
\]

### Final Vertex Form

Thus, the function in vertex form is:

\[
f(x) = 15(x + 2)^2 - 79
\]

The vertex of the parabola is \((-2, -79)\).