Comparing y-intercepts of quadratic functions

The graph of a quadratic function f(x) has a minimum at (-2,-3) and passes through the point (2,13).  The function g(x) is represented by the equation g(x) = -(x + 2)(x - 3).

How much greater is the y-intercept of g(x) than f(x)?

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For f(x):
the vertex is (-2,-3) and it passes through (2,13)
substitute that information into vertex form

f(x) = a(x - h)² + k   {vertex form of a quadratic function}
13 = a[2 - (-2)]² - 3   {substituted point and vertex into vertex form}
13 = a(4)² - 3   {worked inside parentheses}
13 = 16a - 3   {evaluated exponent}
16 = 16a   {added 3 to each side}
a = 1   {divided each side by 16}

f(x) = 1[x - (-2)]² - 3   {substituted vertex and value of a back into vertex form}
f(x) = (x + 2)² - 3   {simplified}
=  (x + 2)(x + 2) - 3   {began transforming into standard form}
= x² + 4x + 4 - 3   {squared the binomial}
f(x) = x² + 4x + 1   {combined like terms}

y-intercept of f(x) is 1

For g(x):
g(x) = -(x + 2)(x - 3)
=  -(x² - x - 6)   {used FOIL method / distributive property}
g(x) = -x² + x + 6   {distributed negative sign}

y-intercept of g(x) is 6

The y-intercept of g(x) is 5 greater than the y-intercept of f(x).

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