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)?

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

**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).

**- Algebra House**