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)?
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). Ask Algebra House Comments are closed.
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