If f(x) = 2x²  8x + 9, which statement regarding the vertex form of f(x) is true?
A.) In vertex form, f(x) = 2(x  2)² + 1 and therefore has a minimum value of 1. B.) In vertex form, f(x) = 2(x  2)² + 1 and therefore has a minimum value of 2. C.) In vertex form, f(x) = 2(x  2)² + 4.5 and therefore has a minimum value of 4.5. D.) In vertex form, f(x) = 2(x  2)² + 4.5 and therefore has a minimum value of 2.
The equation f(x) = 2x²  8x + 9 is in standard form and must be transformed into vertex form. To begin, find the x and y coordinates of the vertex, then substitute them along with a which is 2, into vertex form. The xcoordinate of the vertex is x = b / 2a In f(x) = 2x²  8x + 9 a = 2, b = 8, and c = 9 x = b/2a {the xcoordinate of the vertex} x = (8) / 2(2) {substituted 8 for b and 2 for a into the xcoordinate of the vertex} x = 8/4 {simplified and multiplied} x = 2 {divided} The xcoordinate of the vertex is 2. To find the ycoordinate of the vertex, substitute 2 in for x into the equation f(x) = 2x²  8x + 9. f(x) = 2x²  8x + 9 {original equation} f(2) = 2(2)²  8(2) + 9 {substituted 2 for x} = 2(4)  16 + 9 {evaluated exponent and multiplied} = 8  16 + 9 {multiplied} = 1 {subtracted and added} The ycoordinate of the vertex is 1. The vertex has coordinates of (2,1). In vertex form, these are the coordinates (h,k). Also the value of a is the same as in standard form, which is 2. f(x) = a(x  h)² + k {vertex form of a quadratic equation} f(x) = 2(x  2)² + 1 {substituted a,h, and k into vertex form} In a parabola, which is the graph of a quadratic function, the ycoordinate of the vertex is either the minimum or maximum value. Since the leading coefficient, 2, is positive, this means the parabola opens upward, and thus has a minimum value. A.) In vertex form, f(x) = 2(x  2)² + 1 and therefore has a minimum value of 1. Comments are closed.

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