Quadratics - the Australian Mathematical Sciences Institute

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{14} • Quadratics We can find a simple formula for the value of k in terms of the coefficients of the quadratic. As usual, we complete the square: y = ax 2 + bx + c [ = a x 2 + b a x + c ] a [( = a x + b ) 2 c ( b ) 2 ] + 2a a − . 2a We can now see that the x-coordinate of the vertex is − b . Thus the equation of the axis 2a of symmetry is x = − b 2a . We could also find a formula for the y-coordinate of the vertex, but it is easier simply to substitute the x-coordinate of the vertex into the original equation y = ax 2 + bx + c. Example Sketch the parabola y = 2x 2 +8x +19 by finding the vertex and the y-intercept. Also state the equation of the axis of symmetry. Does the parabola have any x-intercepts Solution Here a = 2, b = 8 and c = 19. So the axis of symmetry has equation x = − b 2a = − 8 4 = −2. We substitute x = −2 into the equation to find y = 2 × (−2) 2 + 8 × (−2) + 19 = 11, and so the vertex is at (−2,11). Finally, putting x = 0 we see that the y-intercept is 19. There are no x-intercepts. y 19 11 –2 0 x
A guide for teachers – Years 11 and 12 • {15} Exercise 3 A parabola has vertex at (1,3) and passes through the point (3,11). Find its equation. The quadratic formula and the discriminant The quadratic formula was covered in the module Algebra review. This formula gives solutions to the general quadratic equation ax 2 + bx + c = 0, when they exist, in terms of the coefficients a,b,c. The solutions are x = −b + b 2 − 4ac 2a provided that b 2 − 4ac ≥ 0. , x = −b − b 2 − 4ac , 2a The quantity b 2 −4ac is called the discriminant of the quadratic, often denoted by ∆, and should be found first whenever the formula is being applied. It discriminates between the types of solutions of the equation: • ∆ > 0 tells us the equation has two distinct real roots • ∆ = 0 tells us the equation has one (repeated) real root • ∆ < 0 tells us the equation has no real roots. Exercise 4 For what values of k does the equation (4k + 1)x 2 − 6kx + 4 = 0 have one real solution A quadratic expression which always takes positive values is called positive definite, while one which always takes negative values is called negative definite. Quadratics of either type never take the value 0, and so their discriminant is negative. Furthermore, such a quadratic is positive definite if a > 0, and negative definite if a < 0. y Δ < 0, a > 0 0 x Δ < 0, a < 0
Page 1 and 2: 1 Supporting Australian Mathematics
Page 3 and 4: Assumed knowledge . . . . . . . . .
Page 5 and 6: A guide for teachers - Years 11 and
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Quadratics - the Australian Mathematical Sciences Institute

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