7Number 7ExplanationIf the quadratic function y=kx2+8x+(k−1)y = kx^2 + 8x + (k-1)y=kx2+8x+(k−1) has an axis of symmetry x=4x = 4x=4, then the maximum value of the function is...−14-14−14141414−8-8−8888161616ExplanationGiven the quadratic function: y=kx2+8x+(k−1)y = kx^2 + 8x + (k-1)y=kx2+8x+(k−1) The axis of symmetry for a quadratic function ax2+bx+cax^2 + bx + cax2+bx+c is given by the formula x=−b2ax = -\frac{b}{2a}x=−2ab. In this case, a=ka = ka=k and b=8b = 8b=8. Given the axis of symmetry x=4x = 4x=4, then: 4=−82k4 = -\frac{8}{2k}4=−2k88k=−88k = -88k=−8k=−1k = -1k=−1 After finding the value of kkk, we substitute it back into the original function: y=−1x2+8x+(−1−1)y = -1x^2 + 8x + (-1 - 1)y=−1x2+8x+(−1−1)y=−x2+8x−2y = -x^2 + 8x - 2y=−x2+8x−2 To find the maximum value, we substitute the axis of symmetry value x=4x = 4x=4 into the function (since the optimum value is always at the axis of symmetry): y=−(42)+8(4)−2y = -(4^2) + 8(4) - 2y=−(42)+8(4)−2y=−16+32−2y = -16 + 32 - 2y=−16+32−2y=16−2y = 16 - 2y=16−2y=14y = 14y=14 Thus, the maximum value of the function is 141414.
7Number 7ExplanationIf the quadratic function y=kx2+8x+(k−1)y = kx^2 + 8x + (k-1)y=kx2+8x+(k−1) has an axis of symmetry x=4x = 4x=4, then the maximum value of the function is...−14-14−14141414−8-8−8888161616ExplanationGiven the quadratic function: y=kx2+8x+(k−1)y = kx^2 + 8x + (k-1)y=kx2+8x+(k−1) The axis of symmetry for a quadratic function ax2+bx+cax^2 + bx + cax2+bx+c is given by the formula x=−b2ax = -\frac{b}{2a}x=−2ab. In this case, a=ka = ka=k and b=8b = 8b=8. Given the axis of symmetry x=4x = 4x=4, then: 4=−82k4 = -\frac{8}{2k}4=−2k88k=−88k = -88k=−8k=−1k = -1k=−1 After finding the value of kkk, we substitute it back into the original function: y=−1x2+8x+(−1−1)y = -1x^2 + 8x + (-1 - 1)y=−1x2+8x+(−1−1)y=−x2+8x−2y = -x^2 + 8x - 2y=−x2+8x−2 To find the maximum value, we substitute the axis of symmetry value x=4x = 4x=4 into the function (since the optimum value is always at the axis of symmetry): y=−(42)+8(4)−2y = -(4^2) + 8(4) - 2y=−(42)+8(4)−2y=−16+32−2y = -16 + 32 - 2y=−16+32−2y=16−2y = 16 - 2y=16−2y=14y = 14y=14 Thus, the maximum value of the function is 141414.
