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The function h(t)=−16t2+v0t+h0h(t) = -16t^2 + v_0 t + h_0 models the height h(t)h(t), in feet, of a model rocket tt seconds after launch, where v0v_0 and h0h_0 are constants. The rocket reaches its maximum height of 100100 feet above the ground 22 seconds after it is launched. Which of the following equations defines h(t)h(t)?

  1. h(t)=−16t2+64t+36h(t) = -16t^2 + 64t + 36Cevap
  2. B
    h(t)=−16t2−64t+36h(t) = -16t^2 - 64t + 36
  3. C
    h(t)=−16t2+64t+100h(t) = -16t^2 + 64t + 100
  4. D
    h(t)=−16t2+64t−164h(t) = -16t^2 + 64t - 164

Cevap

h(t)=−16t2+64t+36h(t) = -16t^2 + 64t + 36
The correct equation is h(t)=−16t2+64t+36h(t) = -16t^2 + 64t + 36. Since the maximum height of 100100 feet is reached at t=2t = 2 seconds, the vertex of the parabola is (2,100)(2, 100). In vertex form, a quadratic function is written as h(t)=a(t−d)2+ch(t) = a(t - d)^2 + c, where (d,c)(d, c) is the vertex. Given that the leading coefficient aa is −16-16, substituting the vertex yields h(t)=−16(t−2)2+100h(t) = -16(t - 2)^2 + 100. Expanding this expression gives h(t)=−16(t2−4t+4)+100=−16t2+64t−64+100=−16t2+64t+36h(t) = -16(t^2 - 4t + 4) + 100 = -16t^2 + 64t - 64 + 100 = -16t^2 + 64t + 36.

Adım Adım Çözüm

1
Identify the vertex from the problem description.
The vertex of the parabola is (2,100)(2, 100), representing the time t=2t = 2 seconds when the maximum height of 100100 feet is reached.
The vertex (h,k)(h, k) of a quadratic function represents the maximum or minimum point of its graph.
2
Write the quadratic equation in vertex form.
h(t)=a(t−2)2+100h(t) = a(t - 2)^2 + 100. Since the coefficient of t2t^2 in the standard form is −16-16, we set a=−16a = -16, giving h(t)=−16(t−2)2+100h(t) = -16(t - 2)^2 + 100.
The vertex form of a quadratic function is y=a(x−h)2+ky = a(x - h)^2 + k, and the leading coefficient aa is the same as in standard form.
3
Expand the vertex form equation into standard form.
h(t)=−16(t2−4t+4)+100=−16t2+64t−64+100=−16t2+64t+36h(t) = -16(t^2 - 4t + 4) + 100 = -16t^2 + 64t - 64 + 100 = -16t^2 + 64t + 36.
Expanding the squared term and distributing the leading coefficient converts the vertex form to standard form y=ax2+bx+cy = ax^2 + bx + c.

Anahtar Kavram

Quadratic Functions and Graphs

Alternatif Yöntem

The axis of symmetry for a quadratic function in standard form y=ax2+bx+cy = ax^2 + bx + c is given by x=−b2ax = -\frac{b}{2a}. For this model, the maximum occurs at t=2t = 2, meaning the axis of symmetry is t=2t = 2. Since a=−16a = -16, we have 2=−b2(−16)2 = -\frac{b}{2(-16)}, which simplifies to b=64b = 64. We can then test the remaining options where the linear coefficient is 6464. Substituting t=2t = 2 into the correct equation h(t)=−16t2+64t+36h(t) = -16t^2 + 64t + 36 yields h(2)=−16(4)+64(2)+36=−64+128+36=100h(2) = -16(4) + 64(2) + 36 = -64 + 128 + 36 = 100, confirming it reaches the correct maximum height.
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Quadratic Functions and Graphs

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