Functions and graphs

2014-12-05 7:18 am

1. Consider the function f(x) = x²+4x-2.
(a) Rewrite the function into form a(x-h)²+k.
(b) Find the coordinates of the vertex of the graph of y=f(x).

2. The height and the base of a triangle are (50+2x) cm and (200-2x) cm respectively.
(a) Express the area of the triangle in terms of x.
(b) Find the maximum area of the triangle and the corresponding height and base.

回答 (1)

用戶26240

2014-12-05 8:06 am

✔ 最佳答案

1.
(a)
f(x) = x² + 4x - 2
f(x) = [x² + 4x]- 2
f(x) = [x² + 2(2x) + 2²] -2² - 2
f(x) = [x + 2]² - 6
f(x) = [x - (-2)]² - 6

(b)
Coordinates of vertex = (-2, 6)

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2.
(a)
Area of the triangle
= (1/2)(50 + 2x)(200 - 2x) cm²
= (50 + 2x)(100 - x) cm²
= 5000 - 50x + 200x - 2x² cm²
= 5000 + 150x - 2x² cm²

(b)
Area of the triangle
= 5000 + 150x - 2x² cm²
= -2[x² -75x] + 5000 cm²
= -2[x² - 2(75/2)x+ (75/2)²] + 2(75/2)² +5000
= -2(x - 37.5)² + 7812.5

When x is any real number, -2(x - 37.5)² ≤0
Hence, area of the triangle is the maximum when (x - 37.5) = 0, i.e. x = 37.5

Maximum area of the triangle = 7812.5cm²
Corresponding height = 50 + 2(37.5) cm = 125 cm
Corresponding base = 200 - 2(37.5) = 125 cm

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