數學函數圖像

數學函數圖像

2011-04-07 2:42 am

(1)若把y=x^2-4x+3的圖像向左方平移3單位,可得出y=g(x)的圖像,求y=g(x)的圖像的x軸截距。
(2)己知f(x)=2^x和g(x)=4^x。試寫出g(x)與f(x)之間的關係。
(3)己知f(x)=x^2-2x和g(x)=x^2-8x+15。描述當f(x)變成g(x)時,y=f(x)的圖像對應的變換。
(4)己知f(x)=x^2-2x和g(x)=x^2+2x,描述當f(x)變換成g(x)時,y=f(x)的圖像對應的變換。

更新1:

可不可以用中文說明

回答 (1)

知識就是力量

2011-04-07 4:10 am

✔ 最佳答案

1. Let f(x) = y = x^2 - 4x + 3
g(x) = f(x+3)
= (x+3)^2 - 4(x+3) + 3
= x^2 + 6x + 9 - 4x - 12 + 3
= x^2 + 2x
= x(x+2)
When g(x) = 0, x = 0 or x = -2
x-intercept: 0, -2

2. g(x) = 4^x = 2^(2x) = (2^x)^2 = [f(x)]^2

3. f(x-3) = (x-3)^2 - 2(x-3) = x^2 - 6x + 9 - 2x + 6 = x^2 - 8x + 15
= g(x)
g(x) comes from the shifting of f(x) to rightwards by 3 units.

4. f(-x) = (-x)^2 - 2(-x) = x^2 + 2x
= g(x)
g(x) comes from the reflection of f(x) along the y-axis

Or f(x+2) = (x+2)^2 - 2(x+2) = x^2 + 4x + 4 - 2x - 4 = x^2 + 2x
= g(x)
g(x) comes from the shifting of f(x) to leftwards by 2 units.

2011-04-06 20:13:34 補充：
Alternative method for Q1:
f(x) = x^2 - 4x + 3 = (x-1)(x-3)
x-intercepts for f(x): 1,3
since the graph shifts to the leftwards by 3 units,
the x-intercepts for g(x) also shifts by 3 units --> x-intercepts: -2,0

參考: Knowledge is power.

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數學 函數圖像

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