新航道

  中國數學的難度相比較于美國難很多，所以在SAT沒改革前，SAT數學滿分考生比比皆是。進入新SAT時代后，從新SAT考試成績的數據統計中不難發現能到800分的考生是非常罕見的，780分以上的也是寥寥無幾，大部分都集中在700—750之間。是什么原因造成這個看似簡單卻拿不到最高分的考試呢?那是因為你沒有找到學習的正確技巧和方法。

  函數圖像

  常見的幾種圖像變換：

  Suppose defines any function:

  1. translates k units vertically (up if k>0; down if k<0);

  2. translates h units horizontally (right if h>0; left if h<0);

  3. stretches (shrinks) f(x) vertically by a factor of |a| if |a|>1 (|a|<1);

  4. stretches (shrinks) f(x) horizontally by a factor of if |a|>1 (|a|<1);

  5. reflects about the x-axis. (the reflection is vertical.);

  6. reflects about the y-axis. (the reflection is horizontal.);

  例題10. The function f(x) is graphed above. If g(x) = f(x) – 1,which of the following statements is TRUE?

  (A) g(x) is greater than or equal to zero.

  (B) g(x) is greater than or equal to negative one.

  (C) g(x) is greater than or equal to negative two.

  (D) g(x) is greater than negative one, but smaller than five.

  Solution:

  g(x) = f(x) – 1，圖像整體向下平移一個單位，選C。

  例題11. Which of the following equations represents the function f(x) = 2x – 1 shifted 2 units to the left on the my-plane?

  (A) g(x) = 2x – 5

  (B) g (x)– 3 = 2x

  (C) g(x) = 2x + 1

  (D) g(x)= 2x – 3

  Solution：

  向左移動，g(x)=f(x+2)=2(x+2)-1=2x+3

  答案：B

  例題12. A system of three equations and their graphs are shown here. How many solutions does the system of equations have?

  (A) none

  (B) one

  (C) two

  (D) three

  Solution：

  方程的解即為方程圖像的交點，由圖中可以看出，圓，直線，拋物線，三者交于一點，所以以上的方程組只有一解。

  答案：B

  例題13. The following figure shows the graph of f(x) from x = -6 to x = 6. If f(3) = a, what is f(a) ?

  (A) -6

  (B) -5

  (C) 0

  (D) 2

  Solution：

  f(3)=a=-5, f(-5)=2

  答案：D。

  例題14. The graph above shows the amount that a new, high-tech video arcade charges its customers.

  What could the y-intercept of this graph represent?

  A) The cost of playing 5 games

  B) The cost per game, which is $5

  C) The entrance fee to enter the arcade

  D) The number of games that are played

  Solution：

  函數圖像的理解：和Y軸的交點時，X值為零，所以這里的含義應該是基礎費用。選C。

  例題15. The graph of g(x) is shown in the figure above. If h(x) = -g(x) + 1, which of the following statements is true?

  A) The range of h(x) is -3 ≤ y ≤ 3.

  B) The minimum value of h(x) is -4.

  C) The coordinates of point A on the function h(x) are (2, 4).

  D) The graph of h(x) is increasing between x = -5 and x = -2.

  Solution：

  h(x) = -g(x) + 1 所對應的函數圖像操作為：先關于X軸做對稱，再向上平移一個單位。

  原函數的值域為 [-2,4]，變化之后為[-3，3]，A正確。

  答案：A。

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