Question

设 $f(x) = 100x^3 - 300x^2 + 200x$ 。有多少个实数 $a$ ，使得 $y = f(x - a)$ 的图像经过点 $(1, 25)$ ？

Let $f(x) = 100x^3 - 300x^2 + 200x.$ For how many real numbers $a$ does the graph of $y = f(x - a)$ pass through the point $(1, 25)?$

$1$

$2$

$3$

$4$

多于 $4$

more than $4$

答案：C

知识点：函数多项式

难度评级：1440

解答：

图像经过 $(1,25)$ ，当且仅当 $f(1 - a) = 25$ 。令 $t = 1 - a$ ，于是只需要数 $f(t) = 25$ 的解的个数。分解得 $f(x) = 100x(x-1)(x-2)$ ，根为 $0, 1, 2$ 。函数在 $(0,1)$ 上为正，且 $f(0.5) = 37.5 \gt 25$ ，所以由连续性可知 $0.5$ 两侧各有一个解。函数在 $(1,2)$ 上为负，而当 $x \gt 2$ 时从 $0$ 单调增加到无穷大，因此还有一个解。于是直线与这个三次函数图像相交 $3$ 次。每个交点给出唯一的 $a$ ，因此共有 $3$ 个值。正确答案是 C。

The graph passes through $(1,25)$ exactly when $f(1 - a) = 25.$ Let $t = 1 - a,$ so we count solutions of $f(t) = 25.$ Factor $f(x) = 100x(x-1)(x-2),$ with roots $0, 1, 2.$ On $(0,1),$ the function is positive and $f(0.5) = 37.5 \gt 25,$ so continuity gives one root on each side of $0.5.$ On $(1,2)$ the function is negative, while for $x \gt 2$ it increases from $0$ to infinity, giving one more root. A cubic equation has at most $3$ real roots, so these are all the solutions. Each gives one $a,$ so there are $3$ values. Thus, C is the correct answer.

2025 AMC 10A 第 9 题

答案：C

难度评级：1440

解答：

其他年份的第 9 题