这部分需要会什么？
- 使用指数运算法则改写表达式
- 用同底数解指数方程
- 理解指数函数的初始值与每单位倍率
核心概念与公式
在表达式有定义的前提下，\(a^ma^n=a^{m+n}\)，\(a^m/a^n=a^{m-n}\)，\((a^m)^n=a^{mn}\)。除法和负指数要求底数非零；涉及任意实数指数时通常取正底数。
指数函数 \(y=Ab^x\)（\(A>0,b>0,b\ne1\)）的初始值为 \(A\)，输入每增加 1，输出乘 \(b\)。当 \(b>1\) 时递增；\(0<b<1\) 时递减。指数方程能改成相同正底数且底数不为 1 时，可以比较指数。
做题时怎样使用
- 先检查底数能否写成同一个数的幂
- 区分“幂相乘”与“指数相乘”，逐步简化
- 解释函数时，找出输入增加 1 对输出造成的倍率
容易混淆的地方
\(a^m+a^n\) 一般不能合并成 \(a^{m+n}\)。\(A\) 是乘在指数项外面的系数，不是底数。
用一个例题串起方法
知识点教学例题解方程 \(8^{x-1}=4^{x+1}\)。
展开解题思路 ＋
统一底数为 2：\(2^{3x-3}=2^{2x+2}\)。由于底数为 2，可令指数相等，得 \(3x-3=2x+2\)，所以 \(x=5\)。
本知识点题库
先独立作答，再查看答案与解析。
指数式・指数方程与函数特征基础
The graph of the exponential function \( f \) is shown, where \( y=f\left(x\right) \). The y-intercept of the graph is \( \left(0,\ y\right) \). What is the value of y?
原库参考解析
指数式・指数方程与函数特征挑战
The table shows three values of x and their corresponding values of y for the equation \( y=4\left(2\right)^x+3 \) . In the table, a is a constant. What is the value of a?
原库参考解析
指数式・指数方程与函数特征挑战
Function f is defined by f(x) =-ax + b , where a and b are constants. In the xy-plane, the graph of y = f(x) -15 has a y-intercept at \( \left(0\ ,\ -\frac{99}{7}\right) \) . The product of a and b is \( \frac{65}{7} \). What is the value of a ?
原库参考解析
指数式・指数方程与函数特征挑战
f(t) =8,000(0.65)t
The given function f models the number of coupons a company sent to their customers at the end of each year, where t represents the number of years since the end of 1998, and \( 0\le t\le5 \). If y = f(t) is graphed in the ty-plane, which of the following is the best interpretation of the y-intercept of the graph in this context?
原库参考解析
指数式・指数方程与函数特征挑战
M = 1,800(1.02)t
The equation above models the number of members, M, of a gym t years after the gym opens. Of the following, which equation models the number of members of the gym q quarter years after the gym opens?
原库参考解析
指数式・指数方程与函数特征挑战
The functions \(f\) and \(g\) are defined by the given equations, where \(x\ge0\). Which of the following equations displays, as a constant or coefficient, the maximum value of the function it defines, where \(x\ge0\)?
I. \(f(x)=33(0.4)^x+3\)
II. \(g(x)=33(0.16)(0.4)^x-2\)
原库参考解析
指数式・指数方程与函数特征挑战
The function h is defined by h(x) = ax + b , where a and b are positive constants. The graph of y = h(x) in the xy- plane passes through the points (0 ,10 ) and (-2 ,\( \frac{325}{36} \) ). What is the value of ab ?
原库参考解析
指数式・指数方程与函数特征挑战
f(x) =9,000(0.66)x
The given function f models the number of advertisements a company sent to its clients each year, where x represents the number of years since 1997, and 0\( \le \) x\( ≤ 5 \). If y = f(x) is graphed in the xy-plane, which of the following is the best interpretation of the y-intercept of the graph in this context?
原库参考解析
指数式・指数方程与函数特征挑战
f(x) = 9(4)x
The function f is defined by the given equation. If g(x) = f (x +2) , which of the following equations defines the
function g ?
原库参考解析
指数式・指数方程与函数特征挑战
For the exponential function f , the value of f (1) is k , where k is a constant. Which of the following equivalent forms of the function f shows the value of k as the coefficient or the base?
原库参考解析
指数式・指数方程与函数特征挑战
For the exponential function f, the table above shows several values of x and their corresponding values of f(x), where a is a constant greater than 1. If k is a constant and f(k) = a29, what is the value of k ?
原库参考解析
指数式・指数方程与函数特征挑战
Function f is defined by f(x) =-ax + b , where a and b are constants. In the xy-plane, the graph of y = f(x) -12 has a y-intercept at \( \left(0,-\frac{75}{7}\right) \) . The product of a and b is \( \frac{320}{7} \). What is the value of a ?
原库参考解析
