1.\;指數函數的定義

設a>0, \space a≠1,\;x是任意實數， \space 則函數
\y=f(x)=a^x稱為以a為底數的指數函數。

2.\;指數函數的圖形

(1)\;a>1時：

(2)\;a>b>1時：

y=a^x

y=a^x

y=b^x

y=a^x

(3)\;0<a<1時：

(4)\;0<a<b<1時：

3.\;指數函數圖形的特性

若\;a>0,\;a≠1,\;則指數函數\;y=f(x)=a^x\;的圖形有以下性質

(1)\;y=a^x\;的圖形恆在x軸上方，\;\
\;\;\;\;\;\;與x軸漸近，\;但不相交。\;\
\;\;\;\;\;\;即：\;∀\;x∈ℝ,\;a^x>0

(2)\;y=a^x\;的圖形恆過點(0,1)

(0,1)

(3)\;y=a^x\;的圖形與平行於x軸\
\;\;\;\;\;\;且在x軸上方的每一條直線\
\;\;\;\;\;\;都恰交於一點。即：\;∀k>0,\;\
\;\;\;\;\;\;滿足a^x=k之x值恰有一解。

(4)\;a>1時，\;y=a^x的圖形由左而右\
\;\;\;\;\;\;逐漸升高，\;為遞增函數。\;\
\;\;\;\;\;\;即當a>1時，\;x_1,x_2∈ℝ,\;則：\;\
\;\;\;\;\;\;a^{x_1}>a^{x_2}⇔x_1>x_2

(5)\;0<a<1時，\;y=a^x的圖形由左而右\
\;\;\;\;\;\;逐漸降低，\;為遞減函數。\;\
\;\;\;\;\;\;即當0<a<1時，\;x_1,x_2∈ℝ,\;則：\;\
\;\;\;\;\;\;a^{x_1}>a^{x_2}⇔x_1<x_2

(6)\;圖形增高時，\;越往右增高越快；\
\;\;\;\;\;\;圖形降低時，\;越往右降低越慢。\
\;\;\;\;\;\;即：\;若x_1,x_2為相異實數，\;則：\
\;\;\;\;\;\;⇒\frac{a^{x_1}+a^{x_2}}{2}>a^{\frac{x_1+x_2}{2}}

(7)\;底數互為倒數之兩指數函數其\
\;\;\;\;\;\;圖形對稱於y軸。即：\;a^{-x}=(\frac{1}{a})^x

(8)\;當x>0時，\;a越大圖形越接近y軸：\
\;\;\;\;\;\;當x<0時，a\;越小圖形越接近y軸。\
\;\;\;\;\;\;即：\;x>0時,\;a^x>b^x⇔a>b：\
\;\;\;\;\;\;⇔x<0時：\;a^x>b^x⇔a<b

Forward

y=k

指數函數的圖形

s = 0 (slider from 0 to 20, step 1)
A = Corner(4)
B = (9.427851680933745, -8.682983355862644)
C: B + (23, 0)
u_1 = Vector(B - (9, 0), B + (9, 0))
u_2 = Vector(C - (9, 0), C + (9, 0))
v_1 = Vector(B - (0, 3), B + (0, 10))
v_2 = Vector(C - (0, 3), C + (0, 10))
f_1(x) = If[x(B) - 9 ≤ x ≤ x(B) + log(1.5, 10),1.5^(x - x(B)) + y(B)]
f_2(x) = If[x(C) - 9 ≤ x ≤ x(C) + log(1.5, 10),1.5^(x - x(C)) + y(C)]
f_3(x) = If[x(C) - 9 ≤ x ≤ x(C) + log(2, 10),2^(x - x(C)) + y(C)]
g_1(x) = If[x(B) - 6.4 ≤ x ≤ x(B) + 9,0.7^(x - x(B)) + y(B)]
g_2(x) = If[x(C) - 6.4 ≤ x ≤ x(C) + 9,0.7^(x - x(C)) + y(C)]
g_3(x) = If[x(C) - 9 ≤ x ≤ x(C) + 9,0.8^(x - x(C)) + y(C)]
a = 1.7 (slider from 0.1 to 10, step 0.1)
f = If(a > 1, If[x(C) - 9 ≤ x ≤ Min[x(C) + 9, x(C) + log(a, 10)],a^(x - x(C)) + y(C)], If[Max[x(C) - 9, x(C) + log(a, 10)] ≤ x ≤ x(C) + 9,a^(x - x(C)) + y(C)])
D_{14}: C + (0, 1)
k = 7.5 (slider from 0.1 to 10, step 0.1)
k_0 = Segment(C + ((-9), k), C + (9, k))
D = Intersect(f, k_0, (40.10764383795186, -1.1829833558851357))
g(x) = f((-x) + (2 * x(C)))
text121: "y=" + (LaTeX[a]) + "^x"