Exponential Functions
Introduced| English | Meaning |
|---|---|
| base/beɪs/ | The number raised to a power in an exponential expression. |
| exponential function/ˌekspəˈnenʃl ˈfʌŋkʃn/ | A function in which the variable appears in an exponent, such as a fixed positive base raised to that variable. |
| exponential growth/ˌekspəˈnenʃl ɡrəʊθ/ | An increase in which a quantity is multiplied by the same factor greater than one over equal intervals. |
| exponential decay/ˌekspəˈnenʃl dɪˈkeɪ/ | A decrease in which a quantity is multiplied by the same factor, between zero and one, over equal intervals. |
| domain/dəˈmeɪn/ | The set of input values for which a function is defined. |
| range/reɪndʒ/ | The set of output values actually attained by a function on its stated domain. |
| horizontal asymptote/ˌhɒrɪˈzɒntl ˈæsɪmptəʊt/ | A horizontal line approached by a function as its input tends to positive or negative infinity. |
| initial value/ɪˈnɪʃl ˈvæljuː/ | The starting value of a quantity, function or system at a specified initial point or time. |
Same base, opposite directions
- Compare $f(x)=3(4)^x$ with $g(x)=-3(4)^x$. They use the same base 底数 of 4, but one rises while the other falls as $x$ increases.
- Both are exponential functions 指数函数. To describe the graph, read the sign of the starting coefficient as well as the base.
State the nonconstant family
- The general form is $f(x)=ab^x$, with $a\ne0$, $b>0$ and $b\ne1$. The input $x$ is in the exponent; in the power function $x^2$, it is in the base instead.
- At $x=0$, $f(0)=a$, since $b^0=1$. The cases $a=0$ or $b=1$ give constant functions, so they are excluded from this nonconstant family. A negative base cannot be raised to every real input and still give a real output.
An exponential function has the form…
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The variable is in the exponent of a fixed positive base. Exclude a zero coefficient and base 1, which give constants.
$f(x)=0(4)^x$ belongs to the nonconstant exponential family.
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It is the constant zero function. The nonconstant family requires a nonzero coefficient.
Equal input gaps give equal factors
- Increasing the input by 1 multiplies the output by $b$. More generally, a gap $h$ multiplies it by $b^h$: $f(x+h)=b^h f(x)$. A factor must always be attached to its input interval.
- For $f(x)=3(4)^x$, the outputs at $x=0,1,2$ are $3,12,48$. At $x=-1$ the output is $3/4=0.75$; at $x=0.5$ it is $3\sqrt4=6$. The function includes negative and fractional inputs, not just whole-number steps.
For $f(x)=3(4)^x$, find $f(-1)$.
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$4^{-1}=1/4$, so $f(-1)=3/4=0.75$.
Growth and decay for a positive start
- With $a>0$, a base $b>1$ gives exponential growth 指数增长: $3(4)^x$ increases. A base $0<b<1$ gives exponential decay 指数衰减: $3(0.25)^x$ decreases toward zero as $x$ increases.
- The pictured curves have positive coefficients. Their growth/decay labels rely on that condition; they do not describe a graph with a negative coefficient.
For $f(x) = a\,b^x$ with $a > 0$, the function shows decay when…
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A base between $0$ and $1$ shrinks the output each step — exponential decay. A base above $1$ grows it.
A negative coefficient reflects the graph
- Multiplying by $-1$ reflects the positive-coefficient graph across the horizontal axis. Thus $-3(4)^x$ decreases: at $x=0,1,2$ its outputs are $-3,-12,-48$.
- With $a<0$ and $0<b<1$, the outputs instead increase toward zero from below. For example, $-3(0.25)^x$ gives $-3,-0.75,-0.1875$ at $x=0,1,2$. The base alone does not determine increasing or decreasing behavior.
Read the coefficient and the base together
f(x) = a·bˣ
Start with f(x)=3·4ˣ and f(2)=48. Compare positive and negative coefficients, both base ranges, and the excluded constant cases.
How does $g(x)=-2(0.5)^x$ change as x increases?
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The positive decay curve is reflected across the horizontal axis. Its negative outputs increase toward zero.
Domain, range and the zero line
- The domain 定义域 is all real inputs. Since $b^x>0$, the range 值域 is $y>0$ when $a>0$, and $y<0$ when $a<0$. A nonzero coefficient means that no output is zero.
- The horizontal asymptote 水平渐近线 is $y=0$. When $b>1$, the graph approaches it as $x$ goes far to the left; when $0<b<1$, it approaches it to the right. The sign of $a$ decides whether it approaches from above or below.
For $a e0$, $b>0$ and $b e1$, $f(x)=ab^x$ has a horizontal ____ at $y=0$.
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The output gets arbitrarily close to $0$ but never reaches it, giving the horizontal asymptote $y = 0$.
Select all true statements about $f(x) = a\,b^x$ (with $a>0,\ b>1$).
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With $a > 0$ the output is always positive, so it never goes negative. The other three are correct.
What is the range of $g(x)=-3(4)^x$ on all real inputs?
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The positive power $4^x$ is multiplied by −3. Every output is negative, and zero is never reached.
Bend and endpoints
- With $a>0$, the graph is concave up, bending upward; with $a<0$, it is concave down, bending downward. This remains true on either side of $b=1$, provided $b\ne1$. A decreasing positive exponential can still be concave up because its fall becomes less steep.
- A nonconstant exponential is strictly increasing or decreasing, with no interior turning point or inflection point. On a closed interval it can have endpoint extrema: $3(4)^x$ on $[-1,2]$ has minimum $0.75$ and maximum $48$.
What is the maximum of $f(x)=3(4)^x$ on the closed interval $[-1,2]$?
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The function is increasing, so the maximum occurs at the right endpoint: $f(2)=48$.
Read a positive model in context
- In a balance model, the initial value 初始值 is the balance at the chosen time zero. If $P(t)=500(1.08)^t$, then $P(0)=500$ and the factor $1.08$ means an 8% increase each year.
- After 10 years, the model predicts $500(1.08)^{10}\approx1079.46$ USD. This is a hypothetical constant-rate model; its physical time domain can be restricted to $t\ge0$ even though the formula is defined for all real $t$.
For $P(t)=500(1.08)^t$, compare the roles of the two numbers. The coefficient 500 is the output at time zero; the base 1.08 multiplies the current balance at each one-year step. It adds 8% of the current balance, rather than adding 8 USD each year.
In $P = 500 \cdot 1.08^{\,t}$, what does the $500$ represent?
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At $t = 0$, $1.08^0 = 1$, so $P = 500$: the coefficient $a$ is the initial value, and $1.08$ is the base (8% growth).
A shifted output can hide the ratio
- The outputs of $H(x)=20+3(2)^x$ at $x=0,1,2$ are $23,26,32$. Their raw ratios are not constant, but subtracting the background of 20 gives $3,6,12$, with constant factor 2.
- This is a shifted exponential: its horizontal asymptote is $y=20$, and its range is $y>20$. The statement that the range is above or below zero applies to the unshifted form $ab^x$. The modeling lesson develops background-adjusted examples.
An expression containing an exponent is not automatically a nonconstant exponential function. The variable must be in the exponent over a fixed positive base, with $a\ne0$ and $b\ne1$. Also check whether the output has been shifted before stating its range or asymptote.
For $ab^x$, use both $a$ and $b$ to describe the graph. The coefficient gives the intercept and the side of zero; the base gives the factor per unit input. Every nonconstant member has a fixed direction and concavity, all-real domain and horizontal asymptote $y=0$.
The horizontal asymptote of $H(x)=20+3(2)^x$ is y = what value?
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As the exponential term tends to zero, H approaches the added background 20.
Explain the value and the reflection
- Let $f(x)=3(4)^x$. Find $f(2)$ and explain the order of the calculations.
- If the coefficient changes from 3 to $-3$, describe the output at $x=2$, the range and the direction of the graph. State which assumptions you use.
For $f(x)=3(4)^x$, calculate $f(2)$ and explain the order of operations. If the coefficient becomes −3, state the output at x=2, the range and the direction of the resulting graph.
A sign change reflects all outputs; it also reverses the direction and changes the side of the asymptote.
For $f(x)=3(4)^x$, what is $f(2)$?
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The power is computed before the coefficient.