An exponential function has the form $y=ka^x$, where $a>0$, $a\ne1$ and $k\ne0$. For $a=e$ and $k=1$, $y=e^x$ is the natural exponential function. The input $x$ can be any real number. The output has the sign of $k$: it is positive when $k>0$ and negative when $k<0$. $y=k{{a}^{x}}$ $k=1$ $k=-1$ Notes $a>1$ It is an exponential graph. $x$ is the power in the function, $y=k{{a}^{x}}$. The curve cuts the $y$-axis at $\left( 0,k \right)$. The $x$-axis is the horizontal asymptote. For $k>0$, the graph increases rapidly as $x$ increases. For $k<0$, the graph decreases rapidly as $x$increases. $0<a<1$ It is an exponential graph. $x$ is the power in the function, $y=k{{a}^{x}}$. The curve cuts the $y$-axis at $\left( 0,k \right)$. The $x$-axis is the horizontal asymptote. For $k>0$, the graph decreases gently as $x$ increases. For $k<0$, the graph increases gently as $x$increases. Relationship among Graphs of Exponential Functions with Different Bases For exponential functions in the form of $y={{a}^{x}}$: For fixed positive $x$, increasing the base $a$ increases $a^x$. For fixed negative $x$, increasing the base $a$ decreases $a^x$. They all pass through the point $\left( 0,1 \right)$. They all have a horizontal asymptote as the $x$-axis.

Exponential Functions and Graphs

IP 3

An exponential function has the form $y=ka^x$, where $a>0$, $a\ne1$ and $k\ne0$. For $a=e$ and $k=1$, $y=e^x$ is the natural exponential function.

The input $x$ can be any real number. The output has the sign of $k$: it is positive when $k>0$ and negative when $k<0$.

$y=k{{a}^{x}}$	$k=1$	$k=-1$	Notes
$a>1$			It is an exponential graph. $x$ is the power in the function, $y=k{{a}^{x}}$. The curve cuts the $y$-axis at $\left( 0,k \right)$. The $x$-axis is the horizontal asymptote. For $k>0$, the graph increases rapidly as $x$ increases. For $k<0$, the graph decreases rapidly as $x$increases.
$0<a<1$			It is an exponential graph. $x$ is the power in the function, $y=k{{a}^{x}}$. The curve cuts the $y$-axis at $\left( 0,k \right)$. The $x$-axis is the horizontal asymptote. For $k>0$, the graph decreases gently as $x$ increases. For $k<0$, the graph increases gently as $x$increases.

Relationship among Graphs of Exponential Functions with Different Bases
	For exponential functions in the form of \(y={{a}^{x}}\): For fixed positive \(x\), increasing the base \(a\) increases \(a^x\). For fixed negative \(x\), increasing the base \(a\) decreases \(a^x\). They all pass through the point \(\left( 0,1 \right)\). They all have a horizontal asymptote as the \(x\)-axis.

Relationship among Graphs of Exponential Functions with Different Bases

For exponential functions in the form of $y={{a}^{x}}$:

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\(y=k{{a}^{x}}\)	\(k=1\)	\(k=-1\)	Notes
\(a>1\)			It is an exponential graph. \(x\) is the power in the function, \(y=k{{a}^{x}}\). The curve cuts the \(y\)-axis at \(\left( 0,k \right)\). The \(x\)-axis is the horizontal asymptote. For \(k>0\), the graph increases rapidly as \(x\) increases. For \(k<0\), the graph decreases rapidly as \(x\)increases.
\(0<a<1\)			It is an exponential graph. \(x\) is the power in the function, \(y=k{{a}^{x}}\). The curve cuts the \(y\)-axis at \(\left( 0,k \right)\). The \(x\)-axis is the horizontal asymptote. For \(k>0\), the graph decreases gently as \(x\) increases. For \(k<0\), the graph increases gently as \(x\)increases.