For exponential and logarithmic graph families, use $a>0$, $a\ne1$, and a nonzero scale $k$. Logarithmic inputs must be positive. Graphs of Exponential Functions in the form of $y = ka^{x}$ $y = ka^{x}$ $k > 0$ $k < 0$ Notes $a > 1$ It is an exponential graph. $x$ is the power in the function, $y = ka^{x}$. The curve cuts the $y$-axis at $(0, k)$. 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, $y = ka^{x}$. The curve cuts the $y$-axis at $(0, k)$. 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 $x>0$, increasing $a$ increases $a^x$. For fixed $x<0$, increasing $a$ decreases $a^x$. They all pass through the point $(0,1)$. They all have a horizontal asymptote as the $x$-axis. Graphs of Logarithmic Functions in the form of $y = k \log_{a} x$ $y = k \log_{a} x$ $k > 0$ $k < 0$ Notes $a > 1$ It is a logarithmic graph. The curve cuts the $x$-axis at $(1, 0)$. The $y$-axis is the vertical asymptote. For $k > 0$, the graph increases gently as $x$ increases. For $k < 0$, the graph decreases gently as $x$ increases. $0 < a < 1$ It is a logarithmic graph. The curve cuts the $x$-axis at $(1, 0)$. The $y$-axis is the vertical 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 Logarithmic Functions with Different Bases For logarithmic functions in the form of $y = \log_{a} x$ : For fixed $x>1$, increasing $a$ decreases $\log_a x$, within either base interval $0<a<1$ or $a>1$. For fixed $0<x<1$, increasing $a$ increases $\log_a x$, within either base interval. They all pass through the point $(1, 0)$ . They all have a vertical asymptote as the $y$ -axis. Relationship between Graphs of Exponential and Logarithmic Functions with the Same Base For exponential and logarithmic functions in the forms of $y = a^x$ and $y = \log_a x$ : Their graphs are symmetrical about the straight line $y = x$. The functions $y = a^x$ and $y = \log_a x$ are the inverse functions of each other.

Graphs of Exponential and Logarithmic Functions

IP 3

For exponential and logarithmic graph families, use $a>0$, $a\ne1$, and a nonzero scale $k$. Logarithmic inputs must be positive.

Graphs of Exponential Functions in the form of $y = ka^{x}$
$y = ka^{x}$	$k > 0$	$k < 0$	Notes
$a > 1$			It is an exponential graph. $x$ is the power in the function, $y = ka^{x}$. The curve cuts the $y$-axis at $(0, k)$. 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, $y = ka^{x}$. The curve cuts the $y$-axis at $(0, k)$. 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 \(x>0\), increasing \(a\) increases \(a^x\). For fixed \(x<0\), increasing \(a\) decreases \(a^x\). They all pass through the point \((0,1)\). 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 $y = a^x$:

Graphs of Logarithmic Functions in the form of $y = k \log_{a} x$
$y = k \log_{a} x$	$k > 0$	$k < 0$	Notes
$a > 1$			It is a logarithmic graph. The curve cuts the $x$-axis at $(1, 0)$. The $y$-axis is the vertical asymptote. For $k > 0$, the graph increases gently as $x$ increases. For $k < 0$, the graph decreases gently as $x$ increases.
$0 < a < 1$			It is a logarithmic graph. The curve cuts the $x$-axis at $(1, 0)$. The $y$-axis is the vertical 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 Logarithmic Functions with Different Bases
	For logarithmic functions in the form of \(y = \log_{a} x\) : For fixed \(x>1\), increasing \(a\) decreases \(\log_a x\), within either base interval \(0<a<1\) or \(a>1\). For fixed \(0<x<1\), increasing \(a\) increases \(\log_a x\), within either base interval. They all pass through the point \((1, 0)\) . They all have a vertical asymptote as the \(y\) -axis.

Relationship among Graphs of Logarithmic Functions with Different Bases

For logarithmic functions in the form of $y = \log_{a} x$ :

For fixed $x>1$, increasing $a$ decreases $\log_a x$, within either base interval $0<a<1$ or $a>1$.
For fixed $0<x<1$, increasing $a$ increases $\log_a x$, within either base interval.
They all pass through the point $(1, 0)$ .
They all have a vertical asymptote as the $y$ -axis.

Relationship between Graphs of Exponential and Logarithmic Functions with the Same Base
	For exponential and logarithmic functions in the forms of \(y = a^x\) and \(y = \log_a x\) : Their graphs are symmetrical about the straight line \(y = x\). The functions \(y = a^x\) and \(y = \log_a x\) are the inverse functions of each other.

Relationship between Graphs of Exponential and Logarithmic Functions with the Same Base

For exponential and logarithmic functions in the forms of $y = a^x$ and $y = \log_a x$ :

Their graphs are symmetrical about the straight line $y = x$.
The functions $y = a^x$ and $y = \log_a x$ are the inverse functions of each other.

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Graphs of Exponential Functions in the form of \(y = ka^{x}\)
\(y = ka^{x}\)	\(k > 0\)	\(k < 0\)	Notes
\(a > 1\)			It is an exponential graph. \(x\) is the power in the function, \(y = ka^{x}\). The curve cuts the \(y\)-axis at \((0, k)\). 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, \(y = ka^{x}\). The curve cuts the \(y\)-axis at \((0, k)\). 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.

Graphs of Logarithmic Functions in the form of \(y = k \log_{a} x\)
\(y = k \log_{a} x\)	\(k > 0\)	\(k < 0\)	Notes
\(a > 1\)			It is a logarithmic graph. The curve cuts the \(x\)-axis at \((1, 0)\). The \(y\)-axis is the vertical asymptote. For \(k > 0\), the graph increases gently as \(x\) increases. For \(k < 0\), the graph decreases gently as \(x\) increases.
\(0 < a < 1\)			It is a logarithmic graph. The curve cuts the \(x\)-axis at \((1, 0)\). The \(y\)-axis is the vertical asymptote. For \(k > 0\), the graph decreases gently as \(x\) increases. For \(k < 0\), the graph increases gently as \(x\) increases.