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The following questions will gauge your understanding of exponentials and how to graph them.

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The exponential you will be working with for this question is \\[y=\\var{b}^x.\\]

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The easiest type of exponential to graph where the base is greater than 1 and no transformations take place.

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The \$y\$-intercept of \$y=\\var{b}^x\$ is the point \$\\large(\$[[0]], [[1]]\$\\large)\$.

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Another easily found point on the curve is \${\\large(}1,\$ [[0]]\$\\large)\$.

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Given \$y=\\var{b}^x\$, everytime \$x\$ increases by 1, \$y\$  [[0]].

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increases by 1.

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decreases by 1.

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increases by {b-1}.

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decreases by {b-1}.

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increases by a factor of {b}.

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decreases by a factor of {b}.

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exponential growth

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exponential decay

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Would \$y=\\var{b}^x\$ best be described as exponential decay or exponential growth?

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The horizontal asymptote of \$y=\\var{b}^x\$ is \$y=\$ [[0]].

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{graph1(1)}

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{graph1(2)}

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{graph1(3)}

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{graph1(4)}

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Which graph best represents \$y=\\var{b}^x\$?

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a) To find the \$y\$-intercept, substitute \$x=0\$ into the equation: \$y=\\var{b}^0=1\$. Therefore, the \$y\$-intercept is the point \$(0,1)\$.

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b) Substitute \$x=1\$ into the equation: \$y=\\var{b}^1=\\var{b}\$. Therefore, another easily found point is \$(1,\\var{b})\$.

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c) Let's investigate what happens to the value of \$y\$ when we add 1 to the value of \$x\$:

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\\[\\var{b}^{x+1}=\\var{b}^x\\var{b}^1=\\var{b}^x\\var{b}\\]  That is, the old \$y\$ value is multiplied by \$\\var{b}\$, so we can say that \$y\$ is increased by a factor of \$\\var{b}\$.

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d) Since \$y=\\var{b}^x\$ is an exponential and as \$x\$ increases \$y\$ increases without bound, we call this exponential growth.

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e) An asymptote is a line or curve that approaches a given curve arbitrarily closely. For the curve \$y=\\var{b}^x\$ the smaller \$x\$ gets, the closer \$y\$ gets to \$0\$. In other words as \$x\$ approaches negative infinity, \$y\$ approaches \$0\$. This means that the asymptote for \$y=\\var{b}^x\$ is the line \$y=0\$ (the \$x\$-axis).

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f) Given all the information above, it should be clear that the graph should look like

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{graph1(1)}

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