// Numbas version: exam_results_page_options {"name": "Graphing exponentials of the form y=b^x with 0Note: if we so desire we can rewrite this exponential with a negative index since $\\left(\\simplify[fractionNumbers]{{b}}\\right)^x=\\left(\\simplify[fractionNumbers]{{1/b}}\\right)^{-x}$.

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

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b) Substitute $x=1$ into the equation: $y=\\left(\\simplify[fractionNumbers]{{b}}\\right)^1=\\simplify[fractionNumbers]{{b}}$. Therefore, another easily found point is $\\left(0,\\simplify[fractionNumbers]{{b}}\\right)$.

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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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\$\\left(\\simplify[fractionNumbers]{{b}}\\right)^{x+1}=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x\\left(\\simplify[fractionNumbers]{{b}}\\right)^1=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x\\left(\\simplify[fractionNumbers]{{b}}\\right)\$  That is, the old $y$ value is multiplied by $\\simplify[fractionNumbers]{{b}}$ (which actually decreases the $y$ value since $\\simplify[fractionNumbers]{{b}}<1$).

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d) Since $y=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x$ is an exponential and as $x$ increases $y$ decreases to 0, we call this exponential decay.

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e) An asymptote is a line or curve that approaches a given curve arbitrarily closely. For the curve $y=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x$ the larger $x$ gets, the closer $y$ gets to $0$. In other words as $x$ approaches infinity, $y$ approaches $0$. This means that the asymptote for $y=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x$ 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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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=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x.\$

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The $y$-intercept of $y=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x$ 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 $y=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x$, 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 $\\simplify[fractionNumbers]{{b}}$.

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decreases by $\\simplify[fractionNumbers]{{b}}$.

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is multiplied by $\\simplify[fractionNumbers]{{b}}$.

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is divided by $\\simplify[fractionNumbers]{{b}}$.

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exponential growth

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exponential decay

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Would $y=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x$ best be described as exponential decay or exponential growth?

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The horizontal asymptote of $y=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x$ 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 $y=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x$?

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Graphing exponentials with a base between 0 and 1 and no transformations take place.

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