// Numbas version: finer_feedback_settings {"name": "Graphing exponentials of the form y=b^x with 0Graphing exponentials with a base between 0 and 1 and no transformations take place.

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

The exponential you will be working with for this question is \\[y=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x.\\]

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Note: if we so desire we can rewrite this exponential with a negative index since $\\left(\\simplify[fractionNumbers]{{b}}\\right)^x=\\left(\\simplify[fractionNumbers]{{1/b}}\\right)^{-x}$.

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 $(0,1)$.

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

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

\\[\\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 $\\simplify[fractionNumbers]{{b}}<1$).

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.

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).

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

{graph1(1)}

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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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The $y$-intercept is the point at which the graph intercepts the $y$-axis. The $y$-axis has equation $x=0$ so the $y$-intercept will always have an $x$ value of $0$, and the $y$ value will be the result of substituting $x=0$ into the equation $y=\\left(\\var[fractionnumbers]{b}\\right)^x$.

Substituting $x=0$ gives $y=\\left(\\var[fractionnumbers]{b}\\right)^0=1$ and therefore the $y$-intercept is the point with coordinates $(0,1)$.

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

Substituting $x=1$ into $y=\\left(\\var[fractionnumbers]{b}\\right)^x$ gives $y=\\var[fractionnumbers]{b}$. Therefore, $\\left(1,\\var[fractionnumbers]{b}\\right)$ is another point on the curve.

Given $y=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x$, everytime $x$ increases by 1, $y$ [[0]].

When we add $1$ to $x$, the power increases by $1$ and therefore the value of $y$ is multiplied by $\\var[fractionnumbers]{b}$ since $\\left(\\var[fractionnumbers]{b}\\right)^{x+1}=\\left(\\var[fractionnumbers]{b}\\right)^x\\times \\left(\\var[fractionnumbers]{b}\\right)$.

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

We saw in the last part of the question that as we add $1$ to $x$, the power increases by $1$ and therefore the value of $y$ is multiplied by $\\var[fractionnumbers]{b}$. Since $\\var[fractionnumbers]{b}$ is between $0$ and $1$, this multiplication makes the $y$ value smaller and we have exponential decay.

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exponential growth

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exponential decay

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The horizontal asymptote of $y=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x$ is $y=$ [[0]].

The horizontal asymptote is $y=0$ (the $x$-axis).

Based on the above, which graph best represents $y=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x$?

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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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