// Numbas version: exam_results_page_options {"name": "Graphing exponentials of the form y=b^x with b>1", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"b": {"group": "Ungrouped variables", "name": "b", "description": "", "templateType": "anything", "definition": "random(2..10)"}}, "ungrouped_variables": ["b"], "parts": [{"type": "gapfill", "variableReplacementStrategy": "originalfirst", "marks": 0, "gaps": [{"mustBeReduced": false, "type": "numberentry", "mustBeReducedPC": 0, "marks": 1, "variableReplacementStrategy": "originalfirst", "minValue": "0", "showFeedbackIcon": true, "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "allowFractions": false, "maxValue": "0", "scripts": {}, "variableReplacements": [], "showCorrectAnswer": true}, {"mustBeReduced": false, "type": "numberentry", "mustBeReducedPC": 0, "marks": 1, "variableReplacementStrategy": "originalfirst", "minValue": "1", "showFeedbackIcon": true, "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "allowFractions": false, "maxValue": "1", "scripts": {}, "variableReplacements": [], "showCorrectAnswer": true}], "prompt": "

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

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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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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.\\]

\n

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

\n

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 $\\var{b}$.

\n

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

\n

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 

\n

{graph1(1)}

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