The easiest type of exponential to graph where the base is greater than 1 and no transformations take place.
"}, "tags": [], "ungrouped_variables": ["b"], "variables": {"b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "", "name": "b"}}, "statement": "The following questions will gauge your understanding of exponentials and how to graph them.
\nThe exponential you will be working with for this question is \\[y=\\var{b}^x.\\]
\n", "extensions": ["jsxgraph"], "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "a) To find the $y$-intercept, substitute $x=0$ into the equation: $y=\\var{b}^0=1$. Therefore, the $y$-intercept is the point $(0,1)$.
\nb) Substitute $x=1$ into the equation: $y=\\var{b}^1=\\var{b}$. Therefore, another easily found point is $(1,\\var{b})$.
\nc) Let's investigate what happens to the value of $y$ when we add 1 to the value of $x$:
\n\\[\\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 $\\var{b}$.
\nd) Since $y=\\var{b}^x$ is an exponential and as $x$ increases $y$ increases without bound, we call this exponential growth.
\ne) 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).
\nf) Given all the information above, it should be clear that the graph should look like
\n{graph1(1)}
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", "variableReplacementStrategy": "originalfirst", "marks": 0, "type": "gapfill", "scripts": {}, "variableReplacements": []}, {"gaps": [{"type": "numberentry", "showFeedbackIcon": true, "minValue": "{b}", "marks": 1, "showCorrectAnswer": true, "allowFractions": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "mustBeReduced": false, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "maxValue": "{b}", "variableReplacements": []}], "showCorrectAnswer": true, "showFeedbackIcon": true, "prompt": "Another easily found point on the curve is ${\\large(}1,$ [[0]]$\\large)$.
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", "decreases by 1.
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", "variableReplacementStrategy": "originalfirst", "marks": 0, "type": "gapfill", "scripts": {}, "variableReplacements": []}, {"type": "1_n_2", "showFeedbackIcon": true, "matrix": ["1", 0], "displayColumns": 0, "marks": 0, "showCorrectAnswer": true, "choices": ["exponential growth
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