Graphing $y=ab^x+c$
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "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=\\simplify{{a}{b}^x+{c}}.\\]
\n\n", "advice": "a) To find the $y$-intercept, substitute $x=0$ into the equation: $y=\\simplify[!collectNumbers,basic]{{a}{b}^0+{c}}=\\simplify[!collectNumbers, basic]{{a}+{c}}=\\var{a+c}$. Therefore, the $y$-intercept is the point $(0,\\var{a+c})$.
\nb) Substitute $x=1$ into the equation: $y=\\simplify[!collectNumbers, basic]{{a}{b}^1+{c}}=\\var{a*b+c}$. Therefore, another easily found point is $(1,\\var{a*b+c})$.
\nc) As $x$ gets larger and larger (increases without bound, or approaches infinity) $y=\\simplify{{a}{b}^x+{c}}$ gets smaller and smaller (decreases without bound, or approaches negative infinity). larger and larger (increases without bound, or approaches infinity).
\nd) An asymptote is a line or curve that approaches a given curve arbitrarily closely. Because as $x$ gets very small, $\\simplify{{b}^x}$ gets close to zero, $\\simplify{{a}{b}^x}$ gets close to zero, and $\\simplify{{a}{b}^x+{c}}$ gets close to $\\var{c}$. That is, for the curve $\\simplify{{a}{b}^x+{c}}$ the smaller $x$ gets, the closer $y$ gets to $\\var{c}$. In other words as $x$ approaches negative infinity, $y$ approaches $\\var{c}$. So the asymptote for $y=\\simplify{{a}{b}^x+{c}}$ is the line $y=\\var{c}$.
\ne) Given all the information above, it should be clear that the graph should look like
\n{graph1(1)}
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", "stepsPenalty": "2", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "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 $y=\\simplify[!noleadingminus]{{a}{b}^x+{c}}$.
\nSubstituting $x=0$ gives
\\begin{align}y&=\\simplify[!collectnumbers,!noleadingminus]{{a}{b}^0+{c}}\\\\&=\\simplify[!collectnumbers,!noleadingminus]{{a}+{c}}\\\\&=\\simplify{{a+c}}\\end{align} and therefore the $y$-intercept is the point with coordinates $(0,\\var{a+c})$.
Another easily found point on the curve is ${\\large(}1,$ [[0]]$\\large)$.
", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Substituting $x=1$ into $y=\\simplify[!noleadingminus]{{a}{b}^x+{c}}$:
\n\\begin{align}y&=\\simplify[basic]{{a}{b}^1+{c}}\\\\&=\\simplify[!collectnumbers,!noleadingminus]{{a}*{b}+{c}}\\\\&=\\simplify[!collectnumbers,!noleadingminus]{{a*b}+{c}}\\\\&=\\var{a*b+c}\\end{align}
\nTherefore, $(1,\\var{a*b+c})$ is another point on the curve.
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", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Since $\\var{b}$ is greater than $1$, as $x$ increases the value of $\\var{b}^x$ grows exponentially. Furthermore, since $\\var{a}$ is positive, $y=\\simplify[!noleadingminus]{{a}{b}^x+{c}}$ increases without bound. Furthermore, since $\\var{a}$ is negative, $y=\\simplify[!noleadingminus]{{a}{b}^x+{c}}$ decreases without bound.
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