Graphing $y=a\\log_{b}(\\pm x+d)+c$
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "The following questions will gauge your understanding of logarithms and how to graph them.
\nThe logarithm you will be working with for this question is \\[y=\\simplify{{a}log({s}x+{d},{b})+{c}}.\\]
", "advice": "This question assumes you understand the definition and the laws of logarithms.
\na) To find the $x$-intercept, substitute $y=0$ into the equation and solve for $x$:
\n$\\begin{align*}&&0&=\\simplify{{a}log({s}x+{d},{b})+{c}}\\\\\\implies&& \\var{-c}&=\\simplify{{a}log({s}x+{d},{b})}\\\\\\implies&&\\simplify[fractionNumbers,simplifyFractions,basic]{-{c/a}}&=\\simplify{log({s}x+{d},{b})}\\\\\\implies&&\\simplify[fractionNumbers,simplifyFractions,basic]{{b}^(-{c/a})}&=\\simplify{{s}x+{d}}\\\\\\implies&&\\simplify[fractionNumbers,simplifyFractions,basic,unitDenominator,unitFactor]{({b}^(-{c/a})-{d})/{s}}&=x\\end{align*}$
\nTherefore, the $x$-intercept is the point $\\left(\\simplify[fractionNumbers,simplifyFractions,basic,unitDenominator,unitFactor]{({b}^(-{c/a})-{d})/{s}},0\\right)$.
\nb) Substitute $x=\\simplify[fractionNumbers]{{(b-d)/s}}$ into the equation: $y=\\simplify[!collectNumbers,basic,fractionNumbers]{{a}log(({(b-d)})+{d},{b})+{c}}=\\simplify[!collectNumbers,basic]{{a}log({b},{b})+{c}}=\\var{a+c}$. Therefore, another easily found point is $\\left(\\simplify[fractionNumbers]{{(b-d)/s}},\\var{a+c}\\right)$.
\n\nc) As $x$ gets larger and larger (increases without bound, or approaches infinity) $y=\\simplify{{a}log({s}x+{d},{b})+{c}}$ gets smaller and smaller (decreases without bound, or approaches negative infinity). larger and larger (increases without bound, or approaches infinity).
\n\nc) As $x$ gets smaller and smaller (decreases without bound, or approaches negative infinity) $y=\\simplify{{a}log({s}x+{d},{b})+{c}}$ gets smaller and smaller (decreases without bound, or approaches negative infinity). larger and larger (increases without bound, or approaches infinity).
\n\n\nd) An asymptote is a line or curve that approaches a given curve arbitrarily closely. For the curve $y=\\simplify{{a}log({s}x+{d},{b})+{c}}$ the closer $\\simplify{{s}x+{d}}$ gets to zero (approaching from the right), the smaller larger $y$ gets (without bound). In other words, as $x$ approaches $\\simplify[fractionNumbers]{{-d/s}}$ from the right left, $y$ approaches negative infinity. This means that the asymptote for $y=\\simplify{{a}log({s}x+{d},{b})+{c}}$ is the line $x=\\simplify[fractionNumbers]{{-d/s}}$.
\ne) Given all the information above, it should be clear that the graph should look like
\n{graph1(1)}
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