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Function Graphs
Graphs are visual representations of functions. If you know how to read graphs, you can say a lot about a function just by looking at its graph. Learn this fine art of mathematical divining.
(i) Translate B by \(-7\) and \(-5\) in the positive directions of the \(x\)-axis and \(y\)-axis, respectively, and stretch by a factor of \(2\) with respect to the \(y\)-axis.
(ii) Translate B by \(-7\) and \(-5\) in the positive directions of the \(x\)-axis and \(y\)-axis, respectively, and stretch by a factor of \(2\) with respect to the \(x\)-axis.
(iii) Translate B by \(7\) and \(5\) in the positive directions of the \(x\)-axis and \(y\)-axis, respectively, and stretch by a factor of \(\frac{1}{2}\) with respect to the \(y\)-axis.
(iv) Translate B by \(-7\) and \(-5\) in the positive directions of the \(x\)-axis and \(y\)-axis, respectively, and stretch by a factor of \(\frac{1}{2}\) with respect to the \(x\)-axis.
Every point \((x,y)\) on the curve \(y = \log_{2}{3x} \) is shifted to a new point by the translation \[(x',y') =(x+m,y+n),\] where \(m\) and \(n\) are integers. The set of \((x',y')\) forms the curve \(y = \log_{2}{(12 x-132)} \). What is the value of \(m + n \)?
\(\quad \text{(1)}\) Translate to the left by 1 and up by \( \ln 4 \), then scale vertically by 2.
\(\quad \text{(2)}\) Translate to the left by \(\frac{1}{2}\) and up by \( \ln 2 \), then scale vertically by 2.
\(\quad \text{(3)}\) Translate to the left by 1 and up by \( \ln 2 \), then scale vertically by 2.
\(\quad \text{(4)}\) Translate to the left by \( \frac{1}{2} \) and up by \( \ln 4 \), then scale vertically by 2.
Note: The above graph is not drawn to scale.
(i) Stretch the given graph by a factor of \(5\) with respect to the \(x\)-axis and translate by \(4\) in the positive direction of the \(x\)-axis.
(ii) Stretch the given graph by a factor of \(5\) with respect to the \(y\)-axis and translate by \(-4\) in the positive direction of the \(y\)-axis.
(iii) Stretch the given graph by a factor of \(\frac{1}{5}\) with respect to the \(x\)-axis and translate by \(4\) in the positive direction of the \(y\)-axis.
(iv) Stretch the given graph by a factor of \(\frac{1}{5}\) with respect to the \(x\)-axis and translate by \(-4\) in the positive direction of the \(y\)-axis.