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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.

Graph Transformation

         

In the above diagram, suppose the equation of curve A is \(y=ax^2.\) Which of the following is a possible equation for curve B? \[\begin{align} \text{(i) } y&=5ax^2 &\qquad \text{(ii) } y&=-2ax^2 \\ \text{(iii) } y&=6ax^2 &\qquad \text{(iv) } y&=-\frac{1}{2}ax^2 \end{align}\]

In the above diagram, A is a circle with radius \(2\) centered at the origin and B is an ellipse with major axis \(4\) and minor axis \(2\) centered at \((7, 5).\) If B is stretched and translated to obtain A, which of the following processes should be applied?

(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 \(m + n \)?

Given the graph \( y = \ln x \), which of these statements describes the transformations to get the graph of \( y = \ln (4x^2 + 4x + 1) \) for \( x > - \frac{1}{2} \)?

\(\quad \text{(1)}\) 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.

Given the graph \( y = 7x^2+7 \), what is the sequence of operations required to obtain the graph of \[ y = 175x^2 +3?\]

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.

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