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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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