Algebra

Function Graphs

Graphical Transformation - Problem Solving

         

In the above diagram, suppose the equation of curve A is y=ax2.y=ax^2. Which of the following is a possible equation for curve B? (i) y=5ax2(ii) y=2ax2(iii) y=6ax2(iv) y=12ax2\begin{aligned} \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{aligned}

In the above diagram, A is a circle with radius 22 centered at the origin and B is an ellipse with major axis 44 and minor axis 22 centered at (7,5).(7, 5). If B is stretched and translated to obtain A, which of the following processes should be applied?

(i) Translate B by 7-7 and 5-5 in the positive directions of the xx-axis and yy-axis, respectively, and stretch by a factor of 22 with respect to the yy-axis.

(ii) Translate B by 7-7 and 5-5 in the positive directions of the xx-axis and yy-axis, respectively, and stretch by a factor of 22 with respect to the xx-axis.

(iii) Translate B by 77 and 55 in the positive directions of the xx-axis and yy-axis, respectively, and stretch by a factor of 12\frac{1}{2} with respect to the yy-axis.

(iv) Translate B by 7-7 and 5-5 in the positive directions of the xx-axis and yy-axis, respectively, and stretch by a factor of 12\frac{1}{2} with respect to the xx-axis.

Every point (x,y)(x,y) on the curve y=log23xy = \log_{2}{3x} is shifted to a new point by the translation (x,y)=(x+m,y+n),(x',y') =(x+m,y+n), where mm and nn are integers. The set of (x,y)(x',y') forms the curve y=log2(12x132)y = \log_{2}{(12 x-132)} . What is the value of m+nm + n ?

Given the graph y=lnx y = \ln x , which of these statements describes the transformations to get the graph of y=ln(4x2+4x+1) y = \ln (4x^2 + 4x + 1) for x>12 x > - \frac{1}{2} ?

(1)\quad \text{(1)} Translate to the left by 1 and up by ln4 \ln 4 , then scale vertically by 2.

(2)\quad \text{(2)} Translate to the left by 12\frac{1}{2} and up by ln2 \ln 2 , then scale vertically by 2.

(3)\quad \text{(3)} Translate to the left by 1 and up by ln2 \ln 2 , then scale vertically by 2.

(4)\quad \text{(4)} Translate to the left by 12 \frac{1}{2} and up by ln4 \ln 4 , then scale vertically by 2.

Given the graph y=7x2+7 y = 7x^2+7 , what is the sequence of operations required to obtain the graph of y=175x2+3? y = 175x^2 +3?

Note: The above graph is not drawn to scale.

(i) Stretch the given graph by a factor of 55 with respect to the xx-axis and translate by 44 in the positive direction of the xx-axis.

(ii) Stretch the given graph by a factor of 55 with respect to the yy-axis and translate by 4-4 in the positive direction of the yy-axis.

(iii) Stretch the given graph by a factor of 15\frac{1}{5} with respect to the xx-axis and translate by 44 in the positive direction of the yy-axis.

(iv) Stretch the given graph by a factor of 15\frac{1}{5} with respect to the xx-axis and translate by 4-4 in the positive direction of the yy-axis.

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