Question

Let $ f\left( x \right)=-\dfrac{{{x}^{2}}}{2} $ . If the graph of $ f\left( x \right) $ is translated 2 units up and 3 units left and the resulting graph is that of $ g\left( x \right) $ , then $ g\left( \dfrac{1}{2} \right) $ is equal to
A. 0
B. $ -\dfrac{1}{8} $
C. $ -\dfrac{2}{8} $
D. $ -\dfrac{33}{8} $
E. $ \dfrac{13}{8} $

Answer 1

Hint: We first express the shift of the graph based on the base change of the coordinates of x and y. We find the changes for particular signs for the change. Then we put the values for the given function $ f\left( x \right)=-\dfrac{{{x}^{2}}}{2} $ . We get $ g\left( x \right) $ . We put the value for $ x=\dfrac{1}{2} $ in $ g\left( x \right) $ .

Complete step by step solution:
The given graph is $ f\left( x \right)=-\dfrac{{{x}^{2}}}{2} $ . If any change in the graph happens then that is because of the base change of the value of x and y.
If we take a general graph of $ y=f\left( x \right) $ , then any change in the vertical direction (upward or downward) is because of the change in value of y.
If the function changes from $ y=f\left( x \right) $ to $ y=f\left( x \right)+a $ , then the curve changes $ a $ units. If the sign of a is positive then the curve goes upward and if the sign is negative then it goes downward.
If we take a general graph of $ y=f\left( x \right) $ , then any change in the horizontal direction (leftward or rightward) is because of the change in value of x.
If the function changes from $ y=f\left( x \right) $ to $ y=f\left( x+a \right) $ , then the curve changes $ a $ units. If the sign of a is positive then the curve goes leftward and if the sign is negative then it goes rightward.
Therefore, for $ f\left( x \right)=-\dfrac{{{x}^{2}}}{2} $ the graph is translated 2 units up and 3 units left and we get $ g\left( x \right) $ .
After the changes we get $ g\left( x \right)=-\dfrac{{{\left( x+3 \right)}^{2}}}{2}+2 $ .
We need to find $ g\left( \dfrac{1}{2} \right) $ . We put value of $ x=\dfrac{1}{2} $ in $ g\left( x \right)=-\dfrac{{{\left( x+3 \right)}^{2}}}{2}+2 $ .
So, $ g\left( \dfrac{1}{2} \right)=-\dfrac{{{\left( \dfrac{1}{2}+3 \right)}^{2}}}{2}+2=2-\dfrac{49}{8}=-\dfrac{33}{8} $ .
The correct option is D.
So, the correct answer is “Option D”.

Note: There are four main types of transformations: translation, rotation, reflection and dilation. The transformation moves a figure in some way on the coordinate plane. A translation is a transformation that slides a figure on the coordinate plane without changing its shape, size, or orientation. In a rotation, the centre of rotation is the point that does not move.