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This is the second of our many quizzes on Reflections. Click on this to access more such competitive quizzes on Geometry.

Reflections Quiz 2

mathematics

 Find the reflection of the function $f\left( y \right)=8{{y}^{2}}-2y$  horizontally?

 Multiply all inputs by $–1$ for a horizontal reflection. Apply it for a given function to get the answer. 

For finding the reflection for any function horizontally multiply inputs with $-1$. $  \\ f\left( -y \right)=8{{\left( -y \right)}^{2}}-2\left( -y \right)=g\left( y \right) \\ \Rightarrow g\left( y \right)=8{{y}^{2}}+2y \\ $ 

 When reflected in the x-axis, the point $Q$ with coordinates $\left( 2,3 \right)$  is mapped onto $Q^{\prime}$, and then $Q^{\prime}$ is mapped onto $Q^{\prime\prime}$ when reflected in the y-axis. Find the coordinates of $Q^{\prime\prime}$. 

 Multiply all outputs with $-1$ to get the horizontal reflection and also multiply all inputs with $-1$ to get the vertical reflection.

Here $Q$ is mapped into $Q^{\prime}$ i.e. reflected by x-axis. $ \\ $ Therefore, $Q^{\prime}$ will become $\left( 2,-3 \right)$. $ \\ $ After that $Q^{\prime}$ is mapped into $Q^{\prime\prime}$ which is reflected by the y axis. $ \\ $ Therefore the point $Q^{\prime\prime}$ will become $\left( -2,-3 \right)$.

 Arkesh found a statement in one of his books as $ \\ $ Statement: $f\left( x \right)=2{{x}^{2}}+{{x}^{6}}$ is an odd function. $ \\ $ He wants to know whether the statement is correct or wrong. Find whether the statement is correct or wrong.

 Check the given function by substituting $x=-x$ and then check $f\left( -x \right)\ne f\left( x \right)$ and then find the statement whether it is true or not.

$\text{If }f\left( x \right)\ne f\left( -x \right)\text{ it is as odd function}\text{.} \\ \text{So,} \\ \Rightarrow f\left( -x \right)=2{{\left( -x \right)}^{2}}+{{\left( -x \right)}^{6}}=2{{x}^{2}}+{{x}^{6}}=f\left( x \right) \\ \Rightarrow f\left( -x \right)=f\left( x \right) \\ \text{So, given statement is false}\text{.} $

 If point $P$ is the addition of two points $Q$ and $R$. If $Q=\left( 2,3 \right)$<sub> </sub>and $\text{{R}'}=\left( 3,-9 \right)$<sub> </sub>be the reflection of point $R$ over the X-axis. Then find point P.

 Find the point $R$ using the given reflection point ${R}^\prime$, here x remains constant and y is flipped over the x axis to be negative. Then find the required point which is the sum of $Q$ and $R$.

Given $P$ is addition of two points $Q$ and $R$ $ \\ $ $Q=\left( 2,3 \right)$ and ${R^\prime}\left( 3,-9 \right)$ $ \\ $ Which is reflection of $R$ $ \\ $ Therefore $R=\left( 3,9 \right)$ $ \\ $  Now from the question $P=Q+R$. $ \\ \Rightarrow P=\left( 2+3,3+9 \right)=\left( 5,12 \right) $

 Identify the vertical reflection for the function $f\left( x \right)=4x+9$

 The vertical reflection of the given function is obtained by multiplying all outputs of a given function by $'-1'$ then find the graph that represents the required function.

$f\left( x \right)=4x+9$ $ \\ $ For finding the reflection for the function vertically multiply the all outputs by $-1$ $ \\ \Rightarrow f\left( -x \right)=-4x+9=g\left( x \right) \\ $ Now plot the graph for the function. $ \\ $ Find the $x$ intercept by making $g\left( x \right)=0$. $ \\ \Rightarrow -4x+9=0 \\ \Rightarrow x=2.22.... \\ $ Find the $y$ intercept by making $g\left( 0 \right)$. $ \\ \Rightarrow g\left( x \right)=9 \\ $ Now it is clear that if $x$ intercept is in between ${{\text{1}}^{st}}$ and ${{\text{2}}^{nd}}$ quadrants and $y$ intercept is in between ${{\text{1}}^{st}}$ and ${{\text{2}}^{nd}}$ quadrant $ \\ $ Which will be true only for option $\left( D \right)$.