# Chapter P - P.1 - Graphs and Models - Exercises - Page 8: 52

The intercept with $y$ axis is $y=6$. The intercept with $x$ axis is $6$. The graph is symmetric with respect to vertical line $x=6$ and is shown on the figure.

#### Work Step by Step

To find the intercept with $y$ axis put $x=0$ and calculate $y$: $$y=|6-0|=|6| = 6.$$ To find the intercept with $x$ axis put $y=0$ and calculate $x$: $$0=|6-x|.$$ The absolute value of something is zero only if that something is equal to zero so we have $$6-x=0$$ or $$x=6.$$ To test the symmetry we will put $-x$ instead of $x$: $$y(-x) = |6-(-x)|= |6+x|.$$ We see that $y(-x)$ is neither equal to $y(x)$ nor $-y(x)$ so this graph is not symmetric. However it is symmetric with respect to the vertical line passing through $x=6$ because when you go, lets say $1$ unit left of $6$ you get $$y=|6-5|=|1|=1$$ and if you go $1$ unit to the right of $6$ you again obtain $$y=|6-7| = |-1|= 1.$$ So if you go the same distance to the left of to the right of $6$ you will calculate the same value for $y$ indicating the symmetry.

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