Graphs of functions with absolute values and axis intersections

Text

Draw the graph of the following functions and find their intersections with the $x$- and $y$-axes:

• (a) $y=|x-2|+x+2$
• (b) $y=\dfrac{x^2}{9}+2|x|+9$
• (c) $|x|+|y|=0$
• (d) $y=|x+2|-3$
• (e) $y=|2x+1|+3|x-2|+x-10$
• (f) $y=x^2-7|x+2|-5$

Solution

(a) $x\ge2:\ y=2x$; $\ x<2:\ y=4$. Horizontal ray at height $4$ up to $(2;4)$, then line $y=2x$. $\cap y:(0;4)$; no $x$-intercept (always $\ge4$). (b) Even function, always $\ge9$: $x\ge0:\ \dfrac{x^2}{9}+2x+9$; $x<0:\ \dfrac{x^2}{9}-2x+9$. $\cap y:(0;9)$; no $x$-intercept (minimum $9$). (c) $|x|+|y|=0$ only if $x=0$ and $y=0$: the ”graph” is the single point $(0;0)$. (d) V with vertex $(-2;-3)$: $\cap y:(0;-1)$; $\cap x:\ x=1,\,x=-5$. (e) Breakpoints $x=-\tfrac12,\,x=2$. $\ x<-\tfrac12:\ y=-4x-5$; $\ -\tfrac12\le x<2:\ y=-3$; $\ x\ge2:\ y=6x-15$. $\cap y:(0;-3)$; $\cap x:\ (-\tfrac54;0)$ and $(\tfrac52;0)$. (f) $x\ge-2:\ y=x^2-7x-19$; $x<-2:\ y=x^2+7x+9$. $\cap y:(0;-19)$; $\cap x:\ x\approx9{,}09$ (branch $x\ge-2$) and $x\approx-5{,}30$ (branch $x<-2$). $\boxed{\text{Each function is studied by removing the absolute values piecewise; see intersections above.}}$