Question

Complete a piecewise function that describes the graph.

Answer 1

Answer:

$y=\left \{ \begin{array}{l}-x^2-4x-1,\ x$

Step-by-step explanation:

Left part of the graph is the graph of the parabola passing through the points (-2,3), (-3,2) and (-4,-1). If the equation of the parabola is $y=ax^2+bx+c,$ then

$3=a\cdot (-2)^2+b\cdot (-2)+c\Rightarrow 3=4a-2b+c\\ \\2=a\cdot (-3)^2+b\cdot (-3)+c\Rightarrow 2=9a-3b+c\\ \\-1=a\cdot (-4)^2+b\cdot (-4)+c\Rightarrow -1=16a-4b+c$

Subtract first two equations  and last two equations:

$-1=5a-b\\ \\-3=7a-b$

Suybtract these two equations:

$-2=2a\Rightarrow a=-1$

So $-1=5\cdot (-1)-b\Rightarrow b=-5+1=-4$

Substitute into the first equation:

$3=4\cdot (-1)-2\cdot (-4)+c\Rightarrow c=3+4-8=-1$

The equation of the parabola is $y=-x^2-4x-1$

The right part of the graph is translated 1 unit to the right and 1 unit down graph of the function $y=\dfrac{1}{2}|x|$, so it has the equation $y=\dfrac{1}{2}|x-1|-1$

Hence, the piece-wise function is

$y=\left \{ \begin{array}{l}-x^2-4x-1,\ x$