y=-2|x|+2 is a transformation of y=x, which is a line of gradient 1 (angle of 45° sloping to the right) passing through the origin. y=x+2 shifts the line two units upward, so the "origin" is now (0,2). y=2x+2 increases the gradient (slope of the line). y=-2x+2 alters the direction of the slope (gradient=-2) of the line so that it slopes downwards instead of upwards from (0,2). When x is positive |x| is the same as x so the right side of the graph is unaffected, but the left side of the graph is a reflection of the right side, as if the y-axis is a mirror, so we have a line of gradient 2 (reverse of gradient=-2) ending at the point (0,2). We now have a complete description of the graph y=-2|x|+2. The domain is all of x from minus infinity to plus infinity, but the range has a maximum value of 2 (like the tip of a mountain) and a minimum value of minus infinity because the lines are semi-infinite.
