This week in precalc 11, we started learning about absolute values and reciprocal functions. As we already know, the absolute value of a number can not be negative so we have to always keep that in mind.

If we have a linear graph y = x, we know that it will look like this

If we have an absolute value graph y = |x|, we know since it can’t be negative, the points of the graph can not go to the negative part of the graph (bottom) which will make the graph look like this

What the graph is basically doing is a reflection. In the picture below, the bottom left part of the graph reflects to the top left of the graph. The x intercept of the graph y = f(x) is a critical point

critical point : the point where the graph of the function changes direction

Those are for linear graphs, but we also have quadratic graphs. As we know each quadratic graph has a vertex (either the highest or lowest part of the graph)

Lets say we have a graph $y = (x+3)^2 - 1$, it would look like this with a vertex of (-3,-1)

But if we have an equation like … $y = | (x+3)^2 - 1 |$, since we know an absolute value can not be negative, any negatives will become a positive which will look like this with a vertex of (-3,1)

Everything we’ve done for the past few units have related including solving equations. As for now, since everything we’ve done is related, there aren’t huge new concepts we have learned. But for this unit we must know about the critical point (where the graph of the function changes direction) and to also know any negatives in an absolute value equation will always become a positive. We must also not get confused about the V graph (absolute value) and the parabola (quadratic). As we go on more things will be added but for now we are basically reviewing everything we’ve done but in a different way.