The following graph is a
transformation of 𝑦 equals the absolute value of 𝑥. What is the function it
represents? Write your answer in a form related
to the function.
Here, we’re looking at the general
𝑦 equals the absolute value of 𝑥 with some transformations. 𝑎, first looking at 𝑎 and what it
would do to the absolute value graph. If it would be positive, the graph
would look like a V; it would be facing upwards. If 𝑎 were negative, it would be
the exact opposite; it would be opening downwards. If the absolute value of 𝑎,
meaning we’re not looking at the sign anymore, if the absolute value of 𝑎 is
greater than one, this will sh- vertically stretch the graph. If 𝑎 is between zero and one, so
if it’s a fraction, it will be a vertical compression of the graph.
Now looking at ℎ, that will be the
horizontal shift, so moving left and right. And finally, 𝑘 will be the
vertical shift, how much it moved up or down.
So first, looking at our graph, it
is upside down. So that means 𝑎 must be
negative. And next, to determine what 𝑎
actually is, so if it’s — the absolute value will be bigger than one or if it will
be between zero and one, it’s essentially like the slope of the line. And we can see, if we go from the
peak to the right, it goes down one, right one. So 𝑎 is one, which this kind of
makes sense. Looking at that, it’s negative one;
it’s the slope.
Next, we look at ℎ, the horizontal
shift. So it’s 𝑥 minus the horizontal
shift. So the original 𝑦 equals the
absolute value of 𝑥 graph would start at zero. So-so if the original graph started
at zero, zero, our new graph started one space to the left, which would be a
negative one shift.
And then finally, how much would it
have moved up? It went up four.
So to simplify this, we have 𝑦
equals negative one absolute value 𝑥 plus one plus four. We could use the commutative
property of addition because we’re adding four to that absolute value. So we could put the four first and
then put minus. We don’t have to put the one in
front of the absolute value. So we could have 𝑦 equals four
minus the absolute value of 𝑥 plus one.