Graph this function: F(x)=$-2^{|x|+1}$.

Question

Graph this function: F(x)=$-2^{|x|+1}$.

when I graph this function and go to check by a graphing calculator I find that I am wrong !

even though I do the horizontal translation

So, what's wrong?

:(

Answer 1

The thing you have to watch out for when the absolute value operation appears in the equation for a curve is that it has the effect of treating the $ \ y-$ axis as a "mirror". The equation $ \ y \ = \ f( \ |x| \ ) \ $ can be understood to mean that we graph $ \ f(x) \ $ for $ \ x \ \ge \ 0 \ $ , but $ \ f( -x ) \ $ for $ \ x \ < \ 0 \ $ . This is equivalent to saying that we plot the portion of the graph for $ \ f(x) \ $ that lies on or "to the right" of the $ \ y-$ axis as usual, then take that part, "flip it around" horizontally, and plot that reflected version "to the left" of the $ \ y-$ axis.

What this will mean for making the horizontal translation $ \ x \ \rightarrow \ x + 1 \ $ is that the portion on the right of the $ \ y-$ axis will be shifted "to the left" by one unit (as expected), and we would "remove" the part of the curve that was carried "to the left" of the $ \ y-$ axis (since that no longer lies where $ \ x \ \ge \ 0 \ $ ) . The part of the original curve where $ \ x \ < \ 0 \ $, however, is "mirror-reflected", so it is going to be translated "to the right" by one unit, and the part that is now "on the right" of the $ \ y-$ axis (where $ \ x \ > \ 0 \ $ ) is what must be "trimmed off".

You can also picture this as if you had made a graph of $ \ f(x) \ $ , held it alongside the edge of a mirror, with the $ \ y-$ axis of the graph right at that edge, and then slid the graph one unit toward the mirror. What you would see, combining the views of the paper graph and what you see in the mirror, would be the curve for $ \ f( \ |x| \ ) \ $ . Upon sliding the graph of the curve so it is carried out of sight behind the edge the mirror, the mirror-image also moves toward the mirror edge: the leftward translation of the graph becomes a "rightward" translation of the mirror image.

In this graph, the curves in red are $ \ y \ = \ -2^x \ $ (the curve decreasing left-to-right) and $ \ y \ = \ -2^{-x} \ $ (the curve increasing left-to-right), which are "upside-down" exponential-growth and -decay curves, respectively. The graph for $ \ -2^{|x|} \ $ is the piece of $ \ -2^{-x} \ $ climbing until it reaches $ \ (0 \ , \ -1) \ $ , together with the piece of $ \ -2^x \ $ descending for $ \ x \ > \ 0 \ $ .

When we perform the horizontal translation on $ \ -2^x \ $ to produce $ \ -2^{x+1} \ $ (the descending blue curve) , the curve does move one unit "to the left" , and we just keep the part for $ \ x \ \ge \ 0 \ $ . But the transformation $ \ -2^{-x+1} \ $ (the ascending blue curve) behaves like $ \ -2^{-(x-1)} \ $ , which is a shift "to the right" when compared with $ \ -2^{-x} \ $ . So the curve for $ \ -2^{|x|+1} \ $ now ascends to $ \ (0 \ , \ -2) \ $ and then decreases beyond that point.

It is a characteristic of exponential functions that a "horizontal shift" behaves like a "vertical stretch", to relate this to LAcarguy's answer. The $ \ y-$ intercept changes from $ \ (0 \ , \ -1) \ $ to $ \ (0 \ , \ -2) \ $ , and the transformed curve rises and falls more steeply by a factor of 2.

Answer 2

Let's write down the steps to complete the graph of this function:

Step 1: Start with the graph of: $y = 2^x$

Step 2: Delete the part of the graph in step 1 where $x < 0$. Reflect the part of the graph of step 1 for $x \geq 0$ about the $y$ axis. The resulting graph looks like a $V$ shape curve that has a vertex at $(0,1)$ and this is the lowest point of the graph in this step 2.

Step 3: Stretching the graph in step 2 by a factor of $2$. This makes the graph in step 2 thinner and taller. Notice the new vertex is $(0,2)$ and this is due to the vertical stretch by a factor of $2$.

Step 4: Reflect the graph in step 3 by the $x$ axis to get the answer.

Answer 3

This is a nice application of the absolute-value function's definition.

If $x \ge 0$, then this function is $-2^{x + 1}$.

If $x < 0 $, then this function is $-2^{1 - x}$.

In graphing, you can work around the minus sign by reflecting a graph. Graphing both of the functions is deemed simple. If not, use the fact that $2^{x + 1} = 2 \cdot 2^{x}$ and $2^{1 -x} = \large \frac{2}{2^x}$.

If there's any trouble, do try marking a few points and do eyeball-analysis to see how the pattern is.