SOLUTION: CAN SOMEONE HELP ME SOLVE THIS GRAPHING? 1. DESCRIBE THE TRANSFORMATIONSON ON A GRAPH OF f(x)=e^x. State the placement of the horizonal asympotote and y-intercept after the trans

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: CAN SOMEONE HELP ME SOLVE THIS GRAPHING? 1. DESCRIBE THE TRANSFORMATIONSON ON A GRAPH OF f(x)=e^x. State the placement of the horizonal asympotote and y-intercept after the trans      Log On


   

Question 116009: CAN SOMEONE HELP ME SOLVE THIS GRAPHING?
1. DESCRIBE THE TRANSFORMATIONSON ON A GRAPH OF f(x)=e^x. State the placement of the horizonal asympotote and y-intercept after the transformation. For example,"left 1" or "rotated about the y-axis" are descriptions
a) g(x)=e^x+3
description of transformation:
Horizonal asymptote:
y-intercept in (x,y) form:
b)H(x)=e^-x
description of transformation:
Horizonal asymptote:
y-intercept in (x,y)form
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
a)

Description of transformation:
Remember, f%28x%29 is the same as y. So this means y=e%5Ex.
So when we say e%5Ex%2B3, we're also saying y%2B3 (replace e%5Ex with y). So this means we're adding 3 to each y value which graphically shows us that we're shifting each y value up 3 units.

----------------------------
Answer:
So the transformation g%28x%29=e%5Ex%2B3 simply shifts the entire curve up 3 units.


Notice if we graph f%28x%29 and g%28x%29, we get

+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+exp%28x%29%2Cexp%28x%29%2B3%29+ Graph of f%28x%29=e%5Ex (red) and h%28x%29=e%5Ex%2B3 (green)

and we can visually verify the transformation


Horizontal Asymptote:
Now if we found the asymptote of y=e%5Ex, we would find that the asymptote is y=0. Since we're translating each point on y=e%5Ex up 3 units, we're also translating the horizontal asymptote up 3 units.

----------------------------
Answer:
So the new horizontal asymptote is y=3.

Also, you can visually verify this answer by looking at the graph above



y-intercept in (x, y) form:

If we let x=0 and plug it into y=e%5Ex, we get

y=e%5E0 Plug in x=0

y=1 Raise e to the zeroth power to get one. Remember any number x to the zeroth power is always one (ie x%5E0=1)

So for f%28x%29 the y-intercept is (0,1)


Now if g%28x%29 translates each y value up 3 units, then simply add 3 to the y-coordinate of the y-intercept to get

(0,1+3)---->(0,4)

----------------------------
Answer:
So the new y-intercept is (0,4)

Once again, you can visually verify this answer by looking at the graph above






b)

Description of transformation:

Looking at h%28x%29=e%5E%28-x%29, notice how the exponent is negated. So let's see what affect this transformation has on f%28x%29=e%5Ex,

h%28x%29=e%5E%28-x%29 Start with the given transformation


h%28-2%29=e%5E%28-%28-2%29%29 Plug in x=-2


h%28-2%29=e%5E%282%29 Negate -%28-2%29 to get 2


Notice if we plug in x=2 into f%28x%29=e%5Ex, we get f%282%29=e%5E2

--------

h%28x%29=e%5E%28-x%29 Start with the given transformation


h%28-1%29=e%5E%28-%28-1%29%29 Plug in x=-1


h%28-1%29=e%5E%281%29 Negate -%28-1%29 to get 1

h%28-1%29=e Remove the exponent of 1

Notice if we plug in x=1 into f%28x%29=e%5Ex, we get f%281%29=e%5E1=e


So if we take the opposite of x (to get -x), and plug that into g(x), we'll get the same f(x) answer.


----------------------------------
Answer:
So what this does is simply reflect the entire graph over the y-axis

Notice if we graph f%28x%29 and g%28x%29, we get

+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+exp%28x%29%2Cexp%28-x%29%29+ Graph of f%28x%29=e%5Ex (red) and h%28x%29=e%5E%28-x%29 (green)

and we can visually verify the transformation



Horizontal Asymptote:

Since we reflected the graph with respect to the y-axis, the horizontal asymptote of h%28x%29=e%5E%28-x%29 is the same as f%28x%29=e%5Ex (you can see this from the graph above)
----------------------------------
Answer:
So the horizontal asymptote is y=0

Note: you can visually verify this answer by looking at the graph above

y-intercept in (x, y) form:

Since the line of symmetry between the two graphs is the line x=0 (ie the y axis), this means that the point that intersects with the y-axis is reflected to itself. So essentially the y-intercept does not change also. Once again, you can visually verify this using the graph above.

----------------------------------
Answer:
So the y-intercept is (0,1)

Once again, you can visually verify this answer by looking at the graph above