Question 977329
Assume that f(2)=3. Assume also that the graph of y=f(x) is symmetric with respect to the line x=-3. Find another value for the function.
<pre>
f(2)=3 means that when you substitute x=2 in the equation

y = f(x)

like this:

y = f(2)

That the value of y is 3 because f(2) = 3

That means that the graph contains the point (x,y) = (2,3)

So we plot that point (2,3) and the line x=-3, which is a vertical line 
through -3 on the x axis, the green line below:

{{{drawing(400,300,-11,5,-3,9,graph(400,300,-11,5,-3,9),
locate(2,3,"(2,3)"),
green(line(-3,-10,-3,10)),


circle(2,3,0.15),circle(2,3,0.13),circle(2,3,0.11),circle(2,3,0.09),circle(2,3,0.07),circle(2,3,0.05),circle(2,3,0.03),circle(2,3,0.01) )}}}

We want to reflect that point across the green line.  Let's draw a dotted line
from that point (2,3) to the green line, like this:


{{{drawing(400,300,-11,5,-3,9,graph(400,300,-11,5,-3,9,

3*( sqrt(sin(7x))/sqrt(sin(7x)) )*
(sqrt(x+3)/sqrt(x+3))*
(sqrt(2-x)/sqrt(2-x))
),
locate(2,3,"(2,3)"),
green(line(-3,-10,-3,10)),

circle(2,3,0.15),circle(2,3,0.13),circle(2,3,0.11),circle(2,3,0.09),circle(2,3,0.07),circle(2,3,0.05),circle(2,3,0.03),circle(2,3,0.01) )}}}

We notice that the dotted line is 5 units.  So to reflect the point (2,3)
across the green line we extend the 5-unit dotted line 5 units to the left
of the green line like this:

{{{drawing(400,300,-11,5,-3,9,graph(400,300,-11,5,-3,9,

3*( sqrt(sin(7x))/sqrt(sin(7x)) )*
(sqrt(x+8)/sqrt(x+8))*
(sqrt(2-x)/sqrt(2-x))
),
locate(2,3,"(2,3)"),

circle(-8,3,0.15),circle(-8,3,0.13),circle(-8,3,0.11),circle(-8,3,0.09),circle(-8,3,0.07),circle(-8,3,0.05),circle(-8,3,0.03),circle(-8,3,0.01),

locate(-9.9,3,"(-8,3)"),


green(line(-3,-10,-3,10)),
circle(2,3,0.15),circle(2,3,0.13),circle(2,3,0.11),circle(2,3,0.09),circle(2,3,0.07),circle(2,3,0.05),circle(2,3,0.03),circle(2,3,0.01) )}}}

So another point on the graph is (-8,3).

Answer: another value is f(-8)=3

[Recall that f(a)=b means that the point (a,b) is on the graph of f(x)]

The graph could look something line this: 

{{{drawing(400,300,-11,5,-3,9,graph(400,300,-11,5,-3,9,

3*( sqrt(sin(7x))/sqrt(sin(7x)) )*
(sqrt(x+8)/sqrt(x+8))*
(sqrt(2-x)/sqrt(2-x))
),
locate(2,3,"(2,3)"),
graph(400,300,-11,5,-3,9,((x+3)^2-10)/5),
circle(-8,3,0.15),circle(-8,3,0.13),circle(-8,3,0.11),circle(-8,3,0.09),circle(-8,3,0.07),circle(-8,3,0.05),circle(-8,3,0.03),circle(-8,3,0.01),

locate(-9.9,3,"(-8,3)"),


green(line(-3,-10,-3,10)),
circle(2,3,0.15),circle(2,3,0.13),circle(2,3,0.11),circle(2,3,0.09),circle(2,3,0.07),circle(2,3,0.05),circle(2,3,0.03),circle(2,3,0.01) )}}}

Edwin</pre>