Stuck with Algebra 2
- Thread starter adrian06
- Start date
HallsofIvy
Elite Member
- Joined
- Jan 27, 2012
- Messages
- 7,763
Suppose you have a graph of f(x). Since "x- 2" is a change in x that is a shift left or right. In particular f(0) is shifted to f(0- 2)= f(-2) so the graph is shifted 2 places to the left. f(x)+ 2, on the other hand has the "2" added after the function is applied: y= f(x) becomes y+ 2= f(x)+ 2 so is a change in y and the y-axis is vertical- the graph is shifted vertically, up two places.
For your second question, I think you are asking if there exists functions, f, such that f(x- 2)= f(2)+ 2. That is a "functional equation" since the "unknown" is a function. Those are generally much more difficult than an algebraic equation because there are so many different kinds of functions. You might try different kinds of equations with unknown coefficients so they become algebraic equations. For example if we try a linear function, f(x)= ax+ b, then f(x- 2)= a(x- 2)+ b= ax- 2a+ b. f(x)+ 2= ax+ b+ 2. So we want ax- 2a+ b= ax+ b+ 2. Subtracting ax+ b from both sides, -2a= 2.
So the answer to your question is "Yes!" In particular, f(x)= -x+ b, where b can be any number, satisfies f(x- 2)= f(x)+ 2: f(x- 2)= -(x- 2)+ b= -2x+ 3+ b= -2x+ b+
2= f(x)+2.
For your second question, I think you are asking if there exists functions, f, such that f(x- 2)= f(2)+ 2. That is a "functional equation" since the "unknown" is a function. Those are generally much more difficult than an algebraic equation because there are so many different kinds of functions. You might try different kinds of equations with unknown coefficients so they become algebraic equations. For example if we try a linear function, f(x)= ax+ b, then f(x- 2)= a(x- 2)+ b= ax- 2a+ b. f(x)+ 2= ax+ b+ 2. So we want ax- 2a+ b= ax+ b+ 2. Subtracting ax+ b from both sides, -2a= 2.
So the answer to your question is "Yes!" In particular, f(x)= -x+ b, where b can be any number, satisfies f(x- 2)= f(x)+ 2: f(x- 2)= -(x- 2)+ b= -2x+ 3+ b= -2x+ b+
2= f(x)+2.
Dr.Peterson
Elite Member
- Joined
- Nov 12, 2017
- Messages
- 16,749
Halls made a little mistake; f(x-2) shifts the graph of f(x) 2 units to the right.
HallsofIvy
Elite Member
- Joined
- Jan 27, 2012
- Messages
- 7,763
left- right
I never could keep those straight!
I never could keep those straight!
Steven G
Elite Member
- Joined
- Dec 30, 2014
- Messages
- 14,598
You probably can do very high level math (Phd course level and higher) but have trouble with left and right! I've seen similar things with brilliant people and it always has amazed me.
HallsofIvy
Elite Member
- Joined
- Jan 27, 2012
- Messages
- 7,763
Therefore, I am a brilliant person!
I love logic!
I love logic!
