Building Functions
Finding an Explicit Expression
Sequence 1: 9, 18, 27, 36, 45, 54
To find the explicit expression for this arithmetic sequence you must use the universal formula for an explicit formula for an arithmetic sequence, which is (an = a1 + (n – 1)d). "a1" represents the first term in the sequence, "n" represents the nth term you are looking for, and "d" represents the common difference between any two number in the sequence consecutively.
To solve for the expression we must substitute in the necessary numbers.
an = a1 + (n – 1)d
an=9+(n-1)9
an=9+9n-9
an=9n
Finding a Recursive Process
Finding a recursive process for a sequence is quite simple. You must use the universal formula which is (an=an-1+d).For our example we will use the sequence above. You must first get the first term in the sequence. Our first term in the sequence above is 9. So to put it algebraically, it would look like this.
t1=9
Then you figure out what is being added to get the next term and in this sequence it is 9.
So to put it algebraically, it would look like this
t1= 9 (term 1 in the sequence)
tn=tn-1+9 (taking the term before the nth term and subtracting 1, then adding 9)
The way I would work a recursive process like this is I would substitute the first term in the sequence for "n-1" and I would add 9. Then I would substitute the sum and add 9 until I got the term I was solving for.
Combining Functions
Combining functions is a process that is simpler than you think.
For our examples, we will use the functions
f(x)=10x+12 and g(x)=5x+6
Find a(x) where a(x)= f(x)+g(x)
(f(x)=10x+12) + (g(x)=5x+6)
(10x+12) + (5x+6)
a(x)=15x+18
Find b(x) where b(x)= f(x)-g(x)
(f(x)=10x+12) - (g(x)=5x+6)
(10x+12) - (5x+6)
b(x)=5x+6
Find c(x) where c(x)= f(x) x g(x)
(f(x)=10x+12) x (g(x)=5x+6)
(10x+12) x (5x+6)
c(x)=50x+72
Find d(x) where d(x)= f(x) / g(x)
(f(x)=10x+12) / (g(x)=5x+6)
(10x+12) / (5x+6)
d(x)= 2x+2
Vertical Translations
A vertical translation is the movement of a line either up or down. To translate a line you must change the y-intercept.
Below will be examples of translations
We will use the function f(x)=4x+7
Sequence 1: 9, 18, 27, 36, 45, 54
To find the explicit expression for this arithmetic sequence you must use the universal formula for an explicit formula for an arithmetic sequence, which is (an = a1 + (n – 1)d). "a1" represents the first term in the sequence, "n" represents the nth term you are looking for, and "d" represents the common difference between any two number in the sequence consecutively.
To solve for the expression we must substitute in the necessary numbers.
an = a1 + (n – 1)d
an=9+(n-1)9
an=9+9n-9
an=9n
Finding a Recursive Process
Finding a recursive process for a sequence is quite simple. You must use the universal formula which is (an=an-1+d).For our example we will use the sequence above. You must first get the first term in the sequence. Our first term in the sequence above is 9. So to put it algebraically, it would look like this.
t1=9
Then you figure out what is being added to get the next term and in this sequence it is 9.
So to put it algebraically, it would look like this
t1= 9 (term 1 in the sequence)
tn=tn-1+9 (taking the term before the nth term and subtracting 1, then adding 9)
The way I would work a recursive process like this is I would substitute the first term in the sequence for "n-1" and I would add 9. Then I would substitute the sum and add 9 until I got the term I was solving for.
Combining Functions
Combining functions is a process that is simpler than you think.
For our examples, we will use the functions
f(x)=10x+12 and g(x)=5x+6
Find a(x) where a(x)= f(x)+g(x)
(f(x)=10x+12) + (g(x)=5x+6)
(10x+12) + (5x+6)
a(x)=15x+18
Find b(x) where b(x)= f(x)-g(x)
(f(x)=10x+12) - (g(x)=5x+6)
(10x+12) - (5x+6)
b(x)=5x+6
Find c(x) where c(x)= f(x) x g(x)
(f(x)=10x+12) x (g(x)=5x+6)
(10x+12) x (5x+6)
c(x)=50x+72
Find d(x) where d(x)= f(x) / g(x)
(f(x)=10x+12) / (g(x)=5x+6)
(10x+12) / (5x+6)
d(x)= 2x+2
Vertical Translations
A vertical translation is the movement of a line either up or down. To translate a line you must change the y-intercept.
Below will be examples of translations
We will use the function f(x)=4x+7
The line above has a y-intercept of 7 and a slope of 4.
The graph above has two lines. The purple line has a positive translation because it slid above the original line. The slope is the same as the original line but the y-intercept is different, because it starts at (0,9) as opposed to the original which starts at (0,7).
The graph above has three lines. The orange line has a negative translation. It is a negative translation because it slid below the original line. Also, it has a y-intercept of (0,5) as opposed to the y-intercept of the original line which is (0,7).
The graph above has 4 line. The red line is a positive translation. It is positive because it slid above the original line. Also, it has a y-intercept of (0,11) as opposed the y-intercept of the original which is (0,7)