1. Quadratic Patterns, Function Notation and Graphing Parabolas

2. Linear Function Probes
How do the slopes and y-intercepts of the graphs below compare?
Slopes
Y-intercepts A.
Q>R>S B.
S>Q>R
Q>S>R C.
R>S>Q D.
R>Q>S

3. Linear Function Probes
2. Which graph has a slope of 1/3?
What do the patterns given by B and D have in common?

4. Starter:
Find the equation of the line that passes through the points (-3,5) and (4, -44).
The equation of a line can be written as y = mx +b.
We need the slope and a point.
The slope is the change in y divided by the change in x:
Now use one of the points to find the y-intercept:
y= -7x + b
-44 = -7(4) + b 5 = -7(-3) +b
-44 = b 5 =+21 + b
= b = b
b = -16 b = -16

5. Starter:
Find the equation of the line that passes through the points (-3,5) and (4, -44).
So the equation is y = -7x-16

6. Patterns
The equations we saw last class represent relationships. These are also seen as patterns. Ex. 4, 7, 10, 13,… You can predict the next term, and for that matter, every term. You’re using the equation of this pattern whether you know it or not!
We can set this up using two variables: the term number and the term value.

7. Patterns
Term #, n 1 2 3 4
Value, tn 7 10 13
We can the COMMON DIFFERENCE between consecutive terms. This is given the symbol ‘d’.

8. Patterns The pattern is: 4 4+1*3 4+2*3 4+3*3 . +0*3 n = 1 n = 2 n = 3
Term #, n 1 2 3 4
Value, tn 7 10 13
The pattern is: 4
4+1*3
4+2*3
4+3*3 .
+0*3
n = 1
n = 2
n = 3
n =4
So:
tn = 4 + (n-1)3
The first term: ‘a’ or t1
The CD, d
In general:
tn = t1 + (n-1)d
n = 100
4+99*3

9. Pattern Examples
1) What is the equation or general rule for the following pattern?
5,1,-3,-7,…
t1 = 5
CD = -4
tn = 5+(n-1) (-4)
tn = 5-4n+4
tn = -4n+9
b) What is the 100th term of the pattern above?
tn = -4n+9 where n = 100
t100 = -4(100)+9
t100 = -391

10. Pattern Examples 2) How many terms are in the following pattern?
-9,-6,-3,0,…,81.
t1 = -9
CD = 3
tn = -9+(n-1) (3)
81 = 3n-12
81+12 = 3n-12+12
tn = -9+3n-3
93 = 3n
tn = 3n-12
There are 31 terms in this pattern.
93 = 3n
31= n

11. Patterns Examples 1) Find the general rule for each pattern.
-8,-10,-12,…
1,5,9,13,…
0.5,1,1.5,…
2) Find the 50th term for the patterns above.
3) Which term in the pattern -4,-1,2,… has a value of 59?
4) Which term in the pattern tn = -2n-1 has a value of -85?

12. Deeper Patterns What’s the pattern? - - - - 8 10 12 14 NOT COMMON
Term #, n 1 2 3 4 5
Value, tn 10 18 28 40 54
NOT COMMON
Let’s find the common difference
But what does this CD mean?

13. Quadratic Patterns In general: In general: tn = t1 + (n-1)d
A quadratic pattern is one where the common difference is found on the second level of difference.
Every quadratic pattern can be represented by the equation:
In general:
tn = t1 + (n-1)d
In general:
tn = an2 + bn + c
(That’s for a linear pattern)
So how does the common difference fit into the equation?

14. Getting the nth term for a quadratic pattern
We need to look at the general pattern of a quadratic
tn = an2 + bn + c
Term #, n 1 2 3 4 5
Value, tn
a + b + c
4a +2b + c
9a +3b + c
16a +4b + c
25a +5b + c
7a +b
9a +b
3a +b
5a +b 2a 2a 2a
This shows us that the common difference of ANY quadratic pattern is equal to 2a!!

15. Quadratic Patterns What’s the pattern? - - - - 8 10 12 14 NOT COMMON
Term #, n 1 2 3 4 5
Value, tn 10 18 28 40 54
NOT COMMON
Let’s find the common difference
But what does this CD mean?
Since CD = 2 and CD = 2a then a =1!!

16. Quadratic Patterns
Term #, n 1 2 3 4 5
Value, tn 10 18 28 40 54
So the equation representing this pattern (the nth term) so far can be written:
tn = n2 + bn +c a 1
So what do ‘b’ and ‘c’ equal?
To find ‘b’ and ‘c’ we need two data points from the pattern.
Where we see an ‘n’ in the equation, we’ll put a 1 and where we see the tn we’ll put the 10.
Where we see an ‘n’ in the equation, we’ll put a 2 and where we see the tn we’ll put the 18.
tn = 1n2 + bn + c
10 = 1(1)2 + b(1) + c
10 = 1 + b + c
9 = b + c
tn = 1n2 + bn + c
18 = 1(2)2 + b(2) + c
18 = 4 + 2b + c
14 = 2b + c

17. Quadratic Patterns 9 = b + c 14 = 2b + c
Let’s subtract these two equations to eliminate the variable ‘c’.
9 = b + c
5 = b
So we have another piece of the equation: tn = 1n2 + 5n +c
We can now use one of these 2 equations to solve for ‘c’.
9 = b + c
9 = 5 + c
4 = c
So here’s the equation that represents the pattern: tn = n2 + 5n + 4

18. 154 180 Not yet… Quadratic Patterns WOW! Let’s test it out!
Term #, n 1 2 3 4 5
Value, tn 10 18 28 40 54
Let’s test it out!
The fourth term in the pattern is 40. Does our equation predict this?
In other words, if n = 4, does t4 = 40?
Can you find the 10th term?
tn = n2 + 5n + 4
t4 = (4) + 4
t4 =
t4 = 40
154
WOW!
Can you find the 11th term?
180
Can you find which term has a value of 270?
Not yet…

19. Ex 2: Quadratic Patterns
What is the 50th term of the pattern: 6,14,28,48,74…
Hint: If we only had a good teacher…
tn = 3n2 + bn + c
When n = 1 t1 = 6 and when n = 2 t2 = 14
Hint: If only the Common Difference was related to one of the constants…
6 = 3(1)2 + b(1) + c 14 = 3(2)2 + b(2) +c
3 = b +c 2 = 2b +c
CD = 2a
6 = 2a
a = 3
1 = -b
b = -1
3 = (-1) + c
c = 4
Hint: If we only knew what type of pattern we have…
Hint: If we only knew the equation of this pattern…

20. Ex 2: Quadratic Patterns
What is the 50th term of the pattern: 6,14,28,48,74…
So tn = 3n2 -1n + 4 represents the above pattern.
To find the 50th term plug in n = 50 and calculate t50
t50 = 3(50)2 -1(50) + 4
= 7454
BUT… how can we find which term has value of 756?
Stay tuned….

21. More quadratic patterns
Find the 12th term in each pattern.
Term #, n 1 2 3 4 5
Value, tn -9
-16
-27
-42
-61
-306
Term #, n 1 2 3 4 5
Value, tn 13 28 49 76
433

22. Function Notation
In our last example, we set n = 50 to find t50 . We will use function notation to simplify writing it out.
The term “function” means a relationship between two variables. One variable is responds to the other. Say that the responding variable depends on the variable x, through the function, f, as in our last example.
Instead of writing tn = 3n2 -1n + 4 or y = 3x2 -1x + 4, we can write f(x) = 3x2 -1x + 4
This allows us to write things like:
f(50) which means what is the y value when x = 50?
Or f(x) = 756 which says “Find the x-value which gives a y value of 756.

23. Function Notation Ex. If f(x) = 4x -5, then calculate f(3) f(0)7 -5
x = 2
x = 5/4
Ex 2. If f(x) = 3x-2 and g(x) = 3-5x, then calculate
g(7)
f(x) =7
g(x)=f(x)
-32
x = 3
x = 5/8

24. More fun (ction notation)
Ex 3. If f(x) = 2x2 + 3x – 8, calculate
the first 5 terms of the pattern
f(-4)
f(0)
-3, 6, 19, 36, 57 12 -8