Quadratic Patterns, Function Notation and Graphing Parabolas

Linear Function Probes1. How do the slopes and y-intercepts of

the graphs below compare?

Slopes Y-intercepts

A. Q>R>S Q>R>S B. S>Q>R Q>S>RC. R>S>Q Q>S>R

D. R>Q>S R>S>Q

Linear Function Probes

2. Which graph has a slope of 1/3?

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:

77

4943

)44(5

x

ym

Now use one of the points to find the y-intercept:

y= -7x + b-44 = -7(4) + b 5 = -7(-3) +b-44 = -28 + b 5 =+21 + b-44+28 = b 5 -21 = bb = -16 b = -16

Starter:

Find the equation of the line that passes through the points (-3,5) and (4, -44).

So the equation is y = -7x-16

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.

Term #, n

1 2 3 4

Value, tn 4 7 10 13

We can the COMMON DIFFERENCE between consecutive terms. This is given the symbol ‘d’.

- - - 3 3 3

Term #, n

1 2 3 4

Value, tn 4 7 10 13

The pattern is:44+1*34+2*34+3*3...

n = 1n = 2n = 3n =4

+0*3 So:tn = 4 + (n-1)3

The CD, dThe first term: ‘a’ or t1

n = 1004+99*3In general:tn = t1 + (n-1)d

1) What is the equation or general rule for the following pattern?

5,1,-3,-7,…

CD = -4

t1 = 5 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

2) How many terms are in the following pattern?

-9,-6,-3,0,…,81.

CD = 3t1 = -9

tn = -9+(n-1) (3)

tn = -9+3n-3

tn = 3n-12

81 = 3n-12

81+12 = 3n-12+12

93 = 3n

93 = 3n 3 3

31= n

There are 31 terms in this pattern.

1) Find the general rule for each pattern.

a)-8,-10,-12,…b)1,5,9,13,…c)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?

Term #, n

1 2 3 4 5

Value, tn 10 18 28 40 54

- - - - 8 10 12 14NOT COMMON

Let’s find the common difference

What’s the pattern?

- - - 2 2 2

But what does this CD mean?

Quadratic PatternsA 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

(That’s for a linear pattern)

In general:tn = an2 + bn + c

So how does the common difference fit into the equation?

Getting the nth term for a quadratic pattern We need to look at the general pattern of a quadratic

Term #, n 1 2 3 4 5

Value, tn

tn = an2 + bn + c

a + b + c 4a +2b + c 9a +3b + c 16a +4b + c 25a +5b + c

- - - -

3a +b 5a +b 7a +b 9a +b

- - -

2a 2a 2a

This shows us that the common difference of ANY quadratic pattern is equal to 2a!!

Term #, n

1 2 3 4 5

Value, tn 10 18 28 40 54

- - - - 8 10 12 14NOT COMMON

Let’s find the common difference

What’s the pattern?

- - - 2 2 2

But what does this CD mean?

Quadratic Patterns

Since CD = 2 and CD = 2a then a =1!!

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 +ca1 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.

tn = 1n2 + bn + c10 = 1(1)2 + b(1) + c10 = 1 + b + c9 = b + c

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 + c18 = 1(2)2 + b(2) + c18 = 4 + 2b + c14 = 2b + c

Quadratic Patterns

9 = b + c 14 = 2b + c

9 = b + cLet’s subtract these two equations to ELIMINATE the variable ‘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 + c9 = 5 + c4 = c

So here’s the equation that represents the pattern: tn = n2 + 5n + 4

Quadratic Patterns

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?

tn = n2 + 5n + 4 t4 = 42 + 5(4) + 4 t4 = 16 + 20 + 4 t4 = 40

Can you find the 10th term?

Can you find the 11th term?

Can you find which term has a value of 270?

Quadratic Patterns

Ex 2: Quadratic PatternsWhat is the 50th term of the pattern: 6,14,28,48,74…

Hint: If we only knew the equation of this pattern…

Hint: If we only knew what type of pattern we have…

Hint: If we only had a good teacher…

8 14 20 26 6 6 6

tn = 3n2 + bn + cWhen n = 1 t1 = 6 and when n = 2 t2 = 14 6 = 3(1)2 + b(1) + c 14 =

3(2)2 + b(2) +c3 = b +c 2 = 2b +c

1 = -bb = -1

Hint: If only the Common Difference was related to one of the constants…

CD = 2a6 = 2aa = 3

3 = (-1) + cc = 4

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….

Ex 2: Quadratic Patterns

More quadratic patterns

Term #, n

1 2 3 4 5

Value, tn -9 -16 -27 -42 -61

Find the 12th term in each pattern.

Term #, n

1 2 3 4 5

Value, tn 4 13 28 49 76

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.

Function NotationEx. If f(x) = 4x -5, then calculate

a.f(3)b.f(0)c.f(x) = 3d.f(x) =0

a. 7b. -5c. x = 2d. x = 5/4

Ex 2. If f(x) = 3x-2 and g(x) = 3-5x, then calculatea.g(7)b.f(x) =7 c.g(x)=f(x)

a. -32b. x = 3c. x = 5/8

More fun (ction notation)

Ex 3. If f(x) = 2x2 + 3x – 8, calculate

a.the first 5 terms of the patternb.f(-4) c.f(0)

a. -3, 6, 19, 36, 57b. 12c. -8