Wednesday, February 26, 2014 Algebra 2 GT

Objective: We will continue to explore rational functions, including higher order polynomials, all types of asymptotes and “holes”. We will work from graph to equation and from equation to graph.

Warm Up:

1. What is the similarity between a VA and a hole? What is the difference?

2. How do you use the values of the VA to “build” the equation?

3. How do you use the value of the x-intercept to “build” the equation?

4. How do you solve for the lead coeff. in the equation?

Weds. 2/26/14 Algebra 2 GT

Complete the CW worksheet and Study check out the “coolmath” website

http://coolmath.com/algebra/23-graphing-rational-functions/index.html

Quiz on 9.3 on Thursday 2/27

Steps for Finding Key Features of a Rational Function and using that information to Create the Graph

1. FACTOR both the numerator and denominator, and REDUCE if possible.

2. If a factor remains in the numerator then its root is the x-intercept [point: (x, 0)]

3. If a factor(s) remains in the denominator then its root is (are) the VA [equation: x = #]

4. If a factor was canceled then its root is the hole [point: (x, y)]

5. Determine HA [equation: y = #]

6. Find the y-intercept by letting x = 0 in the reduced equation [point: (0, y)]

7. Graph all of this information and then use your understanding of the behavior of these graphs to sketch. If necessary, you can use some table values to help.

http://coolmath.com/algebra/23-graphing-rational-functions/index.html

Determining Horizontal AsymptotesThe value of the H.A. is determined by comparing the highest degree

of the numerator with that of the denominator.

1.If numerator > denominator (top-heavy fraction), then there is NO H.A. More on this next year.

2.If numerator < denominator (bottom-heavy fraction), then the H.A. is ALWAYS at y = 0.

3. If numerator = denominator (powers-equal fraction), then the H.A. is ALWAYS at the line with equation y = a/b, where a and be are the lead coefficients of the num. and denom.

€

y=x+6

x 2 + x − 6

€

y=0

€

x=2

€

x=−3

€

y=4x

x 3 − 4x

(0, -1)

€

x=−2

€

x=2

€

y=0

€

−3,4

5

⎛

⎝ ⎜

⎞

⎠ ⎟

€

y=x2 − 3x

2x 2 − x −15

€

3,3

11

⎛

⎝ ⎜

⎞

⎠ ⎟

€

x=−5

2

€

y=1

2

€

−2, − 2( )

x = -3

y = 5

We have a new parent function! The RECIPROCAL FUNCTION:

€

y=1

x

Can you recreate this graph on your graphing calculator?

Variation Vocabulary …

INVERSE Variation – A relationship between variables characterized by the equation

DIRECT Variation – A relationship between variables characterized by the equation

Constant of Variation – the value of k (also, the slope of a line with y-intercept = 0)

€

y =k

x

€

y = k⋅x

JOINT Variation – when one quantity varies directly with respect to two or more other quantities. COMBINED Variation involves multiple variations.Some Translations:

“z varies jointly with x and y”

“z varies jointly with x and y andinversely with the square of w”

“z varies directly with x and inversely with the product wy”€

z = kxy

€

z =kxy

w2

€

z =kx

wy

Common Sense understanding of Variation …

Direct Variation – as one quantity increases, so does the other, by a constant amount.

For example, as the amount of time you drive increases, the distance you drive also increases. The constant of variation is the rate (speed) at which you are driving.

Common Sense understanding of Variation …

Inverse Variation – as one quantity increases, the other decreases.

For example, as the outside temperature increases, the amount of time it takes an ice cube to melt decreases.

Example:A quantity c varies jointly with d and the square of g.

Given c = 30 when d = 15 and g = 2, find k, the constant of variation.

Then, find d when c = 6 and g = 8.