1. Use the discriminant to determine the number and type of roots of: a. 2x 2 - 6x + 16 = 0b. x 2...

Warm Up #7 Feb. 19th 2014DO NOT TURN IN WEEKLY HW!

1. Use the discriminant to determine the number and type of roots of:

a. 2x2 - 6x + 16 = 0 b. x2 – 7x + 8 = 0

2. Solve using the quadratic formula:-3x2 – 8x + 5 = 0

3. Which method would you use to solve this and why? 8x2 - 32 = 0

4. Solve using the square root method (2x-1)2 =6

Systems of Linear/Quad Equations and Word Problems

February 17th, 2015

Quick review…NO WARM UP

Given the points….find the equation!

Sometimes we will be given data in a table or a list of points and asked to write the equation

Steps a=___ b=___ c= ___

STAT - #1 Edit Enter X values in L1 Enter Y values or f(x) in L2 STAT CALC CALC - #5 QuadReg Go down to Calculate Put in the “a” “b” “c” values to form a quadratic

Example-put in a table!

a = ______ b = ______ c = ________

Equation:

(–2, 1), (–1, 0), (0, 1), (1, 4), (2, 9)

You try!

a = ______ b = ______ c = ________

Equation:

Example

The following data forms a parabola, what are the roots? Find the equation.

x y-4 8-3 0-2 -60 -124 05 8

How could you write the equation from looking at the graph?

Find the zeros, write out the factors, and multiply out!

Example:

What if the parabola opens down?

Writing the Equation

If the equation does not have whole number zeros, you can always make a table of points from the graph and find the Quadratic Equation of best fit using “QuadReg” on the calculator

Last year you studied systems of linear equations.

You learned three different methods to solve them.

Elimination, Substitution and Graphing

Graphing Method

To solve is to find the intersections of the graph.

Put each in slope intercept form and graph

This is what we will use to solve a Quadratic/Linear System

Calculator Notes

1. Type equation 1 = y1, equation 2 = y2

2. Push 2ndTRACE5, move cursor to the left intersection push ENTER 3 times.

3. Push 2ndTRACE5, move cursor to the right intersection push ENTER 3 times.

Example 1-ONE Solution

53

1042

yx

yx

Example 2-NO solutions

73

103

xy

xy

Example 3-INFINITE Solutions

264

132

yx

yx

Graphing Method w/ Quadratics & Linear

We do this the same way we do linear equations

To solve is to find the intersections of the graph.

14

32 xxy

xy

Graphs & Solutions

How can we CHECK our answers?

Example #2

Example #3:

6

332

xy

xxy

Example #4 –

6

652

xy

xxy

Quadratic modeling

We can create quadratic functions to model real world situations all around us.

We can use these models to find out more information, such as: Minimum/maximum height Time it takes to reach the ground Initial height How long it takes to reach a height

Example #1:

For a typical basketball shot, the ball’s height (in feet) will be a function of time in flight (in seconds), modeled by an equation such as

h = -16t2 +40 t +6.

a) What is the maximum height of the ball? How long does it take to reach the maximum height?

How do we approach this problem…

To find maximum height:

Are we looking for x or for y?

Graph the function. Adjust the window as needed. (this takes some practice!)

Find the vertex.

Interpreting the question…

The maximum or minimum HEIGHT is represented by the Y VALUE of the vertex.

How long it takes to reach the max/min height is represented by the X VALUE of the vertex.

Example #1:

For a typical basketball shot, the ball’s height (in feet) will be a function of time in flight (in seconds), modeled by an equation such as

h = -16t2 +40 t +6.

b) When will the shot reach the height of the basket? (10 feet)

How do we approach this problem…

To find a time at a given height…

Set the equation equal to the height you want to be at

Let y2 = given height Let y1 = the original equation

Find the intersection of y1 and y2

Interpreting the problem…

The X VALUE always represents TIME How long it takes….

So when you find the intersection, it should have X = time, and Y = height

Example #1:

For a typical basketball shot, the ball’s height (in feet) will be a function of time in flight (in seconds), modeled by an equation such as

h = -16t2 +40 t +6.

c) When will the ball hit the floor if it missed the basket entirely?

How do we approach this problem…

To find the time it takes it hit the ground…

This is asking us when does the height = 0

Let y2 = 0.

Find the intersection of y1 and y2

Interpreting the problem…

When asking when something HITS the GROUND you should think ZERO!

GROUND = ZERO

Find the second zero (not the first!) think left to right…goes up then down

Example #1:

For a typical basketball shot, the ball’s height (in feet) will be a function of time in flight (in seconds), modeled by an equation such as

h = -16t2 +40 t +6.

d) What is the height of the ball when it leaves the player’s hands?

How do we approach this problem..

Interpreting the problem….

Here we want to find the INITIAL HEIGHT….where did the ball start? ON the ground? In someone's hands?

The INITIAL HEIGHT is the Y-INTERCEPT!

Example #1:

For a typical basketball shot, the ball’s height (in feet) will be a function of time in flight (in seconds), modeled by an equation such as

h = -16t2 +40 t +6.

d) What is the height of the ball after 2 seconds?

How do we approach this problem..

Evaluating

I take the x-value (time) and plug it in to find the y-value (height)

h(2) = -16(2)2 + 40(2) + 6 = ____ feet

Example #2:

The distance of a diver above the water h(t) (in feet) t seconds after diving off a platform is modeled by the equation h(t) = -16t2 +8t +30.

a) How long does it take the diver to reach her maximum height after diving off the platform?

b) When will the diver reach a height of 2 feet?

c) What is her maximum height?

d) When will the diver hit the water?

e) How high is the diver after 1.5 seconds?

f) How high is the diving board?

Example #3:

The height, H meters, of a rocket t seconds after it is fired vertically upwards is given by h(t) = -50t2 + 80t.

a) When will the rocket hit the ground?

b) What is the highest point that the rocket reaches?

c) When does it reach the highest point?

c) At what time(s) is the rocket at a height or 25 m?

d) What was the initial height of the rocket?

NC Final exam –can you interpret what this problem is asking?

Challenge problem for candy!

Reasoning What are the solutions of the system y = 2x2 – 11 and y = x2 + 2x – 8? Explain how you solved the system.

Homework

Begin working on the Practice Test for Quadratics!

Due on Thursday (TEST DAY)