Unit 6 Lesson 1: Properties of Parabolas

Example #1: Determine if the function is linear or quadratic. Identify the constant, linear, and quadratic term.

1) y = (2x + 3)(x -4) Constant Term ____________Linear Term ________________Quadratic Term ______________Linear or Quadratic __________________

2) y = 3(x2 – 3x) – 3(x2 – 2)Constant Term ____________Linear Term ________________Quadratic Term ______________Linear or Quadratic __________________

Example #2: Below is the graph of y = 2x2 – 8x + 8. Identify the vertex and axis of symmetry, x & y intercepts, domain & range.

Vertex: _____Axis of Symmetry: _____x-intercept(s): ______________y-intercept: ______________Domain: ____________Range: _____________You Try:

When a is positive,Domain: (−∞ ,∞) Range: [k, ∞)

When a is negative,Domain: (−∞ ,∞) Range: (−∞ ,k]

Vertex Coordinates: (h, k)Axis of symmetry: x = h

Question #1: Determine if f(x) = (x2 + 5x) – x2 is a linear or quadratic function. Identify the quadratic, linear, and constant term.

Question #2: Identify the vertex and axis of symmetry, x & y intercepts, domain & range of the graph below.

Vertex: _____

Axis of Symmetry: _____

x-intercept(s): ______________

y-intercept: ______________

Domain: ____________

Range: _____________

Transformation Rules revised…..

Quadratic Rule Definition/Picture Example

f(x) = (x – h)2 Right h units f(x) = (x – 3)2

f(x) = (x + h)2 Left h units f(x) = (x + 6)2

f(x) = x2 + k Up k units f(x) = x2 + 4

f(x) = x2 - kDown k unit

f(x) = x2 - 5

f(x) = +x2 Positive a; opens up f(x) = x2

f(x) = -x2 Negative a; opens down f(x) = -x2

f(x) = ax2

If |a|>¿ 1 skinny f(x) = 3x2

If |3|>¿ 1 f(x) = ax2

If 0 ¿ ⌈ a ⌉<1wide f(x) = 14x2

If 0 ¿ ⌈ 14 ⌉<1f(x) = ax2

If a = 1no change f(x) = x2

If a = 1

The starting location is always the vertex (h, k)….

Examples: Describe the transformation. Then, sketch the graph.

1) y = (x – 4)2 + 3Description: _______________________________________________________

______________________________________________________________________________________________________________Graph:

2) y = -2 (x + 6)2 – 1Description: _____________________________________________________________________________________________________________________________________________________________________Graph:

Unit 6 Lesson 2: Solving Quadratics

Quadratic Formula:

X = −b+√b2−4 ac2a x = −b−√b2−4ac

2a

Examples: Solve using quadratic formula.

1)3x2 – 5x = 2

2)2x2 = -6x – 7

3)3x2 + 4x + 10 = 0

Unit 6 Lesson 3: Analyzing Quadratic Functions

The coefficient of the quadratic term, a, tells us 2 things Concavity Wide or skinny

Concave Up a is positive, a > o

Concave Down a is negative, a < 0

Wide 0 ¿ |a| ¿ 1

Skinny |a| ¿ 1

Identify the important parts from the equation in standard form: y = ax2 + bx + cVertex (h, k)

h = −b2a , k = a(h)2 + b(h) + c

Axis of symmetry: x = hy-intercept = cx-intercept(s) – factor & solve for xDomain: (−∞ ,∞) Range: [k, ∞), if concave UP, or (-∞ ,k] if concave DOWN!Examples: Analyze the following quadratic function.

1) f(x) = x2 + 8x + 7Vertex: __________________________Axis of Symmetry: ________________y-intercept: _____________________Concave Up or Down: ________________Wide or Skinny: _______________________x-intercepts: __________________________Domain: ______________________Range: ________________________

2) f(x) = -2x2 + 11x – 15Vertex: __________________________Axis of Symmetry: ________________y-intercept: _____________________Concave Up or Down: ________________

Wide or Skinny: _______________________x-intercepts: __________________________Domain: ______________________Range: ________________________

3) f(x) = 4x2 – 4x – 15Vertex: __________________________Axis of Symmetry: ________________y-intercept: _____________________Concave Up or Down: ________________Wide or Skinny: _______________________x-intercepts: __________________________Domain: ______________________Range: ________________________

Unit 6 Lesson 4: Different Forms of Quadratic Functions

Standard Form: y = ax2 + bx + cFactored Form: y = a(x – xi1)(x – xi2)Vertex Form: y = a(x – h)2 + k

Example #1: Write in Standard form. Analyze the quadratic function.

1)f(x) = (x – 4)(x – 5)Standard form: _________________Vertex: ________________________Axis of Symmetry: ________________y-intercept: _____________________Concave Up or Down: ________________Wide or Skinny: _______________________x-intercepts: __________________________Domain: ______________________Range: ________________________

2)f(x) = 3x(x + 4)Standard Form: _________________________________

Steps…

1) Subtract/Add c from both sides

2) Complete the square

3) Add the completed square to

BOTH sides

4) Factor left side; simplify right side.

5) Write the factored expression as a

square.

6) Add simplified right side back to

the Left side.

Vertex: __________________________Axis of Symmetry: ________________y-intercept: _____________________Concave Up or Down: ________________Wide or Skinny: _______________________x-intercepts: __________________________Domain: ______________________Range: ________________________

You Try….3)f(x) = -2(x + 1)(x -5)

Standard Form: ____________________________Vertex: __________________________Axis of Symmetry: ________________y-intercept: _____________________Concave Up or Down: ________________Wide or Skinny: _______________________x-intercepts: __________________________Domain: ______________________Range: ________________________

Example #2 Write in vertex form.1)X2 + 8x + 7 = 0

Add/Subtract C from BOTH sidesX2 + 8x = ____________

Complete the SquareX2 + 8x + __________ = -7 + ___________

Factor Left Side; simplify Right(x + ______)(x + _____) = ________

Write the factored expression as a square(x + _____)2 = _______

Remember, completing the square…

(b2)2

Add/Subtract the simplified right side back to the left side(x + _____)2 + ____________ = 0

2)x2 + 10x – 25 = 36Add/Subtract C from BOTH sidesX2 + 10x = ____________

Complete the SquareX2 + 10x + __________ = 61 + ___________

Factor Left Side; simplify Right(x + ______)(x + _____) = ________

Write the factored expression as a square(x + _____)2 = _______

Add/Subtract the simplified right side back to the left side(x + _____)2 + ____________ = 0

You Try…3)x2 – 6x = -22

Add/Subtract C from BOTH sidesX2 - 6x = ____________

Complete the SquareX2 - 6x + __________ = ___________

Factor Left Side; simplify Right(x - ______)(x - _____) = ________

Write the factored expression as a square(x - _____)2 = _______

Add the simplified right side back to the left side(x - _____)2 + ____________ = 0

Example #3: Write in standard form. Analyze the quadratic.

1)y = (x – 4)2 + 6Standard Form: ________________________________

Vertex: __________________________Axis of Symmetry: ________________y-intercept: _____________________Concave Up or Down: ________________Wide or Skinny: _______________________x-intercepts: __________________________Domain: ______________________Range: ________________________

2)y = -2(x + 1)2 – 5Standard Form: ___________________________Vertex: __________________________Axis of Symmetry: ________________y-intercept: _____________________Concave Up or Down: ________________Wide or Skinny: _______________________x-intercepts: __________________________Domain: ______________________Range: ________________________

You Try… Y = 3(x – 2)2 + 4Standard Form: ________________________________Vertex: __________________________Axis of Symmetry: ________________y-intercept: _____________________Concave Up or Down: ________________Wide or Skinny: _______________________x-intercepts: __________________________Domain: ______________________Range: _______________________

Unit 6 Lesson 5: Quadratic Word Problems

You are either finding the vertex or finding the x-intercepts…

Key words or phrases to look for when finding the vertex… Maximum or minimum

Key word or phrases to look for when finding the x-intercepts...

“Hit the ground” “When time is zero” “How Long”

Examples: Solve the word problem.Question #1:The Big Brick Bakery sell more bagels when it reduces its process, but then its profit changes. The function y = -1000(x - .55)2 + 300 models the bakery’s daily profit in dollars, from selling bagels, where x is the price of a bagel in dollars.

a. Can the domain of the function be negative?

b. What is the price that will maximum the profit?

c. What is the maximum profit?

Question #2:A smoker jumper jumps from a plane that is 1700 ft above the ground. The functions y = -16t2+ 1700 gives the jumper’s height y in feet at t seconds.

a. How long is the jumper in free if the parachute opens at 940 ft?

b. When will the jumper land?

c. What is the jumper maximum height?

Question #3:Suppose you throw a ball straight up in the ground. As the ball moves upward, gravity slows it. The height of the ball after t seconds in the air is modeled by the quadratic h(t) = -16t2+ 80t

a. How high does the ball go?

b. What time will the ball reach its maximum height?

c. When will the ball hit the ground?

Unit 6 Lesson 6 Linear A and Quadratic Systems

A linear – quadratic system of equation is where a linear and a quadratic with the same set of unknowns (x, y). In a linear-quadratic system one of 3 situations may occur:

2 Solutions 1 solution No solution

How do you find the solutions?1st – Solve one of the equations for one of its variable (x or y).2nd – Substitute the expression from the first step into the other equation in order to solve for the other variable.3rd – Substitute the answer from the second step in the one of the original equations and solve.

Examples: Solve the system below using substitution.

1) { y=2 x−4y=x2−4 x+1

2) { y=−x+5y=−x2−4 x+5

3) { y=−12x+4

y=−x2−2 x+2

You Try….

{ y=−2 x−3y=2 x2+4 x−3