Regents Review #3

FunctionsQuadratics & More

Quadratic FunctionsThe graph of a quadratic function is a U-shaped curve called a parabola. The vertex (turning point) is the lowest point (minimum point) of the parabola when it opens up and the highest point (maximum point) when it opens down.

Quadratic FunctionsHow do we graph quadratic functions in standard form?

Standard Form: y = ax2 + bx + c

1)Find the coordinates of the vertex 2)Create a table of values (vertex in the middle)3)Graph the function (parabola)4)Label the graph with the equation

2ab

x

Quadratic FunctionsGraph y = x2 + 4x + 3

Finding the coordinates of the vertex and the axis of symmetry

Finding the x-coordinate

y = x2 + 4x + 3 a = 1, b = 4

x =

a

b

2

22

4

)1(2

4

Finding the y-coordinate

y = x2 + 4x + 3

x =

x = -2

y = (-2)2 + 4(-2) + 3

y = 4 – 8 + 3

y = -1

Vertex = (-2, -1)

Quadratic Functions

x y

-5 8

-4 3

-3 0

-2 -1

-1 00 3

1 8

x-intercept(-3, 0) Root: -3

x-intercept(-1, 0)Root: -1

Axis of Symmetryx = -2 y= x2 + 4x + 3

Domain: {x | x is all Real Numbers}Range: {y | y > -1}

Vertex

Quadratic FunctionsThe roots (zeros) of a quadratic function are the x-values of the x-intercepts.

2 Roots 1 Roots(2 equal roots)

No real roots

Root: 3

Root: -1

Quadratic FunctionsWe can identify the “roots” of a quadratic function by looking at the graph of a parabola.

We can also identify the roots algebraically.

Consider the graph of y = x2 + x – 6

Set y equal to zero and solve for x. x-intercept: (x, 0)

The function expressed in factored form is y = (x + 3)(x – 2)The roots are {-3, 2}

0 = x2 + x – 60 = (x + 3)(x – 2)x + 3 = 0 x – 2 = 0 x = -3 x = 2

Quadratic FunctionsHow does a affect the graph of y = ax2 + bx + c ?

• As the |a| increases, the graph becomes more narrow

• As |a| decreases, the graph becomes wider

• If a > o, the parabola opens up

• If a < 0, the parabola opens down

Quadratic Functions

• opens up or down (a value)• y-intercept (c value)

• opens up or down (a value)• Vertex (h, k)

Rewrite the function f(x) = x2 – 4x + 5 in vertex form by completing the square and identify the vertex of the function.

f(x) = x2 – 4x + 5

y = x2 – 4x + 5

y – 5 = x2 – 4x

y – 5 + 4 = x2 – 4x + 4

y – 1 = (x – 2) 2

y = (x – 2) 2 + 1

f(x) = (x – 2) 2 + 1 Vertex: (2, 1)

(x – 2)(x – 2)

42

4

2

22

b

Quadratic FunctionsParabolic Functions Increase and Decrease

The function is decreasing when x < 2. The function is increasing when x > 2. The function does not increase or decrease at 2.

The function is increasing when x < 1. The function is decreasing when x > 1. The function does not increase or decrease at 1.

Quadratic FunctionsEnd Behavior of Quadratic Functions

a > 0 a < 0 Opens up Minimum point x approaches + ∞ and - ∞ ends (y –values) approach + ∞

Opens down Maximum point x approaches + ∞ and - ∞ ends (y-values) approach - ∞

x y

0 4

2 0

4 4

x y

0 -1

1 0

2 -1

Quadratic FunctionsAn angry bird is launched 80 feet above the ground and follows a path that can be modeled by the function h(t) = -16t2 + 64t + 80 where h(t) represents the birds height after t seconds.A. How long will it take the bird to reach its

maximum height? What is the maximum height the bird will reach?

B. Approximately, how long will it take the bird to hit a target 100 feet above the ground (see graph)?

C. If the bird misses the target, at what time will it hit the ground?

A – Find the Vertex

h(2) = -16(2) 2 + 64(2) + 80h(2) = 144

23264

16)2(64

2ab

x

Vertex: (2, 144)It will take 2 seconds for the bird to reach its maximum height of 144 feet.

B – between 3 and 4 secondsC – The bird will hit the ground at 5 seconds.

ttime

h(t)height

0 80

1 128

2 144

3 128

4 80

5 0

Algebraic ApproachReplace h(t) with the given height and solve for tB. 100 = -16t2 + 64t + 80C. 0 = -16t2 + 64t + 80

Quadratic FunctionsAlgebraic Approach (letters B and C on the previous slide)

B. Approximately, how long will it take the bird to hit a target 100 feet above the ground (see graph)?

}66.3341{.

32

0659.117

32

9341.10

32

...0659.536432

281664

)16(2

)20)(16(46464

2

4

206416

2064160

806416100

2

2

2

2

andx

xandx

x

x

x

a

acbbx

cba

tt

tt

Looking at where the target is located, the bird would have hit the target on its descend (past the vertex) which means the bird would have taken about 3.66 seconds to hit the target located 100 feet above the ground.

C. If the bird misses the target, at what time will it hit the ground?

)(15

0105

)1)(5(160

)54(160

80641602

2

rejecttt

tt

tt

tt

tt

The bird will hit the ground when h(t) = 0 (height = 0 ft). This happens at 5 seconds. We reject -1 because we cannot have negative time.

Absolute Value Functions

Vertex: (0,0)

Domain: {x|x is all Real Numbers}

Range: {y|y > 0}

Function decreases when x < 0

Function increases when x > 0

End Behavior: as x approaches + ∞ and -∞, the ends approach + ∞

Reminder: The graph does not increase or decrease at the vertex

y = xCharacteristics of x y

-2 2

-1 1

0 0

1 1

2 2

y = |x|

Absolute Value Functions

Parent FunctionThe function becomes more narrow (x 2)

The function shifts 3 units right

The function moves 4 units down

Vertex: (3, -4)

Square Root Functions

Domain: {x| x > 0}Range: {y| y > 0}

xy x y

0 0

1 1

4 2

9 3

The domain is restricted because you cannot take the square root of a negative number.

Square Root Functions

Parent Function

Domain: {x| x > 0}Range: {y| y > 0}

Shifts 1 unit right

Domain: {x| x > 1}Range: {y| y > 0}

Shifts 2 units up

Domain: {x| x > 1}Range: {y| y > 2}

xy 1-xy 2 1-xy

Cubic Functions

Domain: {x | x is all Real Numbers} Range: {y | y is all Real Numbers} End Behavior: As x approaches – ∞, the ends of the graph approach – ∞As x approaches + ∞, the ends of the graph approach + ∞

3xy x y

-2 -8

-1 1

0 0

1 1

2 8

Cube Root Functions

Domain: {x | x is all Real Numbers} Range: {y | y is all Real Numbers} End Behavior: As x approaches – ∞, the ends of the graph approach – ∞As x approaches + ∞, the ends of the graph approach + ∞

3 xy x y

-8 -2

-1 -1

0 0

1 1

8 2

Piecewise FunctionsA piecewise function has different rules (equations) for different parts of the domain.

x + 3 x2 3

x + 3

x2 3

x y

-1 2

-2 1

-3 0

-4 -1

x y

-1 1

0 0

1 1

2 4

x y

2 3

3 3

4 3

5 3

Evaluate f (1.5)

f(x) = x2

f(1.5) = (1.5)2

= 2.25

Piecewise FunctionsA STEP FUNCTION is a piecewise function that is discontinuous and constant over a finite number of intervals.

How much does it cost to mail a letter weighing 3 ½ ounces? 45 cents

Jan says that it costs 43 cents to mail a letter weighing 2 ounces. Do you agree or disagree with Jan?Disagree. The point (2, 43) graphed with an open circle tells us that 2 oz. does not cost 43 cents. The closed circle above 2 shows us that the cost of 2 oz. is 41 cents.

Piecewise FunctionsA wholesale t-shirt manufacturer charges the following prices for t-shirt orders:

$20 per shirt for shirt orders up to 20 shirts.$15 per shirt for shirt between 21 and 40 shirts.$10 per shirt for shirt orders between 41 and 80 shirts.$5 per shirt for shirt orders over 80 shirts.

A. Create a graph that models this situation.

B. You've ordered 40 shirts and

must pay shipping fees of $10. How much is your total order?

Solution

If 40 shirts are ordered, each shirt will cost $15.

(40 shirts x $15) + $10 shipping =

$610

Now it’s your turn to review on your own!

Using the information presented today and the study guide posted on halgebra.org,

complete the practice problem set.

Regents Review #4 Thursday, May 22nd

BE THERE!