honors a2-trig unit 5 - lancaster central school … the roots of the following quadratic equations...

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A2-TRIG UNIT 5 PART 1

COMPLETING THE SQUARE (DAY 1)

Warm-Up: Find the roots of 2x2 + x = 15

1) What is a quadratic equation?

2) What are the roots of a quadratic equation? (Algebraically and graphically)

3) What are some methods you know of to find the roots of a quadratic equation?

Procedure 2x2 – 12x + 14 = 0

1. Rearrange the equation:

Get terms with variables on the left hand side.

Get c# by itself on the right hand side

2. If a# > 1 then divide through by a#.

3.

Identify the b#

(half it - square it - add it to both sides) (This will form a perfect square trinomial)

4.

Write Expression as a perfect square trinomial

Simplify the # on the other side

5.

Square root both sides

Put a on the right in front of term

6. Solve for x to find the roots.


Solve each quadratic equation (find the real roots) by completing the square; express

each root in simplest radical form.

1) x2 – 2x = 2 2) 3x2 + 6x – 24 = 0

3) 2x2 – x – 1 = 0 4) x2 + 10x + 3 = -4

5) 05x4x4 2 6) 4x7x2 2


CIRCLES

1) Find the center-radius form of the equation 03y6x4yx 22

What are the center and radius of the circle?

2) In standard form, the equation of a circle is 014y4x3yx 22 . Write the

equation in center-radius form and find the coordinates of the center and the

radius of the circle.

If you are given an equation of a circle in standard form and need center-radius form:

1. Group like variables together – leave a space for a 3rd term for each grouping)

2. Move Constant # to the other side of the = sign

3. Identify the “b#” for each variable group (half it - square it - add it to both sides)

4. FACTOR each trinomial group as a perfect squared binomial

5. Write the equation in center-radius form.


3) In standard form, the equation of a circle is x2 + y2 +10x – 5y – 32 = 0. Write the



4) In standard form, the equation of a circle is x2 + y2 +10x – 5y – 32 = 0. Write the



5) In standard form, the equation of a circle is x2 + y2 – 12y + 25 = 0. Write the equation

in center-radius form and find the coordinates of the center and the radius of the

circle.


THE QUADRATIC FORMULA (DAY 2)

Warm-Up: Find the solution set for:

3a - 5

a+12

Ways to Solve Quadratic Equations Quadratic Formula:

1. Factoring and T-chart

2. Completing the Square

“x =’s negative b, plus or minus the square root of b2 minus 4 a c, all over 2 a”

Solve each quadratic equation (find the real roots) by using the quadratic formula. Express

each root in simplest radical form.

1) x2 – 5x – 3 = 0 2) 2x2 – 4x = 1

3) -9x2 + 2x = -10 4) x

36x2


5) 5x2 – 2 = 3x 6) x2 + x + 2 = 0

7) One leg of a right triangle is 1 centimeter shorter than the other leg and the

hypotenuse is 2 centimeters longer than the longer leg. What are lengths of the

sides of the triangle to the nearest tenth?


METHODS OF FINDING ROOTS OF QUADRATIC EQUATIONS (DAY 3)

Roots of a graph are where _______________________________________________________

METHODS FOR FINDING ROOTS: 1.__________________________

2.__________________________

3.__________________________

4.__________________________

Except for completing the square, the quadratic MUST BE __________________________.

Not all equations can be factored. Completing the square and quadratic formula can

always be used. If work must be shown, you cannot graph it on your calculator.

Find the real roots by FACTORING:

1) x2 – 12x = -27 3) 2x2 + 3 = -7x

2) x2 – 64 = 0 4) xx 155 2

Solve the equation using the QUADRATIC FORMULA. State the exact value of the roots and

to the nearest hundredth.

5) x2 – 5x – 3 = 2 6) 504x3x2


Solve the quadratic GRAPHICALLY.

7) x2 – 12x = -27 8) 2x2 – x – 6 = 0

Find the roots by COMPLETING THE SQUARE:

9) 3x2 - 2x – 4 = 0 10) 0252 2 xx

11) Write the equation of the circle in center-radius form and state the center and radius:

x2 + y2 + 2x – 4y – 11 = 0

12) The endpoints of a diameter of a circle are P(8,3) and Q(-2,-1). Write the equation

of the circle in center-radius form and standard form.


QUADRATIC APPLICATIONS (DAY 4)

The picture below represents the path of an object after being tossed.

1) What are Q and R called?

2) What can you say about the equation of the parabola?

3) If you were given an equation for the

parabola, how can you find Q and R?

4) Which letter represents when the object reaches its maximum height?

5) Which letter represents the objects maximum height?

6) What is the initial height of the object?

7) A ball is thrown straight up at an initial velocity of 54 feet per second. The height of

the ball t seconds after it is thrown is given by the formula h(t) = 54t –12t2.

a) What does the t represent?

b) What does h(t) represent?

c) Find h(1).

d) What does your answer to part c mean?

e) What is the height of the ball when it hits the ground?

P (l, m)

Q R

S


8) An archer shoots an arrow into the air such that its height at any time, t, is given by

the function h(t) = -16t2 + 128t + 3.

a) What does h(t) represent?

b) What is the height of the arrow after a half of a second?

c) How long, to the nearest tenth of a second, will it take for the arrow to reach

the ground?

9) Suppose a ball is thrown upward from a height of 5 feet with an initial velocity

of 35 ft/sec. Using the equation, h(t) = -16t2 + 35t + 5, answer the following questions.

a) Is the ball still in the air after 3 seconds? Explain.

b) When does the ball reach its maximum height? What is the maximum height

of the ball?


10) Brent threw a stone upward at a speed of 10m/sec while standing on a cliff 40m

above the ground. Using the equation, h(t) = -4.9t2 + 10t + 40, answer the following

questions.

a) Find how long it will take for the stone to touch the ground to the nearest

hundredth of a second.

b) At what time will the stone reach its maximum height? What is the maximum

height? Find your answers algebraically.

11) The height of a ball thrown upward is shown as a function of time on the graph.

a) Estimate the initial height of the ball.

b) Approximately when did the ball

reach its maximum height?

c) What was the maximum height?

When was the ball 8m high?


THE DISCRIMINANT AND THE NATURE OF THE ROOTS (DAY 5)

Roots of a quadratic equation:

Find the roots of the following quadratic equations and describe them.

1) 2x2 + 3x – 2 = 0 2) 3x2 + x – 1 = 0

3) 9x2 – 6x + 1 = 0 4) 2x2 + 3x + 4 = 0

REAL

RATIONAL OR IRRATIONAL

EQUAL OR UNEQUAL

VS. NOT REAL

DESCRIBING ROOTS


DISCRIMINANT

When you are asked to DESCRIBE the NATURE OF THE ROOTS, you can just find the

value of the discriminant and use the information from the table below.

DISCRIMINANT

ac4b2 ____

ac4b2 ____

and a

PERFECT SQUARE

ac4b2 ____

and NOT a

Perfect Square

ac4b2 ___

DESCRIPTION

OF THE ROOTS

GRAPH

WILL/LOOKS

LIKE

2

2 1 5


Determine the nature of the roots: a) real, rational, and equal

b) real, rational, and unequal

c) real, irrational, and unequal

d) not real numbers

1) x2 – 3x + 2 = 0 2) x2 + x + 3 = 0 3) x2 + 2x + 1 = 0

4) x2 + 10 = x 5) x2 – 3x = 0 6) 3x2 = 5x – 1

7) If the roots of an equation are real, rational, and equal, the graph will

(1) cross the x-axis at two rational points

(2) be tangent to the x-axis

(3) cross the x-axis at two irrational point

(4) never cross the x-axis

8) The equation 2x2 + 8x + n = 0 has equal. roots when n is equal to

(1) 10 (2) 8 (3) 6 (4) 4

9) Which equation has non-real roots?

(1) x(5 +x) = 8 (3) x(x + 6 ) = -10

(2) x(5 – x) = -3 (4) (2x + 1)(x – 3) = 7

10) Which of the following graphs has a discriminant

equal to zero?


ROOTS OF POLYNOMIAL FUNCTIONS (DAY 6)

Roots (zeros) of a polynomial: Where a _______________ crosses the x-axis.

Degree of a polynomial: Is the _____________ exponent of the polynomial and tells how

many solutions or ___________ the polynomial should have.

Use the picture of the polynomial function to answer the folowing.

1) What is the degree of the function graphed?

2) How many real roots are there?

3) What do the real roots appear to be

from the graph?

4) How can we determine if these are the exact

roots of the equation?

5) Find the real roots of f(x) = x4 – 4x3 – x2 + 16x – 12 graphically.

6) Determine the exact values of the roots of the

function graphed below.


The graph below represents the equation y = x3 + x2 – x – 1.

7) From the equation, how many roots should this relation have?

8) How many are represented in the picture?

Double Root:

9) From the graph below,

a) determine the roots of the polynomial function p(x).

b) determine the equation of p(x).

c) determine the domain and range of the function.

d) determine if it is one-to-one.

10) A function of degree 2 intersects the x-axis at (-3, 0) and (7, 0). Find the equation of

the function h(x).

If “a” is a root of a polynomial function, then __________ is a factor of the polynomial.


Find the exact roots of the following equations.

11) x3 – 25x = 0 12) f(x) = x3 – 2x2 – 9x + 18

13) 2x3 + 14x2 = 10x 14) x4 – 9x3 + 18x2 = 0

15) Using the function f(x) = x4 – 5x2 + x + 2 and d = 2

a) Find f(d) b) Is d a root of the function?


SOLVING SYSTEMS OF EQUATIONS GRAPHICALLY (DAY 7)

Two equations will be given to you with the directions to solve the system graphically.

1) Solve the following system of equations graphically and check.

3x4xy 2

1yx

Graphing a Parabola: You will have to find the _______________ and then create a

____________________ for the quadratic equation

Graphing a Line: State the ___________ & _____________ or create a ___________.

Graphing a Circle: State the _______________ and ______________.

The solution(s) to the system are where the two graphs ________________________.


2) Solve the following system of equations graphically and check.

4x4xy 2

4x2y

3) Solve the following system of equation graphically and check.

x2 + y2 =4

y – x = 2


GRAPHING QUADRATIC INEQUALITIES

4) Graph the following quadratic inequality: y > x2 - 1

5) Graph the following quadratic inequality: y < x2 – 4x - 5


SOLVING SYSTEMS OF EQUATIONS ALGEBRAICALLY (DAY 8)

Two equations will be given to you with the directions to solve the system algebraically.

1) Use the substitution method by getting the x term or y term by itself.

2) Substitute into the other equation.

3) Solve for BOTH variables.

4) Check your solutions into the original equations.

1) Solve the following system algebraically & check: y = x2 – 4x + 3

y + 1 = x

2) Find the solutions algebraically and check: x2 + y2 = 16

x – y = 4


3) Find the solutions algebraically and check:

y = 2x2 – 4x – 5

3x – y = 1

4) Solve the system algebraically & check: xy = -6

x + 3y = 3

honors a2-trig unit 5 - lancaster central school … the roots of the following quadratic equations...

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