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ALGEBRA 1UNIT 8
(10 LESSONS + 2 QUIZ + 3 PRACTICE +1 STUDY GUIDE + 1 TEST = 17 DAYS)
Date Lesson Plan Standard(s) OtherW 3/11 8-N1 Compare Linear vs. Quadratic, F.IF.1, F.IF.4, F.IF.9, COMMON CORE WARM UP
Verify Point as Solution, A.REI.10, AREI.11 & Hand Out HW Set #23 Determine Upward/DownwardAxis of Symmetry, Compare Max/Min
Th 3/12 8-N2 Find Zeros A.SSE.3 Start Warm Ups
F 3/13 8-N3 Polynomial Zeros F.BF.3
M 3/16 PRACTICE Warm Up Quiz
T 3/17 QUIZ COMMON CORE WARM UP & Collect HW Set #23
& Hand Out HW Set #24
W 3/18 8-N4 Quadratic-Linear Systems of Equations A.REI.6 Start Warm Ups
Th 3/19 8-N5 Appropriate Domain
F 3/20 8-N6 Quadratic Word Problems F.BF.1, A.CED.1 Warm Up Quiz& Collect HW Set #24
M 3/23 8-N7 Quadratic Word Problems F.BF.1, A.CED. COMMON CORE WARM UP & Hand Out HW Set #25
T 3/24 8-N8 Quadratic Word Problems F.BF.1, A.CED. COMMON CORE WARM UP
W 3/25 PRACTICE COMMON CORE WARM UP
Th 3/26 8-N9 Rewrite Standard Form to Vertex Form F.IF.8 Start Warm Ups& Collect HW Set #25
F 3/27 STAFF DEVELOPMENT DAY
M 3/30 8-N10 Transformations F.BF.3 Hand Out HW Set #26
T 3/31 PRACTICE Warm Up Quiz
W 4/1 QUIZ COMMON CORE WARM UP
Th 4/2 STUDY GUIDE
F 4/3 TEST Collect HW Set #26
8 – N1Today, you will be able to:
1. Which point is not on the graph represented by y=x2+3 x−6?
(1) (−6 , 12 ) (3) (2 , 4 )
(2) (−4 , −2 ) (4) (3 , −6 )
2. Which ordered pair below is not a solution to ?
(1) (3)
(2) (4)
Recall that the standard form of a quadratic equation is ________________.
If a is positive, the parabola will open ________________ and it will have a _____________________.
If a is negative, the parabola will open _______________, and it will have a ______________________.
3. Given the function f ( x )=−x2+8 x−9 , state whether the vertex represents a maximum or minimum point for the function. Explain your answer.
The y-intercept of a parabola is represented by _____ in the standard form.
State the y-intercept:
4. y=3 x2+6 x−4
5. y=x2+7
6. y =- x2+14 x
7. Using the x/y chart for y=x2+2 x−15
Is this a linear equation or a quadratic equation?
Does the point (1 ,−12 )
lie on the parabola?
What is the vertex?
What is the axis of symmetry?
What is the minimum value?
What are the roots?
8. But, what is the vertex of ?
IF YOU CAN’T FIND IT IN THE x/y CHART, USE ____________!
9. Given
Using the calculator, what is the vertex?
What is the axis of symmetry?
What is the minimum value?
What are the zeros?
10. Which of the following quadratic functions below has the smallest minimum value?
You can also find the axis of symmetry and the vertex _________________.
Recall that the standard form for a quadratic is __________________.
AXIS OF SYMMETRY FORMULA:
Find the axis of symmetry and vertex algebraically:
11. y=x2+6 x+5
Axis of Symmetry:
Vertex:
12.
Axis of Symmetry:
Vertex:
8 – N2Today, you will be able to:
“TOP FIFTEEN” TOPICThe values where the parabola crosses the x-axis are called the
______________________ or ____________________.
These can also be called the _____________________ of the quadratic equation.
1. How many zeros does each of the following quadratic equations have?
2. Graph the parabola y=x2−4
Using your graph, what are the zeros
of the equation x2−4=0 ?
However, you can also find the solutions algebraically!
Find the zeros of algebraically
3. What are the zeros of the 4. What are the zeros of the parabola whose equation is parabola whose equation is f ( x )=x2−6 x ? f ( x )=x2−2 x−3?
5. What are the zeros of the 6. What are the zeros of the parabola whose equation is parabola whose equation is
f ( x )=x2−4 x+5 ?
Sometimes, you can find the zeros using the table on the calculator!
7. The zeros of the function f ( x )=x2−5 x−6 are
(1) 2 and -3 (3) -1 and 6
(2) -2 and 3 (4) 1 and -6
8. The zeros of the function f ( x )=3 x2−3 x−6 are
(1) -1 and -2 (3) 1 and 2
(2) 1 and -2 (4) -1 and 2
And sometimes, you can’t use the calculator!
9. What are the zeros of the function f ( x )=x2+4 x−16 ?
(1) 2±2√5 (3) 2±4√5
(2) −2±2√5 (4) −2±4√5
8 – N3Today, you will be able to:
1. The function r(x) is defined by the expression x2+3 x−18 . Use
factoring to determine the zeros of r(x).
Explain what the zeros represent on the graph of r(x).
2. Explain how to determine the zeros of .
State the zeros of the function.
3. Keith determines the zeros of the function f(x) to be -6 and 5. What could be Keith’s function?
(1) f ( x )=( x+5 ) ( x+6 ) (3) f ( x )=( x−5 ) ( x+6 )
(2) f ( x )=( x+5 ) ( x−6 ) (4) f ( x )=( x−5 ) ( x−6 )
4. Write an equation for the function represented below.
5. If the zeros of a quadratic function, F, are -3 and 5, what is the equation of the axis of symmetry of F? Justify your answer.
6. Based on the graph below, which expression is a possible factorization of p(x)?
(1) ( x+3 ) ( x−2 ) (x−4 ) (3) ( x+3 ) ( x−5 ) ( x−2 ) ( x−4 )
(2) ( x−3 ) ( x+2 ) (x+4 ) (4) ( x−3 ) ( x+5 ) ( x+2 ) ( x+4 )7. A polynomial function contains the factors , , and .
Which graph(s) below could represent the graph of this function?
(1) I, only (3) I and III
(2) II, only (4) I, II, and III
8. Which equation(s) represent the graph below?
I y= (x+2 ) ( x2−4 x−12 )
II y= (x−3 ) ( x2+x−2 )
III y= (x−1 ) ( x2−5 x−6 )
(1) I, only (3) I and II
(2) II, only (4) II and III9. Wenona sketched the graph of the polynomial P(x) as shown on the
axes below.
Which equation could represent P(x)?
(1) P( x )= (x+1 ) ( x−2 )2 (3) P( x )= (x+1 ) ( x−2 )
(2) P( x )= (x−1 ) ( x+2 )2 (4) P( x )= (x−1 ) ( x+2 )
UNIT 8 – PRACTICE #1
Answer the following multiple choice questions.
1. What is the equation for the axis of symmetry of the parabola
represented by y=−x2+12x−4 ?
(1) x=12 (2) y=12 (3) x=6 (4) y=6
2. What are the coordinates of the vertex of the parabola whose equation
is y=2 x2+12 x+10 ?
(1) (−3 , 64 ) (2) (−3 , −8 ) (3) (3 ,−20 ) (4) (3 ,−8 )
3. The roots of the parabola whose equation is y=x2−2 x−48 are
(1) -6 and -8 (2) -6 and 8 (3) 6 and -8 (4) 6 and 8
Answer the following.
4. Given y=x2−8 x+7 :
Is this a linear equation or a quadratic equation?
Is (2 ,−5 ) a point on this parabola?
What is the y-intercept of this parabola?
Will this parabola open upward or downward?
Will this parabola have a maximum or a minimum?
Find the axis of symmetry algebraically.
Find the vertex algebraically.
Find the roots algebraically.
5. Given y=x2−6 x :
Is this a linear equation or a quadratic equation?
Is (1 ,−5 ) a point on this parabola?
What is the y-intercept of this parabola?
Will this parabola open upward or downward?
Will this parabola have a maximum or a minimum?
Find the axis of symmetry algebraically.
Find the vertex algebraically.
Find the roots algebraically.
Answer the following multiple choice questions.
6. Which quadratic function has the largest maximum over the set of real numbers?
7. Which polynomial function has zeros at -3, 0, and 4?
(1) (3)
(2) (4)
Answer the following.
8. The graph of the function is given below.
Could the factors of be and ? Based on the graph, explain why or why not.
8 – N4
Today, you will be able to:
1. Solve the system of equations graphically. y=x2−6 x+6y=x−4
Now, solve that same system of equations algebraically.
y=x2−6 x+6y=x−4
2.y=x2+4 x+4y=x+2
3.y=x2+4y=−4 x
4. Iff ( x )=x2 and g( x )=x , determine the value(s) of x that satisfy the
equation f ( x )=g( x ) .
5. If f ( x )=x2−2 x−8 and g( x )=1
4x−1
, for which values of x is f ( x )=g( x )?
8 – N5
Today, you will be able to:
1. The function h (t )=−16 t 2+144 represents the height, h(t), in feet, of an object from the ground at t seconds after it is dropped. A realistic domain for this function is
(1) −3≤t≤3 (3) 0≤h( t )≤144
(2) 0≤t≤3 (4) all real numbers
2. A toy rocket is launched from the ground straight upward. The height of the rocket above the ground, in feet, is given by the equation
h (t )=−16 t2+64 t , where t is the time in seconds. Determine the domain for this function in the given context. Explain your reasoning.
3. Morgan throws a ball up into the air. The height of the ball above the
ground, in feet, is modeled by the function h ( t )=−16 t2+24 t , where t represents the time, in seconds, since the ball was thrown. What is the appropriate domain for this situation?
(1) 0≤t≤1. 5 (3) 0≤h( t )≤1 . 5
(2) 0≤t≤9 (4) 0≤h( t )≤9
4. Let h ( t )=−16 t2+64 t +80 represent the height of an object above the ground after t seconds. Determine the number of seconds it takes to achieve its maximum height. Justify your answer.
State the interval, in seconds, during which the height of the object decreases. Explain your reasoning.
5. The height, H, in feet, of an object dropped from the top of a building
after t seconds is given by .
How many feet did the object fall between one and two seconds after it was dropped?
Determine, algebraically, how many seconds it will take for the object to reach the ground.
6. When an apple is dropped from a tower 256 feet high, the function
models the height of the apple, in feet, after t seconds. Determine, algebraically, the number of seconds it takes the apple to hit the ground.
7. An Air Force pilot is flying at a cruising altitude of 9000 feet and is forced
to eject from her aircraft. The function h (t )=−16 t2+128 t+9000 models the height, in feet, of the pilot above the ground, where t is the time, in seconds, after she is ejected from the aircraft.
Determine and state the vertex of h(t). Explain what the second coordinate of the vertex represents in the context of the problem.
After the pilot was ejected, what is the maximum number of feet she was above the aircraft’s cruising altitude? Justify your answer.
8. The height of the rocket, at selected times, is shown in the table below.
Based on these data, which statement is not a valid conclusion?
(1) The rocket was launched from a height of 180 feet.
(2) The maximum height of the rocket occurred 3 seconds after launch.
(3) The rocket was in the air approximately 6 seconds before hitting the ground.
(4) The rocket was above 300 feet for approximately 2 seconds.
8 – N6Today, you will be able to:
1. John and Sarah are each saving money for a car. The total amount of money John will save is given by the function f ( x )=60+5 x . The total amount of money Sarah will save is given by the function g( x )=x2+46 . After how many weeks, x, will they have the same amount of money saved? Explain how you arrived at your answer.
2. Sam is 5 years older than Jacob. The product of their ages is 84. How old is Sam?
3. The length of a rectangle is 4 inches more than its width. The area of the rectangle is 45 square inches. What is the length of the rectangle?
4. Stan and Harold have ages that are consecutive odd integers. The product of their ages is 63. If Stan is the younger man, how old is Harold?
5. Sadie and Paige have ages that are consecutive even integers. The product of their ages is 840. Which equation could be used to find Paige’s age, p, if she is the younger girl?
(1) p2+2=840 (3) p2+2 p=840
(2) p2−2=840 (4) p2−2 p=840
6. Abigail’s and Gina’s ages are consecutive integers. Abigail is younger than Gina and Gina’s age is represented by x. If the difference of the square of Gina’s age and eight times Abigail’s age is 17, how old is Gina?
8 – N7Today, you will be able to:
1. A school is building a rectangular soccer field that has an area of 6000 square yards. The soccer field must be 40 yards longer than its width. Determine algebraically the dimensions of the soccer field, in yards.
2. A landscaper is creating a rectangular flowerbed such that the width is half of the length. The area of the flowerbed is 34 square feet. Write and solve an equation to determine the width of the flowerbed, to the nearest tenth of a foot.
3. Joe has a rectangular patio that measures 10 feet by 12 feet. He wants to increase the area by 50% and plans to increase each dimension by equal lengths, x. Which equation could be used to determine x?
(1) (10+x ) (12+x )=120 (3) (15+x ) (18+ x )=180
(2)(10+x ) (12+x )=180 (4) (15 ) (18 )=120+x2
4. A contractor has 48 meters of fencing that he is going to use as the
perimeter of a rectangular garden. The length of one side of the
garden is represented by x, and the area of the garden is 108 square
meters.
Determine, algebraically, the dimensions of the garden in meters.
8 – N8Today, you will be able to:
1. A rectangular garden measuring 12 meters by 16 meters is to have a walkway installed around it with a width of x meters. Together, the walkway and the garden have an area of 396 square meters.
a) Write an equation that can be used to find x, the width of the walkway.
b) Describe how your equation models the situation.
c) Determine and state the width of the walkway, in meters.
2. New Clarendon Park is undergoing renovations to its gardens. One garden that was originally a square is being adjusted so that one side is doubled in length while the other side is decreased by three meters.
The new rectangular garden will have an area that is 25% more than the original square garden. Write an equation that could be used to determine the length of a side of the original square garden.
Explain how your equation models the situation.
Determine the area, in square meters, of the new rectangular garden.
UNIT 8 – PRACTICE #2
Answer the following multiple choice question.
1. The expression represents the height, in meters, of a toy rocket t seconds after launch. The initial height of the rocket, in meters, is
(1) 0 (3) 4.9
(2) 2 (4) 50
Solve the following quadratic-linear systems of equations ALGEBRAICALLY.
2.
y=x2−xy =- x+4
3.
y=x2+2 x+1y−x=3
Solve the following word problems using an algebraic solution.
4. Find two positive consecutive integers whose product is 110.
5. The length of a rectangle is 5 cm more than its width. The area is 66 cm2. Find the dimensions of the rectangle.
Solve the following word problem using an algebraic solution.
6. A rectangular picture measures 4 inches by 6 inches. Hayden wants to build a wooden frame for the picture so that the framed picture takes up an area of 48 square inches on his wall. The pieces of wood that he uses to build the frame all have the same width.
Write an equation that could be used to determine the width of the pieces of wood for the frame Hayden could create.
Explain how your equation models the situation.
Solve the equation to determine the width of the pieces of wood used for the frame.
8 – N9Today, you will be able to:
The standard form of a quadratic is ______________________________.
However, this form makes quadratics difficult to graph without the use of a calculator.
There is another form that makes this easier.
The VERTEX FORM of a quadratic is ____________________________.
We can change a quadratic from standard form into vertex form by
____________________________________!
1. Write y=x2+10 x+22 in vertex form.
2. Write y=x2+8 x+11 in vertex form.
When a parabola is written in vertex form, __________________________,
the point (h , k ) is the _______________________ of the parabola.
3. State the coordinates of the vertex of the parabola whose equation is
y= (x+6 )2+4 .
4. State the coordinates of the vertex of the parabola whose equation is
y= (x−3 )2−8 .
5. State the coordinates of the vertex of the parabola whose equation is
y= (x−1 )2+9 .
6. Find the vertex of the parabola whose equation is y=x2−4 x+1 .
7. Find the vertex of the parabola whose equation is y=x2−16 x+70 .
8. Find the vertex of the parabola whose equation is y=x2+8 x+10 .
9. Let f ( x )= (x+3 )2−7 . Which of the following shows f ( x ) ?
(1) (3)
(2) (4)
10. Given the function , state whether the vertex represents a maximum or a minimum point for the function. Explain your answer.
Rewrite in vertex form by completing the square.
11. In , the minimum value occurs when x is
(1) -2 (3) -4
(2) 2 (4) 4
12. If Lylah completes the square for in order to
find the minimum, she must write in the general form
. What is the value of a for ?
(1) 6 (3) 12
(2) -6 (4) -12
13. The function can be written in vertex form as
(1) (3)
(2) (4)
14. Write y=−x2+14 x−48 in vertex form.
8 – N10Today, you will be able to:
The parent function for a family of quadratic functions is ______________.
Recall, when we add or subtract a constant to the parent function, this results
in a transformation called a _______________________________.
1. Compared to its parent function f ( x )=x2, what effect will we see in
the graph of f ( x )−3 ?
(1) translated 3 units left (3) translated 3 units up
(2) translated 3 units right (4) translated 3 units down
2. Given y =- x2 . If the function is shifted up 5 units, which equation describes the new function?
(1) y =- x2−5 (3) y =- ( x+5 )2
(2) y =- x2+5 (4) y =- ( x−5 )2
In addition to shifting the graph of a parabola up or down, we can also shift the graph of a parabola to the left or to the right.
This is called a ________________________________________.
By f ( x ) changing to f ( x+h ) , the parabola will shift h units to the ________.
By f ( x ) changing to f ( x−h ) , the parabola will shift h units to the ________.3. Compared to its parent function f ( x )=x2
, what effect will we see in the graph of f ( x+2 ) ?
(1) translated 2 units left (3) translated 2 units up
(2) translated 2 units right (4) translated 2 units down
4. Compared to its parent function f ( x )=x2, what effect will we see in
the graph of f ( x−4 )?
(1) translated 4 units left (3) translated 4 units up
(2) translated 4 units right (4) translated 4 units down
REFLECTION:Recall that the standard form of a quadratic equation is ________________.By changing the sign of a, you ________________ over the x-axis!
STRETCHING AND SHRINKING:By changing the coefficient of a, your parabola can stretch or shrink!
If a>1 , then the parabola will vertically ___________________.
If 0<a<1 , then the parabola will vertically _____________________.
5. How does the graph of f ( x )=4 ( x+5 )2+3 compare to the graph of
g ( x )=x2?
(1) wider and its vertex is moved to the left 5 units and up 3 units
(2) wider and its vertex is moved to the right 5 units and up 3 units
(3) narrower and its vertex is moved to the left 5 units and up 3 units
(4) narrower and its vertex is moved to the right 5 units and up 3 units
6. How does the graph of f ( x )=1
2( x−4 )2
compare to the graph of
g ( x )=x2?
(1) wider and its vertex is moved to the left 4 units
(2) wider and its vertex is moved to the right 4 units
(3) narrower and its vertex is moved to the left 4 units
(4) narrower and its vertex is moved to the right 4 units
7. How does the graph of f ( x )=−x2+7 compare to the graph of
g ( x )=x2?
(1) opens upward and its vertex is moved up 7 units
(2) opens upward and its vertex is moved down 7 units
(3) opens downward and its vertex is moved up 7 units
(4) opens downward and its vertex is moved down 7 units
8. The vertex of the parabola represented by f ( x )=x2−4 x+3 has
coordinates (2 ,−1 ) . Find the coordinates of the vertex of the parabola
defined by g ( x )=f ( x−5 ) . Explain how you arrived at your answer.
9. The graph of is shown below.
What is the graph of ?
10. Richard is asked to transform the graph of b(x) below.
The graph of b(x) is transformed using the equation h ( x )=b ( x−2 )−3 . Describe how the graph of b(x) changed to form the graph of h(x).
11. The graph of the function is represented below. On the same set of
axes, sketch the function .
UNIT 8 – PRACTICE #3
Answer the following multiple choice questions.
1. What is the equation for the axis of symmetry of the parabola
represented by y=−x2−8 x+1?
(1) x=17 (2) y=17 (3) x=−4 (4)y=−4
2. What are the coordinates of the vertex of the parabola whose equation
is y=3 x2+6 x−4 ?
(1) (−1 , −7 ) (3) (1 , 5 )
(2) (−1 , −13 ) (4) (1 , −7 )
3. The roots of the parabola whose equation is y=x2−2 x−24 are
(1) -6 and -4 (2) -6 and 4 (3) 6 and -4 (4) 6 and 4
Answer the following.
4. Given y=x2+14 x−15 :
Is this a linear equation or a quadratic equation?
Is (2 , 17 ) a point on this parabola?
What is the y-intercept of this parabola?
Will this parabola open upward or downward?
Will this parabola have a maximum or a minimum?
Find the axis of symmetry algebraically.
Find the vertex algebraically.
Find the roots algebraically.
Answer the following multiple choice question.
5. The height of a ball Doreen tossed into the air can be modeled by the
function , where x is the time elapsed in
seconds, and is the height in meters. The number 5 in the function represents
(1) the initial height of the ball
(2) the time at which the ball reaches the ground
(3) the time at which the ball was at its highest point
(4) the maximum height the ball attained when thrown into the air
Solve the following quadratic-linear system of equations ALGEBRAICALLY.
6.
y=x2+5 x−30y=2 x+10
Solve the following word problem ALGEBRAICALLY.
7. The length of a rectangle is 8 feet more than its width. If the area of the rectangle is 20 square feet, what is the length, in feet?
Answer the following questions.
8. Write y=x2+10 x+27 in vertex form and state the vertex.
9. Write y=x2−12 x+34 in vertex form and state the vertex.
Answer the following multiple choice questions.
10. How does the graph of f ( x )= 1
4( x+2 )2+1
compare to the graph of g( x )=x2
?
(1) The graph of f(x) is wider than the graph of g(x), and its vertex is moved to the left 2 units and up 1 unit.
(2) The graph of f(x) is narrower than the graph of g(x), and its vertex is moved to the right 2 units and up 1 unit.
(3) The graph of f(x) is narrower than the graph of g(x), and its vertex is moved to the left 2 units and up 1 unit.
(4) The graph of f(x) is wider than the graph of g(x), and its vertex is moved to the right 2 units and up 1 unit.
Answer the following question.
11. The graph of the function is represented below. On the same set
of axes, sketch the function .
UNIT 8 STUDY GUIDE
8-N1 through 8-N2
1. Which of the following points lies on the parabola whose equation is y=x2−9 x+17?
(1) (−3 , 50 ) (3) (0 , 17 )
(2) (−1 , 25 ) (4) (2 ,−1 )
2. Given the equation, y=x2−4 x−5
Is this a linear equation or a quadratic equation?
Does the parabola open upward or open downward?
Does the parabola have a maximum point or a minimum point?
What is the y-intercept?
What is the axis of symmetry?
What are the coordinates of the vertex?
What are the roots?
3. The equation of a function f is f ( x )= (x−2 )2+4 .
The equation of a function g is g ( x )=x2+2 x−3 .
Which function has the lower minimum value? Justify your answer.
8-N3
4. Based on the graph below, which expression is a possible factorization of p(x)?
(1) ( x+3 ) ( x−2 ) (x−4 ) (3) ( x+3 ) ( x−5 ) ( x−2 ) ( x−4 )
(2) ( x−3 ) ( x+2 ) (x+4 ) (4) ( x−3 ) ( x+5 ) ( x+2 ) ( x+4 )
8-N4
5. Solve the quadratic-linear system of equations algebraically.
y=x2−xy−x=3
8-N5
6. A toy rocket is launched from the ground straight upward. The height of the rocket above the ground, in feet, is given by the equation h ( t )=−16 t2+64 t , where t is the time in seconds. Determine the domain for this function in the given context. Explain your reasoning.
8-N6 through 8-N8
7. The product of two positive consecutive odd integers is 35. Find the integers. [Only an algebraic solution will be accepted.]
8. A rectangular garden measuring 12 meters by 16 meters is to have a walkway installed around it with a width of x meters. Together, the walkway and the garden have an area of 572 square meters.
a) Write an equation that can be used to find x, the width of the walkway.
b) Describe how your equation models the situation.
c) Determine and state the width of the walkway, in meters.
8-N9
9. Write y=x2+22 x+118 in vertex form.
10. What is the vertex of the parabola whose equation is y= (x−4 )2−12 ?
8-N10
11. The vertex on the graph of the equation y= f ( x ) is (4,7 ) . What is the vertex on the graph of the equation y= f ( x )−6 ? Explain your answer.
12. The vertex on the graph of the equation y= f ( x ) is (2,3 ) . What is the vertex on the graph of the equation y= f ( x+8 ) ? Explain your answer.
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