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1 ALGEBRA 1 Teacher’s Name: Unit 6 Chapter 9 & 10 This book belongs to: 2015 – 2016
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ALGEBRA 1
Teacher’s Name:
Unit 6
Chapter 9 & 10
This book belongs to:
2015 – 2016
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Unit 6 Part 1
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Algebra 1 Section 9.1 Day 1 Notes: Parts of a Quadratic Graph Warm-up Graph y = x2 by completing the table of values. What observations do you have about the graph? Vocabulary: Quadratic Function: a non-linear function that can be written in the form f(x) = ax2 + bx + c, where a 0. Parabola: graph of a quadratic function Vertex: the maximum or minimum value of the parabola and the point where the axis of symmetry intersects the parabola Axis of Symmetry: a central vertical line that the parabola is symmetric about Graph of a Quadratic Equation The graph of is a parabola in standard form. If a is ____________________, the parabola opens up
If a is____________________, the parabola opens down The vertex has an x-coordinate of
The axis of symmetry is the vertical line
The y-intercept is __________
Domain is
Range is
X
Y
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9.1 Day 1 Notes Continued – Finding the parts of a parabola
1.
Find the Vertex first Step 1: a = b = c =
Step 2: 2
bx
a
Step 3: Evaluate for y.
Step 4: ( , ) Now find – Axis of symmetry: ___________ y‐intercept: _________ Domain: _______ Range: _______
2. 2( ) 1 4 5f x x x
Find the Vertex first Step 1: a = b = c =
Step 2: 2
bx
a
Step 3: Evaluate for y.
Step 4: ( , ) Now find – Axis of symmetry: ___________ y‐intercept: _________ Domain: _______ Range: _______
3. 2 6 3y x x
Vertex: ( , ) Axis of symmetry: ___________ y‐intercept: _________ Domain: _______ Range: _______
4. 2( ) 2 8 7f x x x
Vertex: ( , ) Axis of symmetry: ___________ y‐intercept: _________ Domain: _______ Range: _______
22 4 1y x x
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5. 22 8 1y x x Vertex: ( , ) Axis of symmetry: ___________ y‐intercept: _________ Domain: _______ Range: _______
6. 21 2 1y x x Vertex: ( , ) Axis of symmetry: ___________ y‐intercept: _________ Domain: _______ Range: _______
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Algebra 1 Section 9.1 Day 1 Worksheet #1 Find the vertex, the equation of the axis of symmetry, the y-intercept, domain, and range of the graph of each function. 1) 2)
3) 23 6 1y x x 4) 2 2 1y x x
5) 2 4 5y x x 6) 24 8 9y x x
Determine whether the function has a minimum or a maximum value.
7) 2 4 3y x x 8) 2 2 1y x x
vertex:__________
axis of symmetry:__________
y-intercept: __________
domain: __________
range: __________
vertex:__________
axis of symmetry:__________
y-intercept: __________
domain: __________
range: __________
vertex:__________
axis of symmetry:__________
y-intercept: __________
domain: __________
range: __________
vertex:__________
axis of symmetry:__________
y-intercept: __________
domain: __________
range: __________
vertex:__________
axis of symmetry:__________
y-intercept: __________
domain: __________
range: __________
vertex:__________
axis of symmetry:__________
y-intercept: __________
domain: __________
range: __________
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Algebra 1 Section 9.1 Day 2 Notes: Graphing a Quadratic Warm-up Given the quadratic equation 2 3 , 1. Does the parabola open up or down? 4. Find the y-intercept. 2. Find the coordinates of the vertex , . 5. Find the domain. 3. Find the equation for the axis of symmetry. 6. Find the range. Steps to graph a quadratic function in standard form. 1. Find the 2. Put the vertex in the of your 5 point table 3. Find x-values to the and of your vertex.
Evaluate each to find the remaining y values. 4. the points and connect with a curve. 5. Find remaining pieces: Axis of symmetry, y-intercept, domain and range. Example 1: Use a table of values to graph f(x) = x2 + 4x – 4. State the vertex, axis of symmetry, y-intercept, domain and range. Axis of Symmetry: Vertex: y-intercept: Domain: Range:
x y
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Example 2: Use a table of values to graph y = –2x2 – 8x – 2. State the vertex, axis of symmetry, y-intercept, domain and range. Axis of Symmetry: Vertex: y-intercept: Domain: Range: Example 3: Use a table of values to graph y = x2 + 2x – 3. State the vertex, axis of symmetry, y-intercept, domain and range. Axis of Symmetry: Vertex: y-intercept: Domain: Range:
x y
x y
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Algebra 1 Section 9.1 Day 2 Worksheet #2 Use a table of values to sketch each graph.
1) 2)
x
y
3) 2 2 1y x x 4) 22 4 1y x x
x
y
22 4 6y x x 23 6 3y x x
vertex:__________
axis of symmetry:__________
y-intercept: __________
domain: __________
range: __________
vertex:__________
axis of symmetry:__________
y-intercept: __________
domain: __________
range: __________
x
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vertex:__________
axis of symmetry:__________
y-intercept: __________
domain: __________
range: __________
vertex:__________
axis of symmetry:__________
y-intercept: __________
domain: __________
range: __________
x
y
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Algebra 1 Section 9.1 Day 3 Notes: Graphing a Quadratic in Intercept Form Warm-up Find the x-intercepts in the quadratic equation 4 3 Hint: Think about what the y-value is if you are looking for an x-intercept ). Graph of a Quadratic Equation The graph of is a parabola in intercept form.
The vertex is between the intercepts.
The x-coordinate of the vertex is
The axis of symmetry is the vertical line 1. y = (x + 4)(x – 2) 2. y = ½ (x – 1)(x – 9) Step 1: Find your x-intercepts (p, 0) and (q, 0) Step 1: Find your x-intercepts Step 2: Find the axis of symmetry Step 2: Find the axis of symmetry and half the vertex and half the vertex
p+qx=
2 Step 3: Find the other half of the vertex Step 3: Find the other half of the vertex (Plug x into the original equation) Step 4: Graph Step 4: Graph Step 5: Find Domain and Range Step 5: Find Domain and Range
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3. y = -(x – 6) (x – 2) 4. y = ½ (x – 5) (x + 3) Step 1: Step 1: Step 2: Step 2: Step 3: Step 3: Step 4: Step 4: Step 5: Step 5: 5. y = (x + 3) (x – 1)
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Algebra 1 Section 9.1 Day 3 Worksheet #3 Sketch each graph.
1) ( ) ( 2)( 4)f x x x 2) 2( 3)( 1)y x x
3) ( 2)( 4)y x x 4) 1
( ) ( 2)( 4)2
f x x x
5) ( ) 2( 5)( 1)f x x x 6) ( 3)( 5)y x x
vertex:__________
x-intercepts:___________
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vertex:__________
x-intercepts:___________
vertex:__________
x-intercepts:___________
vertex:__________
x-intercepts:___________
vertex:__________
x-intercepts:___________ vertex:__________
x-intercepts:___________
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Algebra 1 Section 9.1 Day 4 Notes: Factor to Graph in Intercept Form Warm-up 1. If your equation is in standard form , what is the formula to find the vertex? How many points do you need to graph?
2. If your equation is in intercept form , what is the formula to find the vertex? How many points do you need to graph?
Factor the following quadratic expressions.
3. 10 21 4. 2 35
1. 2 3 2. 4 3
Step 1: Factor the equation Step 1: Factor the equation
Step 2: Find the x-intercepts (p, 0) and (q, 0) Step 2: Find the x-intercepts Step 3: Find the axis of symmetry and the Step 3: Find the axis of symmetry and the
x-coordinate of the vertex x-coordinate of the vertex
p+qx=
2
Step 4: Find the y-coordinate of the vertex Step 4: Find the y-coordinate of the vertex
Step 5: Graph Step 5: Graph
Step 6: Find Domain and Range Step 6: Find Domain and Range
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3. 6 5 4. 2 8 6
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Step 6:
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Algebra 1 Section 9.1 Day 4 Worksheet #4 Sketch each graph using intercept form.
1) 2( ) 2 3f x x x 2) 22 5 2y x x
3) 2 8 15y x x 4) 2( ) 8 12f x x x
5) 2( ) 2 8f x x x 6) 22 4 6y x x
vertex:__________
x-intercepts:___________
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vertex:__________
x-intercepts:___________
vertex:__________
x-intercepts:___________ vertex:__________
x-intercepts:___________
vertex:__________
x-intercepts:___________ vertex:__________
x-intercepts:___________
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Algebra 1 Section 9.1 Extra Review: Best Form to Use 1) When do you want to leave the quadratic equation in standard form versus changing it to intercept form? Why? Example 1 Example 2 Graph 5 7 Graph 8 7 2) In Example 2, what if the problem asked you to graph 8 7 AND
a) Find the y-intercept
-OR-
b) Find the x-intercepts
3) Graph 8 7 in standard form 4) Graph 8 7 in intercept form x
y
Vertex: ____________ Vertex: ______________
A of S: ____________ A of S: ______________
y-intercept: __________ x-intercepts: _______________
Domain: ____________ Domain: _____________
Range: _____________ Range: ______________
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Algebra 1 Name:
9.1 Side by Side Comparison of Graphing in Standard Form vs. Intercept Form
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Graph the equation above using standard form methods. Graph the equation above using intercept form methods.
a =
b = Factored form:
c =
x-intercepts:
Vertex:
Axis of Symmetry: Vertex:
y-intercept: Axis of Symmetry:
Domain: Domain:
Range: Range:
x y
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Algebra 1 Section 9.3 Day 1 Notes: Shifting a Quadratic
Warm-up Graph using your method of choice.
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Standard Form Intercept Form
Vertex = Find x-intercepts
Make a table Vertex =
Find y-int, Axis of Sym Find y-int, Axis of Sym
Domain, range Domain, range
Examples: Describe how the graph of each function is related to the graph of . Then state the vertex. 1. 7 2. 1 3. 4 4. 3 5. 2 6. 3
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The quadratic function in the form is called vertex form. The vertex of the parabola is ______________.
(+) means ( ) means
(+) means ( ) means
If a is negative, it opens If a is positive, it opens If it’s stretched. If it’s compressed.
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7. 8. 4 9.
10. 4 1 11. 5 3 12. 2 6
13. 4 1 8 14. 3 15. 8 1
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Algebra 1 Section 9.3 Day 1 Worksheet #5
Describe how the graph of each function is related to the graph of f(x) = . Then state the vertex.
1. g(x) = + 2 2. g(x) = 1 3. g(x) = – 8
4. g(x) = 7 5. g(x) = 6. g(x) = –6
7. g(x) = – + 3 8. g(x) = 5 – 9. g(x) = 4 1
Match each equation to its graph.
10. y = 2 – 2 A. C.
11. y = – 2
12. y = – + 2
13. y = –2 + 2 B. D.
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Algebra 1 Section 9.3 Day 2 Notes: Graphing in Vertex Form
Warm-up Describe how the graph of the following function is related to the graph of . Then state the vertex.
1. 4 5 2. 2 3. 1
Steps to graph a quadratic equation using vertex form: 1. Identify the vertex: 2. Put the vertex in the middle of the table and find two values and
two values . 3. Substitute each x-value into the equation to find the . 4. Plot points and connect them with a curve. 5. To find the y-intercept, substitute in for x. Your answer will be . 6. Domain: All real numbers Range: y the y-value of the vertex. Examples: 1. 2 2. 2 3 Vertex: Vertex: y-int: y-int: Domain: Domain: Range: Range: 3. 7 Vertex:
y-int: Domain: Range:
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Algebra 1 Section 9.3 Day 2 Worksheet #6 Graph the quadratic equation using vertex form. State the vertex, y-intercept, domain, and range.
1. y = – 3 2. 1 – 2
Vertex: Vertex:
y-int.: y-int.:
Domain: Domain:
Range: Range:
3. y = 1 4. y = 4 –
Vertex: Vertex:
y-int.: y-int.:
Domain: Domain:
Range: Range:
5. y = 2 + 1 6. 3
Vertex: Vertex:
y-int.: y-int.:
Domain: Domain:
Range: Range:
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Algebra 1 Section 9.3 Day 3 Notes: Practice
Warm-up 1. Describe how the graph of the following function is related to the graph of . Then state the vertex. 2 3 1 2. Graph 2 3 and find the following:
vertex
y-intercept domain range axis of symmetry.
1. Write an equation of a graph that is narrower than the graph of 10.
2. Write an equation of a graph that possesses function values that are 2 less than 5 .
3. Write an equation of a graph that is wider than the graph of 7.
4. Write an equation of a graph that possesses function values that are 4 more than those of 9 .
5. Is the graph of 5 2 compressed or stretched vertically compared to the graph of 2?
What else is done to the graph?
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6. Is the graph of 7 compressed or stretched compared to the graph of 7 ?
7. What happens when the graph of is changed to 2 3 6?
8. Write an equation in vertex form that 9. Graph using vertex form method.
has the same vertex as the following graph. 2 1 3
Vertex:
y-int.:
Domain:
Range:
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Algebra 1 Section 9.3 Day 3 Worksheet #7
1. Write an equation of a graph that is wider than the graph of 6.
2. Write an equation of a graph that possesses function values that are 7 more than those of 4 .
3. Write an equation of a graph that is narrower than the graph of 4.
4. Write an equation of a graph that possesses function values that are 2 less than 6 .
5. Is the graph of 5 compressed or stretched compared to the graph of 5?
6. Is the graph of 3 2 compressed or stretched compared to the graph of 2 ?
7. What happens when the graph of is changed to 1 4?
8. Write an equation in vertex form that 9. Graph using vertex form method.
has the same vertex as the following graph. 1 2
Vertex:
y-int.:
Domain:
Range:
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Extra Practice Worksheet 1) Use a table of values to sketch a graph of 2 12 10. State the vertex, axis of symmetry, domain, and range and y-intercept.
x
y
Vertex:_________
Axis of Symmetry:_________
Domain:_____________
Range:___________
y – intercept: ___________
2) Sketch a graph of 4 1 . State the vertex, x-intercepts, domain, range, and axis of symmetry.
Vertex:__________
x-intercepts:___________
Domain:__________
Range:____________
Axis of Symmetry:_________
3) Sketch a graph of 2 7. State the vertex, axis of symmetry, domain, and range.
Vertex:__________
Axis of Symmetry:___________
Domain:__________
Range:____________
x y
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Algebra 1 9.1, 9.3 Review
Warm-up 1. If a parabola is given by the equation 2 4, what would change (if anything) if the equation was changed to 2 4?
Choose all that apply. a) the vertex b) the axis of symmetry c) the shape of the parabola d) the direction the parabola opens e) nothing would change
2. What are the x-intercepts of 10 24?
9.1, 9.3 Review – BINGO!
FREE
SPACE
Work goes down here:
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Algebra 1 9.1 and 9.3 Review Day Worksheet #9 1. Using your notes, write the general form of each equation below.
Standard Form:
Intercept Form:
Vertex Form:
2. In the table below, list the formula needed to find the vertex. Or for vertex form, explain how you would identify the vertex.
Standard Form Intercept Form Vertex Form
Vertex formula:
Vertex formula:
3. Find the maximum/minimum 4. Find the vertex 5. Find the vertex of 4 1. of 1 3 . of 1 . 6. Find the x-intercepts of 7 18. 7. If a parabola is given by the equation 4 7, what would change (if anything) if the equation was changed to 2 7? Multiple Choice - Circle all that apply. A) the y-intercept B) the axis of symmetry C) the shape of the parabola D) the direction the parabola opens E) the vertex 8. Describe how the graph of the following function is related to the graph of . Then state the vertex. These are four separate questions – it is not multiple choice.
A) 4 7 B) y = 5 C) 3 D) 6 1 3
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9) Use a table of values to sketch a graph of 4 2. State the vertex, axis of symmetry, domain, and range and y-intercept.
x
y
Vertex:_________
Axis of Symmetry:_________
Domain:_____________
Range:___________
y – intercept: ___________
10) Sketch a graph of 6 2 . State the vertex, x-intercepts, domain, range, and axis of symmetry.
Vertex:__________
x-intercepts:___________
Domain:__________
Range:____________
Axis of Symmetry:_________
11) Sketch a graph of 1 4. State the vertex, axis of symmetry, domain, and range.
Vertex:__________
Axis of Symmetry:___________
Domain:__________
Range:____________
x y
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Algebra 1 9.1 and 9.3 Quiz Day
Warm-up 1. Complete the table below.
Name
Form
Step 1 to graph
Vertex = Find ___________ Vertex =
Vertex =
2. If your vertex is 2, 3 and your parabola is going down, find:
a) Axis of Symmetry: b) Range:
3. In general, how do you find the y-intercept?
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Algebra 1 Section 9.2 Notes: Solving a Quadratic by Graphing Review
Warm-up Find the x-intercepts.
1. 1 3
2. 3 2
3. 5 6
The solutions or roots to a quadratic equation are (Also called )
Example 1: What are the solutions of the quadratic equations that are graphed? a. b. Solutions:___________________ Solutions:___________________ Example 2: Solve x2 – 6x + 8 = 0 by graphing.
How many x-intercepts are there?
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Example 3: Double Root Solve x2 + 8x = –16 by graphing. Example 4: Solve 4 5 0 by graphing. Example 5: Solve x2 – 2x – 8 = 0 by graphing. Example 6: Solve x2 + 2x = –1 by graphing.
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Algebra 1 Section 9.2 Worksheet #10 Use the graph below to find the solutions.
1. 2.
Solution(s): Solution(s):
Solve each equation by graphing.
3. 2 3 0 4. 6 8 0
Solution(s): Solution(s):
5. 2 1 6. 7 10
Solution(s): Solution(s):
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Algebra 1 Chapter 9 Word Problem Notes
Example 1: The graph below represents a baseball’s height (in feet) x seconds after a player hits the ball.
a) At what height was the ball hit?
b) When did the ball reach its maximum height?
c) What was the maximum height of the baseball?
d) After how many seconds did the ball hit the ground? Example 2: Christine hits a birdie in the air during a badminton match. The function
6 7 represents the height (in feet), y, of the birdie x seconds after it is hit in the air.
a) At what height was the birdie hit?
b) When did the birdie reach its maximum height?
c) What was the maximum height of the birdie?
d) After how many seconds did the birdie hit the ground?
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Algebra 1 Chapter 9 Word Problem Homework Show all work for full credit. 1. Ellie hit a tennis ball into the air. The path of the ball can be modeled by 8 2, where y represents the height in feet of the ball x seconds after it is hit into the air.
a) At what height was the ball hit?
b) When did the ball reach its maximum height?
c) What was the maximum height of the tennis ball?
2. Carter throws a paper airplane in the air. The function 4 5 represents the height (in feet), y, of the paper airplane x seconds after it is thrown into the air.
a) At what height was the paper airplane thrown?
b) When did the airplane reach its maximum height?
c) What was the maximum height of the airplane? d) After how many seconds did the airplane hit the ground?
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-10 -8 -6 -4 -2 2 4 6 8 10
-10
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-10 -8 -6 -4 -2 2 4 6 8 10
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Algebra 1 Review Day
Warm-up 1. The graph of 4 has an axis of symmetry at 6. Based on this information, what value of b will satisfy the equation 4 0?
Hint: Use the formula . You know what a is, and based on the information given, you know what x is. Find b.
2. The graph of 4 has a vertex at 3, 5 . Based on this information, what value of b will satisfy the equation
4 0?
Hint: Use the formula . You know what a is, and based on the information given, you know what x is. Find b.
1) Use a table of values to sketch a graph of 2 2 5y x x , including the axis of symmetry. State the vertex, axis of
symmetry, domain, and range and y-intercept.
x
y
Vertex:_________
Axis of Symmetry:_________
Domain:_____________
Range:___________
y – intercept: ___________
2) Sketch a graph of ( 2)( 4)y x x , including the axis of symmetry. State the vertex, x-intercepts, domain, range,
and axis of symmetry.
Vertex:__________
x-intercepts:___________
Domain:__________
Range:____________
Axis of Symmetry:_________
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-10 -8 -6 -4 -2 2 4 6 8 10
-10
-8
-6
-4
-2
2
4
6
8
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3) Sketch a graph of 2( 3) 7y x , including the axis of symmetry. State the vertex, x- intercepts, domain, and range.
Vertex:__________
Axis of Symmetry:___________
Domain:__________
Range:____________
x y
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Algebra 1 Free Response Test Day
Warm-up 1. When given an equation in standard form :
a) What’s your first step to graph?
b) What formula do you use?
c) How many points will you plot to graph?
2. When given an equation in intercept form
a) What’s your first step to graph?
b) What formula do you use to find your vertex?
c) How many points will you plot to graph?
3. When given an equation in vertex form :
a) What’s your first step to graph?
b) Do you need a formula to find the vertex?
c) How many points will you plot to graph?
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Algebra 1 MC Test Day
Warm-up 1. What are the coordinates of the maximum or minimum value for the graph of the function 6 7 ?
a) 3, 2 b) 3, 2 c) 3, 2 d) 3, 2
2. The graph below represents the function 10 20. Which statement is true?
a) There are no y-intercepts
b) There are no x-intercepts
c) There is a y-intercept at 3, 0 .
d) There is an x-intercept at 7,0 .
3. Which quadratic function opens downward and has its vertex at 2?
a) 2 8 1
b) 4 16 7
c) 3 12 5
d) 5 20 3
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Unit 6 Part 2
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Algebra 1 10.2 Day 1 Simplifying Radicals Warm-up Evaluate each expression. Use your calculator if needed. 1. √25 2. √100
3. √16 ∙ √4 4. ∙ √9
5. √
What is a “Radical”? It’s another word for . Square Root of a Number If b2 = a, then , when a and b are positive numbers. Example: If 3 9 then 3 is a square root of 9. Simplest Form A radical expression is in if the following are true:
No factors are underneath the square root.
There are no underneath the square root.
There are no square roots in the of the fraction.
Example 1: Simplify. No Decimal answers. <Refer to Pg. 62 to have a complete perfect square list> a) √48 b) √60 c) √147 d) √72
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Simplifying with Variables If a variable is under a square root, and the exponent is > 1, take the variable out of the square root by the exponent by . Leave any variable under the square root. Example 2
a) 90 b) √32
c) 56 Product Property
√ ∙ √ √ Ex: √18 ∙ √2 √36 6 Example 3 a) √6 ∙ √6 b) √10 ∙ √30
c) 3√63 ∙ √4 d) √2 ∙ √12
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List of Perfect Squares 22 =
32 =
42 =
52 =
62 =
72 =
82 =
92 =
102 =
112 =
122 =
132 =
142 =
152 =
List of Common Square Roots
9
4
225
196
169
144
121
100
81
64
49
36
25
16
65
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Practice Simplifying Square Roots
1.) √27
2.)
80
3.) √98
4.) √108
5.) √40
6.)
162
7.)
28
8.) √45
9.) √96
10.) √180
11.) √150
12.)
125
13.) √52
14.)
63
15.) √128
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Algebra 1 10.2 Day 1 Homework
Simplify. No decimal answers.
1. √44 2. √27
3. √48 4. √75
5. 90 6. √125
7. 200 8. 80
9. √112 10. √54
11. √2 ∙ √8 12. √6 ∙ √8
13. 2√3 ∙ 3√15 14. √6 ∙ 4√24
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Algebra 1 10.2 Day 2 Simplifying Radicals
Warm-up
Simplify. No decimal answers.
1. √108 2. √243
3. √5 ∙ √8 4. √6 ∙ √10
Quotient Property
√
√
Example: √
√
Example 1 Simplify. No decimal answers.
a) b)
c) √
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When the denominator is not a perfect square we need to RATIONALIZE OUR DENOMINATOR! Steps: 1. Simplify the as much as possible. 2. numerator and denominator by square root in the denominator. 3. Multiply straight across. 4. Simplify.
Example 2 Simplify. No decimal answers.
a) b) 3
c) √ √
√
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Algebra 1 10.2 Day 2 Homework Simplify each expression. 1. √28 2. √72 3. √2 ⋅ √10 4. 3√5 ⋅ √5 5. 2√3 ⋅ 3√15 6. √81
7. 75 8.
9. 10.
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Algebra 1 10.2 Day 2 Practice
Warm-up
Simplify.
1. √54 2. 3√24 ∙ 4√6 3. √60
Simplify each expression. 1. √40 2. √99
3. √5 ⋅ √60 4. √6 ⋅ 4√24
5. √16 6. 40
7. 8. ⋅
9. 10.
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Algebra 1 10.2 Day 3 Notes Solving Quadratic Equations by Finding Square Roots
Quadratic Equation: an equation that can be written in the following standard form: ax2 + bx + c = 0, where a 0
In standard form, a is the leading coefficient. When , this equation becomes 0. Solving x2 = d by finding square roots. If d 0, then x2 = d has real solutions: √ .
If d = 0, then x2 = d has real solution: x = 0.
If d 0, then x2 = d has real solution.
Steps to solve quadratic equations in the form √ 1. Isolate on one side. 2. Then find the of each side. Example 1: Solve each equation. Leave answer as a simplified root. a) x2 = 16 b) x2 – 32 = 0 c) 3x2 – 16 = 0 d) 5x2 + 20 = 0 e) 1 16 f) 2 3 49
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Algebra 1 10.2 Day 3 Homework
Simplify.
1. √24 2. √108
3. √7 ⋅ √14 4. 4√3⋅ 3√18
5. 50 6. 56
7. 8. ⋅
9. √
√ 10.
Solve for x.
11. 2 25 12. 2 1 36
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Simplifying Radicals – Extra Practice Name_______________________________ Coloring Date______________ Simplify. (Leave in radical form. No Decimals!!!) 1. Green 250 2. Purple 44
3. Red 20
5 4. Purple
6
2
5. Orange 234 6. Pink 2786
7. Orange 20
4 8. Blue
120
4
9. Green 1112
2 10. Yellow
11
16
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Now, Find your solutions to the problems on the front and color them with the corresponding color assigned.
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Algebra 1 9.4 Day 1 Solving Quadratic Equations by Completing the Square Warm Up Factor the following: 1. 20 100 2. 18 81 3. 4 4
Steps to solving a quadratic equation by completing the square:
1. Get the equation in the form ______ ______. (Move the c value to the other side, leaving blanks as shown above)
2. To fill in the blanks: Divide the b value by and it.
3. the perfect square trinomial
4. the equation for x using square roots
2 22
2 2
b bx bx c
Example 1: Find the value of c that makes x2 – 12x + c a perfect square trinomial. Example 2: Find the value of c that makes x2 + 14x + c a perfect square trinomial. Example 3: Solve the quadratic equations using completing the square. A. 10 3 0 B. 6 8 0
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C. 6 5 12
If the coefficient of x2 is not 1…we have to do one extra step! FIRST: each side of the equation by the of x2 4. 3x2 – 6x – 12 = 0 5. 5x2 – 10x + 30 = 0 6. –2x2 + 36x – 10 = 24
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Algebra 1 9.4 Day 1 Homework Find the value of c that makes each trinomial a perfect square. 1. – 24x + c 2. + 28x + c 3. + 40x + c 4. + 3x + c 5. – 9x + c 6. – x + c Solve each equation by completing the square. Round to the nearest tenth if necessary. 7. – 14x + 24 = 0 8. + 12x = 13 9. + 8x + 9 = 0 10. 2 10 22 4 11. 3 + 15x – 3 = 0 12. 4 – 72 = 24x
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Algebra 1 9.4 DAY 2 Notes
Warm-up
Solve for x by completing the square: 3 – 6x = 10
Real-life Application Example 1: BUSINESS Jaime owns a business making decorative boxes to store jewelry, mementos, and other valuables.
The function y = + 50x + 1800 models the profit y that Jaime has made in month x for the first two years of his business.
a. Write an equation representing the month in which Jaime’s profit is $2400. b. Use completing the square to find out in which month Jaime’s profit is $2400. Rewrite equations in VERTEX FORM by Completing the Square
Write a quadratic function from Standard Form to Vertex Form
to khxay 2)( Steps to solving a quadratic equation by completing the square:
1. the x2 term and the x term (subtract the constant to the same side as y)
2. Divide the x term by , it and it to both sides of the equation.
3. the perfect square trinomial on the side with x
4. Isolate y.
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Example 2: Write the quadratic function 6 16 in vertex form. Identify the vertex.
Example 3: Write the quadratic function 4 8 in vertex form. Identify the vertex. Example 4: Write the quadratic function y = x2 + 26x + 108 in vertex form. Identify the vertex. Example 5: Write the quadratic function y = x2 – 8x + 11 in vertex form. Identify the vertex.
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Algebra 1 9.4 Day 2 Homework 1. PHYSICS From a height of 256 feet above a lake on a cliff, Mikaela throws a rock out over the lake. The
height H of the rock t seconds after Mikaela throws it is represented by the equation H = –16 + 32t + 256. To the nearest tenth of a second, how long does it take the rock to reach the lake below? (Hint: Replace H with 0.)
Find the value of c that makes each trinomial a perfect square. 2. 4 3. 2 Solve each equation by completing the square. 4. 8 15 0 5. 2 4 30 Put the equation in vertex form and identify the vertex. 6. 6 10 7. 12 8 <Over for 8 & 9>
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8. 20 95 9. 4 5
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Algebra 1 10.2 & 9.4 Review Warm-up #1
1. Simplify √243 2. Simplify √
3. Solve by completing the square: 6 2 0 4. Rewrite in vertex form: 4 1
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Algebra 1 Name:____________________________ 10.2 & 9.4 Review Simplify each radical completely.
1) 80 2) 320 3) 4 324x y z
4) 3
19 5) 8 956a b 6)
20
81
xy
7)8 5 5 15 8) 7 4 9 100 9) What value of c will make the expression a perfect square trinomial? x2 + 14x + c
10) Which step is not performed in the process of solving 3 + 18x + 12 = 0 by completing the square?
A) Divide everything by 3 B) Add 81 to each side. C) Factor 2 + 6x + 9 D) Take the square root of each side. E) Add 9 to each side 11) Solve the equation. Write your answer in simplified radical form. 23 7 28x
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Solve by completing the square. Write your answer in simplified radical form. 12) 2 2 14 0x x 13) 23 12 81 15x x 14) 22 12 14x x Rewrite each equation in vertex form and identify the vertex. 15) 2 6 16y x x 16) 2 8 9y x x 17) The freshman class buys t-shirts to wear on Fridays for school spirit. The cost of the shirts can be modeled by the equation C = 0.1x2 + 2.4x + 25, where C is the amount it costs to buy x shirts. How many shirts can the class purchase for $430? Round your answer to the nearest whole shirt they can buy.
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Algebra 1 10.2 & 9.4 Review Warm-up #2
1. Simplify √147 2. Simplify √
3. Solve by completing the square: 8 1 0 4. Rewrite in vertex form and identify the vertex: 6 3
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Algebra 1 Name: 10.2, 9.4 Additional Review WS Period: Simplify.
1.) √50 2.) 18 3.) 4.) 5.) 2√3 ∙ 3√8
Solve the equation. 6. 81 7. 3 5 17 8. 2 9 9. Find the value of c that makes 8 a perfect square trinomial. Solve by completing the square. 10. 12 21 10 11. 2 4 30 12. Elias hits a baseball into the air. The equation h = -16t2 + 32t + 4 models the height h in feet of the ball after t seconds. Use completing the square to determine how long the ball stays in the air.
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10.2 & 9.4 Quiz Day Warm-up
1. Simplify
2. Solve the equation: 2 5 67 3. Solve by completing the square: 3 30 30
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Algebra 1 9.5 Day 1 Notes: The Discriminant & Quadratic Formula Warm-up Evaluate the expression 4 for the given values. 1. 2, 8, 1 2. 3, 6, 3 3. 5, 3, 6 We haven’t learned the Quadratic Formula yet, but in this formula the expression inside the radical is the discriminant.
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What does it do? It tells the number of solutions to a quadratic equation. If the discriminant equals…….
4 = any
positive number 4 = 0
4 = any negative number
Number and Type of Solutions
Number of x-intercepts
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Example 1:
Find the value of the discriminant and use the value to tell if the equation has two solutions, one solution, or no solution.
a) x2 – 2x + 4 = 0 b) –3x2 + 5x – 1 = 0 c) –x2 – 10x – 25 = 0
Example 2:
Use the related equation to find the number of x-intercepts of the graph of the function. Then match the equation to the graph.
a) y = x2 + 6x + 3 b) y = x2 + 6x + 10 c) y = x2 + 6x + 9
In previous sections we learned how to solve quadratic equations of the form ax2 + c = 0 by finding square roots. Now we will learn to solve quadratics in the form ax2 + bx + c = 0. The solutions of the quadratic equation ax2 + bx + c = 0 where a ≠ 0 are given by the QUADRATIC FORMULA:
Steps to Solve:
1. Get the quadratic equation in form 2. Evaluate for the - how many and what type of solution do you have? 3. Factor or continue quadratic formula.
Example 1: Solve x2 + 5x – 6 = 0 using the quadratic formula.
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Example 2: Solve: 3 5 using the quadratic formula. Example 3: Solve: 8 5 2 using the quadratic formula.
Example 4: Solve 8 16 using the quadratic formula.
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Extra Examples 1. Find the x-intercepts of the graph of y = x2 + 3x – 8. Remember what we learned in 9.2? What do we know about the x-intercepts of a quadratic function?. . . The x-intercepts occur when y = 0, they are solutions to the quadratic function. So we can now use the quadratic formula. 2. Find the x-intercept of the graph of 4x2 – x – 7 = y.
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Algebra 1 9.5 Day 1 Homework
State the value of the discriminant for each equation. Then determine the number of real solutions of the equation.
1. + 4x + 3 = 0 2. + 2x + 1 = 0 3. – 4x + 10 = 0
First find the discriminant, then use the quadratic formula to solve. 1. + 2x – 3 = 0 2. + 8x + 7 = 0 3. – 4x + 6 = 0 4. – 6x + 7 = 0 5. 2 + 9x – 5 = 0 6. 3 + 12x + 10 = 0
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Algebra 1 9.5 Day 2 Practice Warm-up First find the discriminant, then use the quadratic formula to solve. 3 – 7x – 6 = 0 First find the discriminant, then use the quadratic formula to solve. 1. 2 5 3 0x x a = b = c = b2 – 4ac = What type of # is the discriminant?
x
Solution(s):
2. 2 42 0x x a = b = c = b2 – 4ac = What type of # is the discriminant?
x
Solution(s):
3. 23 2 0x x 4. 22 6 3 0x x
5. What are the x-intercepts of 2 4 7y x x ? Explain.
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Algebra 1 9.5 Day 2 Homework First find the discriminant, then use the quadratic formula to solve. 1. 3 – 1 = –8x 2. 4 + 7x = 15 3. + 8x + 16 = 0 4. + 3x + 12 = 0 5. 2 + 12x = –7 6. 2 + 15x = –30 7. 4 + 9 = 12x
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Algebra 1 9.5 Day 3 Notes: Quadratic Formula Applications Warm-up Solve using the quadratic formula. 1. 11 30 2. 5 16 3 0 Vertical Motion Model: h = -16t2 + vt + s Can be thrown upward ( velocity) or downward ( velocity) is the initial velocity and is the initial height and is the height at a time of seconds Falling Object: When something is dropped it has no velocity (v = 0) 1. You drop a set of keys from a window to your friend 30 feet below. Your friend misses the keys and they fall to the ground. How long were the keys in the air? 2. A lacrosse player throws a ball upward from her playing stick with an initial height of 7 feet, at an initial velocity of 90 feet per second. How long does the ball take to reach her teammate’s playing stick, which is at a height of 5 feet? 3. During a chemistry experiment, the cork in a 0.5 foot tall beaker with an effervescent solution pops off with an initial velocity of 20 feet per second.
How many seconds does it take for the cork to hit the table?
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4. Fireworks are shot upward with an initial velocity of 125 feet per second from a platform 3 feet above the ground. Use the vertical motion model h = -16t2 + vt + s to find out how long it will take the rocket to hit the ground. Remember h is the ending height, t is the time the firework is in the air, v is the initial velocity, and s is the initial height.
5. PHYSICS Lupe tosses a ball up to Quyen, waiting at a third-story window, with an initial velocity of 30 feet per second. She releases the ball from a height of 6 feet. The equation h = –16 + 30t + 6 represents the height h of the ball after t seconds. If the ball must reach a height of 25 feet for Quyen to catch it, does the ball reach Quyen? Explain. (Hint: Substitute 25 for h and use the discriminant.)
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Algebra 1 9.5 Day 3 Homework Name ___________________________ 1. A roofer tosses a piece of roofing tile from a roof onto the ground 30 feet below. He tosses the tile with an
initial downward velocity of 10 feet per second. a. Write an equation to find how long it takes the tile to hit the ground. Use the model for vertical motion, H = –16 + vt + h, where H is the height of an object after t seconds, v is the initial velocity, and h is the
initial height. (Hint: Since the object is thrown down, the initial velocity is negative.) b. How long does it take the tile to hit the ground? 2. A goalie kicks a soccer ball with an upward velocity of 65 feet per second, and her foot meets the ball 1 foot off the ground. The quadratic function 16 65 1 represents the height of the ball h in feet after t seconds. Approximately how long is the ball in the air? 3. The height of a golf ball in the air can be modeled by the equation h = -16t2 + 60t + 3, where h is the height in feet of the ball after t seconds. How long was the ball in the air? <Over for 4 & 5>
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4. Bob tosses his basketball onto the ground from his tree house. He tosses the basketball with an initial downward velocity of 8 feet per second. The equation h = –16 – 8t + 20 represents the height h of the basketball after t seconds. How long does the basketball take to hit the ground?
5. Use the formula h = – 16 + 250t to model the height h in feet of a model rocket t seconds after it is launched. Determine when the rocket will reach a height of 900 feet.
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Algebra 1 9.6 Analyzing Functions with Successive Differences Warm-up
In previous sections we learned to graph linear, exponential and quadratic functions. Now we want to be able to identify linear, quadratic, and exponential functions from given data.
Example 1: Choose a Model Using Graphs A. Graph the ordered pairs. Determine whether the ordered pairs represent a linear, quadratic, or exponential function.
(1, 2), (2, 5), (3, 6), (4, 5), (5, 2) B. 2 21,6 , 0,2 , 1, , 2,
3 9
YOU TRY! Graph the set of ordered pairs. Determine whether the ordered pairs represent a linear, quadratic, or exponential function. A. (–2, –6), (0, –3), (2, 0), (4, 3) B. (–2, 0), (–1, –3), (0, –4), (1, –3), (2, 0)
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Choose a Model Using Differences or Ratios 1. Subtract each consecutive y-values. Look for the differences to be the SAME number. This is the first
difference. If they are equal, it is a LINEAR function. 2. If not equal subtract each difference again. Look for the same number, this is the second difference. If
equal, it is a QUADRATIC function. 3. If still not equal look for a ratio of the subtracted y-values. If this is equal it is an EXPONENTIAL
function. Look for a pattern in the table of values to determine which kind of model best describes the data. If the function is LINEAR, write the equation to fit the data. A. B.
You Try!
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Algebra 1 9.6 Homework Graph each set of ordered pairs. Determine whether the ordered pairs represent a linear function, a quadratic function, or an exponential function.
1. (2, 3), (1, 1), (0, –1), (–1, –3), (–2, –5) 2. (–1, 0.5), (0, 1), (1, 2), (2, 4) 3. (–2, 4), (–1, 1), (0, 0), (1, 1), (2, 4) 4. (–3, 5), (–2, 2), (–1, 1), (0, 2), (1, 5) Look for a pattern in each table of values to determine which model best describes the data. Then write an equation for any LINEAR function that models the data. 5. 6.
x –3 –2 –1 0 1 2
y 32 16 8 4 2 1
x –1 0 1 2 3
y 7 3 –1 –5 –9
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7. 8. 9.
x –3 –2 –1 0 1
y –27
–12
–3 0 –3
x 0 1 2 3 4
y 0.5 1.5 4.5 13.5 40.5
x –2 –1 0 1 2
y –8 –4 0 4 8
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Review Warm-up #1 1. Solve 2 16 20 by completing the square. 2. Solve 2 16 20 by using the quadratic formula. Review Warm-up #2
1. Alex solved an equation using the quadratic formula and got this: √
Can you help him simplify it?
2. Which quadratic equation has √ as its roots?
A) 5 7 B) 5 7 C) 5 7 D) 5 7
