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This document contains information about a polynomial function P(x)=8x^6+2x^5+7x^4+2x^3+x^2+x+9. It provides the domain and range, leading coefficient, y-intercept, roots, and end behavior of the function's graph. The author enjoyed exploring how manipulating the signs of terms in the polynomial would transform the graph and sought to craft a visually appealing presentation through an exploratory design process.
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D.e.v
The document summarizes how various Avengers calculate and solve different math problems related to their superpowers and equipment. Captain America finds the area of his shield as a function of circumference. Iron Man solves a quadratic equation to power his arc reactor. The Hulk smashes a wall and calculates how many pieces it breaks into. Hawkeye finds the equation of a parabola based on its vertex and a point. Thor analyzes the domain and graph of a skipping hammer's movement. Black Widow determines the domain and range of a radical function.
DirectionsUse what you have learned in this course to answer th.docx
Directions:
Use what you have learned in this course to answer the following questions. Justify your responses completely. Each question is worth 5 points.
1.
Solve for
n
:
–6(
n
– 8) = 4(12 – 5
n
) + 14
n
.
2
. For
f(
x
) = 2|
x
+3| – 5
, name the type of function and describe each of the three transformations from the parent function
f(
x
) = |
x
|
.
3.
Determine whether
f
(
x
) = –5 – 10
x
+ 6
has a maximum or a minimum value. Find that value and explain how you know.
4.
The median weekly earnings for American workers in 1990 was $412 and in 1999 it was $549. Calculate the average rate of change between 1990 and 1999.
5.
Find the roots of the parabola given by the following equation.
2
x
2+ 5
x
- 9 = 2
x
6.
Describe the end behavior and determine whether the graph represents an odd-degree or an even-degree polynomial function. Then state the number of real zeros.
7.
GEOMETRY
Recall the formula for finding the area of a rectangle. Define a variable for the width and set up an equation to find the dimensions of a rectangle that has an area 144 square inches, given that the length is 10 inches longer than its width.
DIMENSIONS:
Length: Width:
8.
The amount
f
(
t
) of a certain medicine, in milligrams, in a patient’s bloodstream
t
minutes after being taken is given by
f
(
t
) =
.
Find the amount of medicine in the blood after 20 minutes.
9.
Graph
f(
x
) =
x
2 + 2
x
- 3
, label the function’s x-intercepts,
y
-intercept and vertex with their coordinates. Also draw in and label the axis of symmetry.
Image result for x y axis
10.
Determine whether the relation shown is a function. Explain how you know.
73-1.jpg
11.
Solve the inequality and graph the solution on a number line.
–3(5
y
– 4) ≥ 17
12.
Assume that the wooden triangle shown is a right triangle.
a. Write an equation using the Pythagorean Theorem and the measurements provided in the diagram.
Hint: (leg 1)2 + (leg 2)2 = (hypotenuse)2
b. Transform each side of the equation to determine if it is an identity.
13.
Use long division or synthetic division to find the quotient of .
14.
Simplify
(9 + 8 – 6)(4 – 5)
.
15.
Find the inverse of
h(
x
) = .
16.
If
f(
x
) = 2
x
– 1
and
g(
x
) = – 2
, find
[g
◦ f](
x
).
17.
Graph the function
y
= – 2
. Then state the domain and range of the function.
Domain:
Range:
18.
If
f(
x
) = 3
x
2 – 2
and
g(
x
) = 4x + 2
, what is the value of
f
+ g 2
?
The price of a sweatshirt at a local shop is twice the price of a pair of shorts. The price of a T-shirt at the shop is $4 less than the price of a pair of shorts. Brad purchased 3 sweatshirts, 2 pairs of shorts, and 5 T-shirts for a total cost of $136.
19.
Let
w
represent the price of one sweatshirt,
t
represent the price of one T-shirt, and
h
represent the price of one pair of shorts. Write a system of three equations that represents the prices of the clothing.
20.
Solve the system. Find the cost of eac.
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This document provides instructions for calculating the volume of an undersea mountain in order to clear it for an oil pipeline. It gives the equation f(x)=-(x-8)^2 + 4 that defines the shape of the mountain. It explains that the volume must be calculated between the mountain equation and a clearance line at g(x)=0. The steps shown are to find the intersection points where the two equations are equal, which are x=6 and x=10. It then explains how to set up and evaluate the integral of the mountain equation between these bounds to calculate the volume, which is found to be 29.6*Pi cubic units.
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2. Key examples show how to pair terms in an arithmetic progression to simplify summing the total. It also illustrates summing fractions in a geometric progression as areas in a rectangle.
3. Formulas are provided for finding the k-th term and summing the total terms of an arithmetic or geometric progression, depending on whether the difference or ratio is constant between terms.
Building Functions
This document discusses building new functions from existing functions. It provides examples of replacing f(x) with f(x) + k, kf(x), f(kx), and f(x + k) and examining the effects on the graph. Students are asked to find inverse functions and verify functions are inverses of one another through composition. Examples are provided of evaluating composite functions f(g(x)) from graphs and algebraic expressions by first applying the inner function g(x) and then the outer function f(x).
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The document outlines the daily lesson log for a Grade 10 mathematics class. It includes 4 sessions on the topic of polynomial functions. The objectives are to illustrate, find intercepts of, and graph polynomial functions. Examples are provided for identifying key features of polynomials like degree, leading coefficient, constant term, and finding x- and y-intercepts by setting the polynomial equal to 0 or using the factored form. The behavior of graphs is also discussed based on whether the degree is odd or even, and the sign of the leading coefficient.
Recommended
D.e.v
The document summarizes how various Avengers calculate and solve different math problems related to their superpowers and equipment. Captain America finds the area of his shield as a function of circumference. Iron Man solves a quadratic equation to power his arc reactor. The Hulk smashes a wall and calculates how many pieces it breaks into. Hawkeye finds the equation of a parabola based on its vertex and a point. Thor analyzes the domain and graph of a skipping hammer's movement. Black Widow determines the domain and range of a radical function.
DirectionsUse what you have learned in this course to answer th.docx
Directions:
Use what you have learned in this course to answer the following questions. Justify your responses completely. Each question is worth 5 points.
1.
Solve for
n
:
–6(
n
– 8) = 4(12 – 5
n
) + 14
n
.
2
. For
f(
x
) = 2|
x
+3| – 5
, name the type of function and describe each of the three transformations from the parent function
f(
x
) = |
x
|
.
3.
Determine whether
f
(
x
) = –5 – 10
x
+ 6
has a maximum or a minimum value. Find that value and explain how you know.
4.
The median weekly earnings for American workers in 1990 was $412 and in 1999 it was $549. Calculate the average rate of change between 1990 and 1999.
5.
Find the roots of the parabola given by the following equation.
2
x
2+ 5
x
- 9 = 2
x
6.
Describe the end behavior and determine whether the graph represents an odd-degree or an even-degree polynomial function. Then state the number of real zeros.
7.
GEOMETRY
Recall the formula for finding the area of a rectangle. Define a variable for the width and set up an equation to find the dimensions of a rectangle that has an area 144 square inches, given that the length is 10 inches longer than its width.
DIMENSIONS:
Length: Width:
8.
The amount
f
(
t
) of a certain medicine, in milligrams, in a patient’s bloodstream
t
minutes after being taken is given by
f
(
t
) =
.
Find the amount of medicine in the blood after 20 minutes.
9.
Graph
f(
x
) =
x
2 + 2
x
- 3
, label the function’s x-intercepts,
y
-intercept and vertex with their coordinates. Also draw in and label the axis of symmetry.
Image result for x y axis
10.
Determine whether the relation shown is a function. Explain how you know.
73-1.jpg
11.
Solve the inequality and graph the solution on a number line.
–3(5
y
– 4) ≥ 17
12.
Assume that the wooden triangle shown is a right triangle.
a. Write an equation using the Pythagorean Theorem and the measurements provided in the diagram.
Hint: (leg 1)2 + (leg 2)2 = (hypotenuse)2
b. Transform each side of the equation to determine if it is an identity.
13.
Use long division or synthetic division to find the quotient of .
14.
Simplify
(9 + 8 – 6)(4 – 5)
.
15.
Find the inverse of
h(
x
) = .
16.
If
f(
x
) = 2
x
– 1
and
g(
x
) = – 2
, find
[g
◦ f](
x
).
17.
Graph the function
y
= – 2
. Then state the domain and range of the function.
Domain:
Range:
18.
If
f(
x
) = 3
x
2 – 2
and
g(
x
) = 4x + 2
, what is the value of
f
+ g 2
?
The price of a sweatshirt at a local shop is twice the price of a pair of shorts. The price of a T-shirt at the shop is $4 less than the price of a pair of shorts. Brad purchased 3 sweatshirts, 2 pairs of shorts, and 5 T-shirts for a total cost of $136.
19.
Let
w
represent the price of one sweatshirt,
t
represent the price of one T-shirt, and
h
represent the price of one pair of shorts. Write a system of three equations that represents the prices of the clothing.
20.
Solve the system. Find the cost of eac.
Modul bimbingan add maths
The document is a mathematics textbook for Additional Mathematics Form 4. It covers topics on functions, simultaneous equations, quadratic equations, and quadratic functions. It contains examples and practice questions for students to work through with answers. The questions range from simple calculations to solving equations and inequalities involving quadratic expressions.
Adam Swenski's calculus presentation
This document provides instructions for calculating the volume of an undersea mountain in order to clear it for an oil pipeline. It gives the equation f(x)=-(x-8)^2 + 4 that defines the shape of the mountain. It explains that the volume must be calculated between the mountain equation and a clearance line at g(x)=0. The steps shown are to find the intersection points where the two equations are equal, which are x=6 and x=10. It then explains how to set up and evaluate the integral of the mountain equation between these bounds to calculate the volume, which is found to be 29.6*Pi cubic units.
Lesson 5: Functions and surfaces
A function of two variables is defined similar to a function of one variable. It has a domain (in the plane) and a range. The graph of such a function is a surface in space and we try to sketch some.
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1. This document provides examples and definitions related to arithmetic and geometric progressions. It defines arithmetic and geometric progressions and gives formulas for finding individual terms and summing the total terms.
2. Key examples show how to pair terms in an arithmetic progression to simplify summing the total. It also illustrates summing fractions in a geometric progression as areas in a rectangle.
3. Formulas are provided for finding the k-th term and summing the total terms of an arithmetic or geometric progression, depending on whether the difference or ratio is constant between terms.
Building Functions
This document discusses building new functions from existing functions. It provides examples of replacing f(x) with f(x) + k, kf(x), f(kx), and f(x + k) and examining the effects on the graph. Students are asked to find inverse functions and verify functions are inverses of one another through composition. Examples are provided of evaluating composite functions f(g(x)) from graphs and algebraic expressions by first applying the inner function g(x) and then the outer function f(x).
G10_Daily Lesson Log_Second QUARTER.docx
The document outlines the daily lesson log for a Grade 10 mathematics class. It includes 4 sessions on the topic of polynomial functions. The objectives are to illustrate, find intercepts of, and graph polynomial functions. Examples are provided for identifying key features of polynomials like degree, leading coefficient, constant term, and finding x- and y-intercepts by setting the polynomial equal to 0 or using the factored form. The behavior of graphs is also discussed based on whether the degree is odd or even, and the sign of the leading coefficient.
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- It describes how to write quadratic functions in standard form and use that form to sketch the graph and find features like the vertex and axis of symmetry.
- Examples are provided to demonstrate how to graph quadratic functions in standard form and how to find the minimum or maximum value of a quadratic function by setting its derivative equal to zero.
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This document provides a lesson on transforming linear equations between the forms Ax + By = C and y = mx + b. It begins with reviewing key concepts of linear equations, slopes, x-intercepts and y-intercepts. Examples are given of transforming equations between the two forms, including the equations 1.4x - 5y = 20 and 2.3x + 7y = 14 transformed to y = mx + b form, and the equations y = x + 3 and y = 5 - 3x transformed to Ax + By = C form. Students are then given a quiz to practice transforming equations between the two linear equation forms.
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The document provides step-by-step instructions for graphing different types of rational functions and polynomials. It includes two examples of graphing rational equations, showing how to find the greatest common factor, factor the numerator and denominator, identify holes and asymptotes, and graph. It also gives an example of solving a word problem involving perimeter and area of a rectangular field enclosed by fencing. Finally, it discusses how to graph polynomials based on their degree and signs.
April 7, 2014
1) The document outlines a class agenda that includes reviewing quadratics and new solving methods in preparation for a Friday test. It also notes a third quarter exam on Friday covering quadratics, factoring, exponents, and polynomials.
2) The class will begin with warm-up problems reviewing graphing quadratics and finding axes of symmetry. Steps for graphing the equation y + 6x = x^2 + 9 are provided.
3) Additional examples are given for graphing a quadratic function modeling the height of a basketball after being shot, finding the maximum height and time to reach it by using the vertex of the parabola.
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留信网认证的作用:
1:该专业认证可证明留学生真实身份
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3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
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8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
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我们所定的价格是非常合理的，而且我们现在做得单子大多数都是代理和回头客户介绍的所以一般现在有新的单子 我给客户的都是第一手的代理价格，因为我想坦诚对待大家 不想跟大家在价格方面浪费时间
对于老客户或者被老客户介绍过来的朋友，我们都会适当给一些优惠。
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The document provides information about graphing trigonometric functions such as sine and cosine. It defines key concepts like periodic functions, amplitude, and period. It explains that the graphs of sine and cosine functions are symmetrical and repeat in a wave-like pattern. The document also provides steps for graphing more complex trigonometric functions involving amplitude and period, including finding key points and repeating the wave pattern over the given interval. Examples are included to demonstrate how to graph functions like y=2sin(-3x).
Modeling with Quadratics
- The document discusses quadratic functions and their graphs. It explains that the graph of a quadratic function is a parabola, which is a U-shaped curve.
- It describes how to write quadratic functions in standard form and use that form to sketch the graph and find features like the vertex and axis of symmetry.
- Examples are provided to demonstrate how to graph quadratic functions in standard form and how to find the minimum or maximum value of a quadratic function by setting its derivative equal to zero.
modul 2 add maths 07
This document provides practice problems for additional mathematics Form 4 students in Terengganu, Malaysia. It covers topics on quadratic equations and quadratic functions, with multiple choice and short answer questions. The problems are divided into three sections: quadratic equations, quadratic functions for paper 1, and quadratic functions for paper 2. The document is copyrighted material from the Terengganu State Education Department.
PPT for Demo.pptx
This document provides a lesson on transforming linear equations between the forms Ax + By = C and y = mx + b. It begins with reviewing key concepts of linear equations, slopes, x-intercepts and y-intercepts. Examples are given of transforming equations between the two forms, including the equations 1.4x - 5y = 20 and 2.3x + 7y = 14 transformed to y = mx + b form, and the equations y = x + 3 and y = 5 - 3x transformed to Ax + By = C form. Students are then given a quiz to practice transforming equations between the two linear equation forms.
DEV
The document provides step-by-step instructions for graphing different types of rational functions and polynomials. It includes two examples of graphing rational equations, showing how to find the greatest common factor, factor the numerator and denominator, identify holes and asymptotes, and graph. It also gives an example of solving a word problem involving perimeter and area of a rectangular field enclosed by fencing. Finally, it discusses how to graph polynomials based on their degree and signs.
April 7, 2014
1) The document outlines a class agenda that includes reviewing quadratics and new solving methods in preparation for a Friday test. It also notes a third quarter exam on Friday covering quadratics, factoring, exponents, and polynomials.
2) The class will begin with warm-up problems reviewing graphing quadratics and finding axes of symmetry. Steps for graphing the equation y + 6x = x^2 + 9 are provided.
3) Additional examples are given for graphing a quadratic function modeling the height of a basketball after being shot, finding the maximum height and time to reach it by using the vertex of the parabola.
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Untitled presentation.pptx
- 1. BIRTHDAY POLYNOMIAL BY: MA. KRISTIANNE GAYLE MONANA 9-LITHIUM
- 2. GRAPH BIRTHDAY: AUGUST 27, 2009 POLYNOMIAL: P(x)=8x^6+2x^5+7x^4+2x^3+x^2+x+9 If you manipulate the polynomial expression by adjusting the signs of its terms, you end up with the following arrangement:
- 3. GRAPH PROPERTIES 1 DOMAIN AND RANGE 2 leading coefficient 3 y intercept 4 roots 5 end behavior D= R= LC= 8 Y INTERCEPT= (0,9) R= NO SOLUTION EB= RISES TO THE LEFT = RISES TO THE RIGHT
- 4. THINKING PROCESS This project was intriguing as it demanded a considerable amount of time and dedication to complete. I embarked on a journey of exploration manipulating the signs of the polynomial function ( P(x)=8x^6+2x^5+7x^4+2x^3+x^2+x+9) to craft a distinctive graph. Witnessing the substantial transformations resulting from these alterations was a source of genuine enjoyment. Moreover, my approach to design was primarily exploratory as I sought novel elements that would enhance the visual appeal of my presentation.
- 5. End of presentation thank you for taking your time for this!