5.8 Modeling with Quadratic Functions
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This document discusses modeling with quadratic functions. It provides examples of writing quadratic functions in vertex form, intercept form, and standard form given certain information about the parabola. It also discusses finding a quadratic model from a data set and using it to find the maximum point.
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SAMPLE QUESTIONExercise 1 Consider the functionf (x,C).docx
SAMPLE QUESTION:
Exercise 1: Consider the function
f (x,C)=
sin(C x)
Cx
(a) Create a vector x with 100 elements from -3*pi to 3*pi. Write f as an inline or anonymous function
and generate the vectors y1 = f(x,C1), y2 = f(x,C2) and y3 = f(x,C3), where C1 = 1, C2 = 2 and
C3 = 3. Make sure you suppress the output of x and y's vectors. Plot the function f (for the three
C's above), name the axis, give a title to the plot and include a legend to identify the plots. Add a
grid to the plot.
(b) Without using inline or anonymous functions write a function+function structure m-file that does
the same job as in part (a)
SAMPLE LAB WRITEUP:
MAT 275 MATLAB LAB 1 NAME: __________________________
LAB DAY and TIME:______________
Instructor: _______________________
Exercise 1
(a)
x = linspace(-3*pi,3*pi); % generating x vector - default value for number
% of pts linspace is 100
f= @(x,C) sin(C*x)./(C*x) % C will be just a constant, no need for ".*"
C1 = 1, C2 = 2, C3 = 3 % Using commans to separate commands
y1 = f(x,C1); y2 = f(x,C2); y3 = f(x,C3); % supressing the y's
plot(x,y1,'b.-', x,y2,'ro-', x,y3,'ks-') % using different markers for
% black and white plots
xlabel('x'), ylabel('y') % labeling the axis
title('f(x,C) = sin(Cx)/(Cx)') % adding a title
legend('C = 1','C = 2','C = 3') % adding a legend
grid on
Command window output:
f =
@(x,C)sin(C*x)./(C*x)
C1 =
1
C2 =
2
C3 =
3
(b)
M-file of structure function+function
function ex1
x = linspace(-3*pi,3*pi); % generating x vector - default value for number
% of pts linspace is 100
C1 = 1, C2 = 2, C3 = 3 % Using commans to separate commands
y1 = f(x,C1); y2 = f(x,C2); y3 = f(x,C3); % function f is defined below
plot(x,y1,'b.-', x,y2,'ro-', x,y3,'ks-') % using different markers for
% black and white plots
xlabel('x'), ylabel('y') % labeling the axis
title('f(x,C) = sin(Cx)/(Cx)') % adding a title
legend('C = 1','C = 2','C = 3') % adding a legend
grid on
end
function y = f(x,C)
y = sin(C*x)./(C*x);
end
Command window output:
C1 =
1
C2 =
2
C3 =
3
More instructions for the lab write-up:
1) You are not obligated to use the 'diary' function. It was presented only for you convenience. You
should be copying and pasting your code, plots, and results into some sort of "Word" type editor that
will allow you to import graphs and such. Make sure you always include the commands to generate
what is been asked and include the outputs (from command window and plots), unless the pr.
April 3, 2014
The document provides examples and explanations for graphing quadratic functions. It begins with an overview of how the a, b, and c values in the quadratic function y=ax2 + bx + c impact the graph. Examples are then worked through step-by-step to show how to find the axis of symmetry, vertex, y-intercept, and additional points to graph the function. An application example models the height of a basketball shot as a quadratic function to find the maximum height and time to reach it. The document concludes with a check your understanding example modeling the height of a dive.
6.4 intercept form
The document discusses intercept form of quadratic functions. Intercept form is written as y = a(x - p)(x - q), where p and q are the x-intercepts. The axis of symmetry is halfway between the intercepts. Examples are provided of writing quadratic functions in intercept form, finding the zeros or x-intercepts, and graphing functions in intercept form by plotting the intercepts and axis of symmetry. Methods are also described for writing a quadratic function equation from a graph in intercept form.
Advanced matlab codigos matematicos
This document discusses various ways to work with polynomials in MATLAB. It covers representing polynomials as vectors, evaluating polynomials, finding roots, computing coefficients from roots, adding, subtracting, multiplying, dividing, taking derivatives and integrals of polynomials, fitting curves to data using polynomial regression, and interpolating values between data points using polynomials.
2 3 Bzca5e
The document discusses linear functions and slopes. It provides examples of finding the slope of a line between two points, writing the equation of a line in point-slope and slope-intercept form, graphing linear equations, finding the x- and y-intercepts of a line, and applications of linear functions including using a graphing calculator.
REPRESENTATION OF FUNCTIONS.pptx
This document discusses functions and their representations. It begins by defining relations and functions, and providing examples of each. It then discusses different types of functions including linear, quadratic, constant, identity, absolute value, and piecewise functions. Examples are provided for each type. The document ends with exercises asking the reader to determine if given relations are functions, and to identify what type of function is being described.
STA003_WK2_L.pptx
Here are the key steps to solve this problem algebraically:
Let x = number of units of product X
Let y = number of units of product Y
Equation for process A: 3x + y ≤ 1750
Equation for process B: 2x + 4y ≤ 4000
Solve the two equations simultaneously using elimination:
3x + y = 1750
2x + 4y = 4000
Eliminate y by subtracting the equations:
x = 1250
Substitute x = 1250 into one of the original equations to find y:
3(1250) + y = 1750
y = 500
Therefore, the maximum number of units of X is 1250 and
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.
STA003_WK2_L.pdf
Here are the key steps to solve this problem algebraically:
Let x = number of units of product X
Let y = number of units of product Y
Write equations relating the process hours to the number of units:
3x + 2x = Hours used in A
1y + 4y = Hours used in B
Solve the simultaneous equations to find the maximum number of each product that can be made.
1.Find the coordinates of the vertex for the parabola define.docx
1.
Find the coordinates of the vertex for the parabola defined and give
the domain of the function in set-builder notation:
f
(
x
) = −3(
x
+ 2)
2
+ 12
2.
Use the vertex and intercepts to graph the quadratic function. Give
the equation of the parabola’s axis of symmetry. Use the graph to
determine the Domain and Range of the function (also in set-builder
notation):
y
− 3 = (
x
− 1)
2
3. Use the vertex and intercepts to graph the quadratic function. Give
the equation of the parabola’s axis of symmetry. Use the graph to
determine the Domain and Range of the function (also in set-builder
notation):
g
(
x
) = 2
x
2
− 7
x
− 4
4.
Determine, without graphing and the use of a calculator, whether
the function has a Minimum value or a Maximum value. Find the
Minimum or Maximum value and determine where it occurs for:
h
(
x
) = −2
x
2
− 12
x
+ 3
5.
Give the domain and range in set-builder notation for the given quadratic
function whose graph is described:
·
the vertex is (−3
,
−4)
·
and the parabola opens down.
6.
Give the domain and range in set-builder notation for the given quadratic
function whose graph is described:
. the minimum = 18 at
x
= 10
7.
(*) Write an equation, in standard form, of the parabolic function
that has the shape as the graph of
f
(
x
) = 2
x
2
, but with the vertex as
the given point: (-8
,
-6)
8.
Determine whether
f
(
x
) =
α
4
- 2
α
2
+ 1 is even, odd, or neither.
Describe the symmetry, if any, for the graph of
f
.
9. An athlete whose event is the shot-put releases the shot with the SAME
initial velocity, but at different angles (35
°
,
65
°
) is based on parabolic
functions that model parabolic paths. When the shot whose path is released at an angle of 65
°
, its height,
g
(
x
), in feet, can be modeled
by:
g
(
x
) = —0
.
04
x
2
+ 2
.
1
x
+ 6
.
1
where
x
is the shot’s horizontal distance, in feet, from its point of
release. Use this model to solve the following:
(a)
What is the maximum height, to the nearest tenth of a foot, of
the shot and how far from its point of release does this occur?
(b)
What is the shot’s maximum horizontal distance, to the nearest
tenth of a foot, or the distance of the throw?
(c)
from what height was the shot released?
10. You have 200 feet of fencing to enclose a rectangular plot that borders
on a river. If you do not fence the side along the river, find the length
and width of the plot that will maximize the area. What is the largest
area that can be enclosed?
11. Does the following make sense? State Yes, or No, and why you believe
it to be so (This is NOT a guessing question, you need to support your
argument here):
The graph of
h
(
x
) = -2(
x
+ 4)
2
-8 has one
y
-intercept and
two
x
-intercepts.
12. Given
f
(
x
) =
4
5
- (
x
-
1
2
)
2
, find the following information and graph
on a rectangular coordinate system:
(a)
y
-intercept:
(b)
x
-intercept(s):
(c)
Orientation:
(d)
Dis.
Natural and Clamped Cubic Splines
This document describes the process of using natural and clamped cubic splines to approximate functions based on data points. It presents the mathematical formulas for natural and clamped cubic splines and their derivatives. Code functions are provided to calculate the splines and plot the results. The document demonstrates applying this process to example functions and data, showing the natural and clamped cubic splines accurately fit the original functions.
Due Week 10 and worth 250 pointsIn preparation for this assignme.docx
Due Week 10 and worth 250 points
In preparation for this assignment, please view the Jurisville scenarios and resulting simulations from Weeks 8 through 10 in the Corrections unit.
In the scenarios and resulting simulations, Robert Donovan, a Jurisville probation officer, discusses the intricacies of probation. Kris, the defendant, is offered an intensive supervised probation plan to follow. Brennan Brooke, a senior criminologist, discusses the tailoring of the inmate to the appropriate facility. Finally, Orlando Boyce, a sergeant at the fictional Deephall correctional facility, discusses measures that could conceivably make prison life effective and thus decrease the likelihood of recidivism.
Write a three to four (3-4) page paper in which you:
1. Outline your findings from your review of the file of Kris, for whom Robert is considering probation. State whether or not your results from the file review match Robert’s. Explain two (2) instances in which your views and those of Robert are both similar and different.
2. Develop a profile of the so-called perfect candidate to participate in an intensive supervised probation program. The profile should contain at least three (3) attributes that you believe make this defendant the perfect candidate for this type of probation.
3. Defend or critique the strategy of matching the inmate to the correctional facility as a response to the legal concept of cruel and unusual punishment. Provide a rationale for your position with concrete examples.
4. Defend or critique whether programs and amenities geared to making prison life effective—which run the gamut from hiring extra officers, to counseling and therapy, to building a garden—are time and taxpayer money well spent.
5. Use at least three (3) quality resources in this assignment. Note: Wikipedia and similar Websites do not qualify as quality resources.
Your assignment must follow these formatting requirements:
· Be typed, double spaced, using Times New Roman font (size 12), with one-inch margins on all sides; citations and references must follow APA or school-specific format. Check with your professor for any additional instructions.
· Include a cover page containing the title of the assignment, the student’s name, the professor’s name, the course title, and the date. The cover page and the reference page are not included in the required assignment page length.
The specific course learning outcomes associated with this assignment are:
· Outline the major characteristics and purposes of prisons, including prisoners’ rights and prison society.
· Use technology and information resources to research issues in criminal justice.
· Write clearly and concisely about criminal justice using proper writing mechanics and APA style conventions.
Grading for this assignment will be based on answer quality, logic / organization of the paper, and language and writing skills, using the following rubric.
MULTIPLE CHOICE. Choose the one alternative that best complete ...
Fst ch2 notes
The document defines key terms related to functions including univariate and bivariate data, independent and dependent variables, domain and range, and linear, exponential, quadratic, and step functions. It provides examples of evaluating various functions and finding linear and quadratic models to describe relationships between variables from sets of data points. The overall content describes different types of mathematical functions and how to analyze and model real-world data using functions.
Language Language Models (in 2023) - OpenAI
1. Emergent Abilities with Scale: The presentation underscores the significance of viewing the development of language models with a perspective of “yet”, highlighting that many ideas may not work now but could become viable as models scale. This perspective challenges traditional scientific experimentation by suggesting that axioms in the field of language models are subject to change with advancements in model capabilities.
2. The Need for Constant Unlearning: As models scale, previously held intuitions may become outdated. The presentation discusses the necessity for researchers to unlearn invalidated ideas, noting that newcomers may sometimes have an advantage due to having fewer entrenched misconceptions.
3. Scaling Challenges and Techniques: The presentation elaborates on the technical challenges and complexities involved in scaling LLMs, using examples from training processes that include unexpected loss spikes. It also touches upon the importance of documenting experiments that fail due to insufficient model “intelligence” and retesting them as models evolve.
4. Instruction Fine-Tuning and RLHF: The presentation discusses instruction fine-tuning as a method to improve model performance across a wide range of tasks by framing tasks in natural language. However, it also points out the limitations of instruction fine-tuning and the potential of reinforcement learning from human feedback (RLHF) to address some of these challenges by learning the objective function.
5. Technical Insights on Transformer Models: Detailed technical insights into the functioning of Transformer models are provided, including tokenization, embedding, and the sequential processing that underpins these models’ ability to understand and generate language.
6. Scaling Infrastructure: The presentation gives an overview of the infrastructure considerations for scaling LLMs, including the use of tensor processing units (TPUs) and the role of software tools like JAX for parallelizing model training across multiple hardware units.
7. The Bitter Lesson and Future Directions: Reiterating “the bitter lesson” in AI research—that progress often comes from scalable general methods rather than specialized approaches—the presentation hints at ongoing and future directions in LLM research, emphasizing scalability, the reduction of inductive biases, and the exploration of novel training paradigms.
Computer Graphics in Java and Scala - Part 1
Computer Graphics in Java and Scala - Part 1.
Continuous (Logical) and Discrete (Device) Coordinates,
with a simple yet pleasing example involving concentric triangles.
Scala code: https://github.com/philipschwarz/computer-graphics-50-triangles-scala
Errata:
1. Scala classes TrianglesPanel and Triangles need not be classes, they could just be objects.
3rd Semester Computer Science and Engineering (ACU-2022) Question papers
3rd Semester Computer Science and Engineering (ACU-2022) Question papersBGS Institute of Technology, Adichunchanagiri University (ACU)
Unix and Shell Programming,
Q P Code: 60305.
Additional Mathematics I
Q P Code: 60306
Computer Organization and Architecture
Q P Code: 62303
Data Structures Using C
Q P Code: 60303
Discrete Mathematical Structures
Q P Code: 60304
Engineering Mathematics - III
Q P Code: 60301
Soft Skill Development
Q P Code: 60307Ch 7 tutoring notes quadratics
The document discusses transformations of quadratic functions in vertex form f(x) = a(x-h)2 + k. It explains how changing the coefficients a, h, and k affects the graph of the quadratic function. Specifically, it states that changing a widens or narrows the graph, changing h shifts the graph left or right, and changing k shifts the graph up or down. It also provides examples of writing equations for quadratic functions based on given graphs and finding the vertex of a quadratic function in standard form.
Similar to 5.8 Modeling with Quadratic Functions (20)
SAMPLE QUESTIONExercise 1 Consider the functionf (x,C).docx
SAMPLE QUESTIONExercise 1 Consider the functionf (x,C).docx
1.Find the coordinates of the vertex for the parabola define.docx
1.Find the coordinates of the vertex for the parabola define.docx
Due Week 10 and worth 250 pointsIn preparation for this assignme.docx
Due Week 10 and worth 250 pointsIn preparation for this assignme.docx
3rd Semester Computer Science and Engineering (ACU-2022) Question papers
3rd Semester Computer Science and Engineering (ACU-2022) Question papers
More from hisema01
7.3 rational exponents
Nth roots can be expressed using rational exponents, where the denominator of the exponent is the index of the radical. To evaluate expressions with rational exponents, one should rewrite them in radical form, evaluate the radical, and then raise the expression to the power indicated by the original exponent. Examples are provided of evaluating various expressions with rational exponents.
6.6 quadratic formula
The document introduces the quadratic formula as a method for solving any quadratic equation in the standard form of ax^2 + bx + c = 0. It provides the quadratic formula, and shows an example of using it to solve for x in a quadratic equation. Users are then prompted to try solving for x in a quadratic equation using the quadratic formula.
6 7 new look at conics
This document discusses classifying second-degree equations as different types of conic sections. It provides the general form of a second-degree equation, explains how to use the discriminant to determine if the graph is a circle, ellipse, parabola, or hyperbola, and gives examples of identifying the conic section from different equations. Key aspects covered include degenerate conics, the general form containing terms for x2, xy, y2, x, y, and the constant, and using the discriminant calculated from the equation coefficients to classify the conic section.
6 6 systems of second degree equations
This document discusses methods for solving systems of second-degree equations, including algebraic substitution or elimination methods and graphical methods. It provides 3 examples of solving systems of equations algebraically and graphically, including solving systems of quadratic equations and systems involving circles.
4.4 multi step trig problems
This document provides instructions for solving for the length of a shared side (x) of two triangles. It instructs the user to write two equations, one for each triangle, relating the lengths of the sides including x. It also describes an approach for solving when the triangles form one large non-right triangle, which is to draw the altitude splitting it into two triangles and using the fact that triangle angles sum to 180 degrees. The user is then told to solve the two equations to find the value of x.
6 5 parabolas
This document provides an overview of parabolas including their key properties and examples of how to graph and write equations of parabolas. It defines that every point on a parabola is equidistant from the focus and directrix, with the directrix perpendicular to the line of symmetry and the vertex halfway between them. Examples are given of finding the focus, directrix, and graphing parabolas given these features as well as writing the equation of a parabola given its directrix.
4.3 finding missing angles
Inverse trig ratios, including sin-1, cos-1, and tan-1, are used to solve for missing angle measures. When given a decimal value for sin, cos, or tan, the corresponding inverse trig function is used to find the angle measure to the nearest degree. When given a triangle, the appropriate trig ratio is set up and the inverse function is then used to find the measure of the indicated angle to the nearest degree.
4.2 solving for missing sides
This document provides instructions for solving for a missing side of a triangle using trigonometric functions. It tells the reader to label the sides of the triangle, set up an equation using the information given, and solve for the variable to find the missing side, making sure a calculator is in degree mode.
4.1 trig ratios
This document defines and explains the three main trigonometric ratios - sine, cosine, and tangent - which are used to solve for missing angles and sides of right triangles. It identifies the hypotenuse as the longest side, and the adjacent and opposite sides in relation to a given angle. The document provides the ratio equations and examples of calculating trig ratios for various angles in a right triangle.
R.4 solving literal equations
This document defines a literal equation as an equation with two or more different variables. It provides examples of literal equations such as A = bh, where b is solved for, and P = 2L + 2W, where W is solved for. The document demonstrates how to solve various literal equations for the indicated variable, such as solving for P in I = Prt, solving for x in A = x + y^2, and solving for x and y in two additional equations.
R.3 solving 1 var inequalities
This document provides instructions for solving inequalities and graphing their solutions. It notes that when multiplying or dividing an inequality by a negative number, the inequality symbol is reversed. Open circles are used for < and >, while closed circles are used for ≤ and ≥. Examples are provided of solving various inequalities algebraically and graphing the resulting solution sets on a number line.
R.2 solving multi step equations
Multi-step equations involve more than two steps to solve. To solve them, simplify the side with variables by distributing and combining like terms, then solve as a single-step or two-step equation. The key steps are to distribute any parentheses, collect like variable terms on one side and constants on the other, and use opposite operations to solve for the variable.
R.1 simplifying expressions
Algebraic expressions contain numbers, variables, and operators and can be simplified using the distributive property and by combining like terms. The document provides examples of algebraic expressions that can be simplified by first distributing all terms inside parentheses and then combining like terms, such as 3(4n + 2) + n, which is simplified to 12n + 6 + n = 13n + 6.
10 4 solving trig equations
This document discusses two methods for solving trigonometric equations: graphing and algebraically. Graphing allows for visually seeing solutions but may result in decimal answers, while algebraic solving provides exact solutions but can become messy. It provides tips for algebraic solving based on whether the equation contains functions of 2x and x, or only 2x. Examples are given to demonstrate each method.
7 2 adding and subtracting polynomials
The document discusses several methods for adding and subtracting polynomials: using algebra tiles, the horizontal method, and the vertical method. It provides examples of adding and subtracting polynomials with one and two variables. Terms with the same variables are combined by adding the coefficients. For subtraction, the signs of the terms in the subtracted polynomial are changed before adding.
7 3 multiplying polynomials
This document discusses four methods for multiplying binomial expressions: algebra tiles, the distributive property, a table or "box" method, and FOIL. It explains that every term in the first binomial must be multiplied by every term in the second binomial. Examples are provided of multiplying binomials such as (x + 2)(x + 5), along with practice problems like (x + 3)(x - 4) and (2p - 4)(3p + 2).
7.2 simplifying radicals
The document discusses simplifying radical expressions. It defines key terms like the radical sign and radicand. It provides two methods for simplifying radicals: using the product property to rewrite with the largest perfect square factor or making a factor tree to pull out factors. Examples are provided to demonstrate simplifying radicals of various forms, including those with variables and higher root expressions. The document also contains practice problems for simplifying radical expressions.
10 3 double and half-angle formulas
1) This document discusses double-angle and half-angle formulas for trigonometric functions like sine, cosine, and tangent.
2) It derives formulas that relate trig functions of double and half angles to trig functions of the original angle, such as sin(2x) = 2sin(x)cos(x) and sin(x/2) = ±√(1-cos(x))/2.
3) Examples are provided to demonstrate applying these formulas to simplify trigonometric expressions and derive new identities.
6.7 other methods for solving
This document discusses two additional methods for solving quadratic equations: the quadratic formula and graphing. The quadratic formula can be used to solve any quadratic equation in standard form (ax^2 + bx + c = 0) by inputting the a, b, and c coefficients. Graphing involves changing the equation to y= form, plotting it on a calculator to find the x-intercepts, which are the solutions. Both methods are demonstrated with examples.
10 2 sum and diff formulas for tangent
1) Formulas are derived for tangent of the sum and difference of two angles using previous trigonometric identities: tan(α + β) = (tanα + tanβ) / (1 - tanαtanβ) and tan(α - β) = (tanα - tanβ) / (1 + tanαtanβ).
2) These formulas are valid when the tangents of the individual angles and their sum/difference are defined, which excludes cases where the cosine of any angle is 0.
3) Examples are provided to demonstrate applying these formulas to find the exact value of tangents of sums/differences of known angles.
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5.8 Modeling with Quadratic Functions
- 1. 5.8 Modeling with Quadratic Functions
- 2. Writing a Quadratic in Vertex Form: y=a(x-h)2 + k Given the vertex and a point on the parabola, we can write the function. Example: (4, 1) vertex: (2, -3)
- 3. Your Turn! Write an equation of the parabola with vertex (-1, 4) that goes through the point (2, 7).
- 4. Writing a Quadratic in Intercept Form: y=a(x-p)(x-q) Given the x-intercepts and a point on the parabola, we can write the function. Example: (-1, 2)
- 5. Your Turn! Write an equation of the parabola with x-intercepts 1 and 4 that goes through the point (2, -6).
- 6. Writing a Quadratic in Standard Form: y = ax2 + bx + c Given three points on the parabola, we can write the function. 1. Substitute each ordered pair into y = ax2 + bx + c to get three equations in a, b, and c. 2. Solve the 3-variable system to find a, b, and c. 3. Write in standard form.
- 7. Example Write a function in standard form for the parabola passing through the points (2, -2), (3, 4), and (0, -2).
- 8. Your Turn! Write a function in standard form for the parabola passing through the points (-2, 0), (-1, 2), and (3, 0).
- 9. Finding a Quadratic Model for a Data Set You can use a graphing calculator Example: A study compared the speed x (in mph) and the avg. fuel economy y (in miles per gallon) for cars. Find a quadratic model for the data and find the speed that maximizes fuel economy. Speed, x 15 20 25 30 35 40 45 50 55 60 65 70 Fuel 22.3 25.5 27.5 29.0 28.8 30.0 29.9 30.2 30.4 28.8 27.4 25.3 Economy, y