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# a real life application of a linear function

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From Wikipedia

Linear function

In mathematics, the term linear function can refer to either of two different but related concepts:

• a first-degree polynomial function of one variable;
• a map between two vector spaces that preserves vector addition and scalar multiplication.

## Analytic geometry

In analytic geometry, the term linear function is sometimes used to mean a first-degree polynomialfunction of one variable. These functions are known as "linear" because they are precisely the functions whose graph in the Cartesian coordinate plane is a straight line.

Such a function can be written as

f(x) = mx + b
(y-y1) = m(x-x1)
0= Ax + By + C

(called slope-intercept form), where m and b are realconstants and x is a real variable. The constant m is often called the slope or gradient, while b is the y-intercept, which gives the point of intersection between the graph of the function and the y-axis. Changing m makes the line steeper or shallower, while changing b moves the line up or down.

Examples of functions whose graph is a line include the following:

• f_{1}(x) = 2x+1
• f_{2}(x) = x/2+1
• f_{3}(x) = x/2-1.

The graphs of these are shown in the image at right.

## Vector spaces

In advanced mathematics, a linear function means a function that is a linear map, that is, a map between two vector spaces that preserves vector addition and scalar multiplication.

For example, if x and f(x) are represented as coordinate vectors, then the linear functions are those functions f that can be expressed as

f(x) = \mathrm{M}x,

where M is a matrix. A function

f(x) = mx + b

is a linear map if and only if b = 0. For other values of b this falls in the more general class of affine maps.

Question:Using the Library, web resources, and/or other materials, find a real-life application of a linear function. State the application, give the equation of the linear function, and state what the x and y in the application represent. Choose at least two values of x to input into your function and find the corresponding y for each. State, in words, what each x and y means in terms of your real-life application.

Answers:You might impress your teacher with a linear function that has a negative correlation - that is, as x increases, y decreases. For example - the value of your car as it increases in age.

Question:If a derivative means to find the "rate of change" then: (1) What is a differential equation mean, or used for? (2) What is linear algebra used for?

Answers:Yes... the derivative of a function F(x) is another function f(x) that outputs the instantaneous slope of F(x) at x. Does that make sense? if y = x then y' = 2x... The parabola y=x , at every value x, has a slope of 2x I dont know how much you know about derivative calculus... but you seem to understand what it does and whats its for already. === 1) What are differential equations? DEs are equations, functions, that relate a function f(x) and its varying degrees of differentiations to each other and to other functions of x. Often times, the objective of a DE is to find out what f(x) is, explicitly, in terms of 'x'; without relating f(x) to any of its derivatives Like, for example, if the DE was: y + 2y' - x = 4 And I asked you to figure out what the function y(x) was in terms of x... The answer would not be y(x) = 4 + x - 2y'(x)... because that is just the same DE rearranged. In many instances, from physics to statistics to economics... differential equations show up. DE's are equations that relate a functions output, quantity and its rate of change over the course of time 't' in a single equation... these relationships might be easy to determine with the data they have... but the mathematician might want to figure out what the quantity is at time t, or what its rate of change is... without having to know the other first

Question:Hey everyone, can you help me with this problem? I have an exam next week and am just going through some problems to help me practice. The problem reads: The quarterback of a football team releases a pass at a height of 7 feet above the playing field, and the football is caught by a receiver 30 yards directly down field at a height of 4 feet. The pass is released at an angle of 35 degrees with the horizontal. a.) what's the speed of the ball when released? b.) max height of the football? c.) time the receiver has to reach the proper position after the quarterback releases the ball? I think my main problem is my lack of knowledge of football, because i cannot seem to make a good enough picture in my mind. Any help as to where to go with this problem? Thank you guys so much!

Answers:This is a classic ballistics problem and has nothing to do with football or real life. You can treat it as primarily a math problem or as a physics problem, but you need at least some bits from both. Assuming no air friction, the path of football is a parabola. You are given two points on the parabola: <0, 7> and <90, 4>. You are also given the tangent at the point <0, 7> (note that the angle is positive - y increases as x increases from 0). That defines the parabola. Part b asks what is the maximum value of y. Now you have to bring some physics into it. In the vertical direction, the path of a projectile is defined by: Vf = Vi + A T D = Vi T + (1/2) A T^2 Where: T is the time interval Vi is the initial vertical velocity Vf is the final vertical velocity D is the vertical distance traveled A is the vertical acceleration. In this case, the only acceleration is that due to gravity - approximately 32 ft/s^2 At the same time, the horizontal velocity is constant. So you have y as a function of x and y as a function of t so you can determine x as a function of t. This will also give you the initial vertical component of the velocity of the ball, and the total time the ball takes to make its trip.

Question:This is gonna be an application question on my test. What are some real life applications of the oxidation of alcohols? Alcohols + {O} = (Aldehyde, Ketone, or Carboxylic acid)

Answers:One of the major processes for making formaldehyde commercially is the oxidation of methanol. Formaldehyde is an important industrial chemical used in things like plastics and insulation.