What is the unit of linear displacement
Properties of linear functions
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In these explanations you will learn what properties linear functions have and how you can recognize them using their graphical representation or the function equation.
The straight line as a graph of a linear function
The term linear is derived from the Latin linea = "leash, cord, thread". The graph of a linear function is, so to speak, a "taut line", i.e. a straight line. You can distinguish the graph of a linear function from the graphs of other functions.
The lines f, g and q are the graphs of linear functions.
The straight line k is not a graph of a linear function.
The functional equation of a linear function
You can also use the function equation to distinguish linear functions from others. A curve runs in a straight line if, with a uniform increase (or decrease) in the x-values (arguments), the y-values (function values) also increase (or decrease) uniformly. This is exactly the case when the variable x in the function term is only multiplied by a factor (the slope). This indicates how much the function values increase or decrease when x changes.
The graph of the function f is described by the straight line equation.
The equation can also contain another summand, the so-called absolute term. This indicates at which point the straight line intersects the y-axis and is therefore also called the y-axis intercept.
The graphs of the functions h, g and i are described by the straight line equations:
The equation of a linear function always has the form. It is also called the normal form of the straight line equation, where m is the slope and b is the y-axis intercept of the function.
Influence of the parameters m and b and special cases
The parameters m and b in the function equation determine the respective course of the graph. Here, m is the measure of the slope and b causes a shift along the y-axis. Depending on the assignment of the parameters m and b, there are different special cases.
The figure shows all of these special cases in a coordinate system.
The slope behavior of the graph of a linear function
The gradient behavior of the straight line depends directly on the value for m: - The greater the amount of m, the steeper the straight line runs - The smaller the value of m, the flatter the straight line runs - If m is positive, the straight line rises Straight line from bottom left to top right - If m is negative, then the straight line falls from top left to bottom right.
The linear functions f, g, h and p are given with f (x) =, g (x) =, h (x) = and p (x) =.
Which statements are true?
If two straight line equations have the same value for m and different y-axis intercepts, then the straight lines have the same slope, they are parallel.
Straight line equations in normal form and in implicit form
The normal form of the straight line equation corresponds to the functional equation of a linear function. With the help of this equation you can draw the graph of the function, i.e. the straight line, because you can take the important parameters for the straight line directly from the equation (m and b). A straight line equation can also be given in the so-called implicit form:. You cannot take the slope and the y-axis intercept directly from this equation.
Convert the equation to normal form.
Linear functions in factual situations
You can describe many situations in everyday life with the help of linear functions. But how do you know? The most common terms used in verbal description include continuous, even, regular, daily, weekly, etc.
Lena has in her piggy bank and from now on she wants to save at the end of each week. Sum in the piggy bank in euros (f (x)) can be described by a linear function.
Mr. Meier pays his telephone provider a basic fee of and for each started minute of conversation. The allocation of the number of minutes started (x) Invoice amount in euros (f (x)) can be described by a linear function.
The basic fee, a starting credit or any initial value always corresponds to the y-axis intercept, because this is the value that was already available at time 0 or when conversation minutes started or must be paid anyway.
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