Examples of constant in the following topics:
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- Indirect variation is used to describe the relationship between two variables when their product is constant.
- Knowing that the relationship between the two variables is constant, we can show that their relationship is:
- where k is a constant known as the constant of proportionality.
- Revisiting the example of the decelerating car, let's say it starts at 50 miles per hour and slows at a constant rate.
- Other constants can be incorporated into the equation for the sake of accuracy, but the overall form will remain the same.
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- Functions can either be constant, increasing as $x$ increases, or decreasing as $x
$ increases.
- A function is a constant function if $f(x)=c$ for all values of $x$ and some constant $c$.
- Example 1: Identify the intervals where the function is increasing, decreasing, or constant.
- There are no intervals where this curve is constant.
- Identify whether a function is increasing, decreasing, constant, or none of these
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- This is accomplished by multiplying either $x$ or $y$ by a constant, respectively.
- Multiplying the entire function $f(x)$ by a constant greater than one causes all the $y$ values of an equation to increase.
- where $f(x)$ is some function and $b$ is an
arbitrary constant.
- Multiplying the independent variable $x$ by a constant greater than one causes all the $x$ values of an equation to increase.
- where $f(x)$ is some function and $c$ is an arbitrary constant.
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- The number $e$ is an important mathematical constant, approximately equal to $2.71828$.
- The number $e$, sometimes called the natural number or Euler's number, is an important mathematical constant, approximately equal to 2.71828.
- The number $e$ is very important in mathematics, alongside $0, 1, i, \, \text{and} \, \pi.$ All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler's identity, which (amazingly) states that $e^{i\pi}+1=0.$ Like the constant $\pi$, $e$ is irrational (it cannot be written as a ratio of integers), and it is transcendental (it is not a root of any non-zero polynomial with rational coefficients).
- Jacob Bernoulli discovered this constant by asking questions related to the amount of money in an account after a certain number of years, if the interest is compounded $n$ times per year.
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- Since we know that the relationship between two values is constant, we can give their relationship with:
- Knowing that the relationship between the
two variables is constant, we can show that their relationship is:
- where
$k$ is a constant known as the constant of proportionality.
- Thus, an inverse relationship cannot be represented by a line with constant slope.
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- A linear function is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.
- For example, a common equation, $y=mx+b$, (namely the slope-intercept form, which we will learn more about later) is a linear function because it meets both criteria with $x$ and $y$ as variables and $m$ and $b$ as constants.
- In the linear function graphs below, the constant, $m$, determines the slope or gradient of that line, and the constant term, $b$, determines the point at which the line crosses the $y$-axis, otherwise known as the $y$-intercept.
- Vertical lines have an undefined slope, and cannot be represented in the form $y=mx+b$, but instead as an equation of the form $x=c$ for a constant $c$, because the vertical line intersects a value on the $x$-axis, $c$.
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- Knowing that the relationship between two variables is constant, we can show their relationship as :
- where k is a constant known as the constant of proportionality.
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- An arithmetic sequence is a sequence of numbers in which the difference between the consecutive terms is constant.
- An arithmetic progression, or arithmetic sequence, is a sequence of numbers such that the difference between the consecutive terms is constant.
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- Make sure that one side of the equation is only variables and their coefficients, and the other side is just constants.
- Solving a system of linear equations using the inverse of a matrix requires the definition of two new matrices: $X$ is the matrix representing the variables of the system, and $B$ is the matrix representing the constants.
- To solve a system of linear equations using an inverse matrix, let $A$ be the coefficient matrix, let $X$ be the variable matrix, and let $B$ be the constant matrix.
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- where $a$, $b$, and $c$ are constants and $x$ is the independent variable.
- The constants $b$and $c$ can take any finite value, and $a$ can take any finite value other than $0$.
- When all constants are known, a quadratic equation can be solved as to find a solution of $x$.
- With a linear function, each input has an individual, unique output (assuming the output is not a constant).