What similarities and differances do you see between functions and linear equations studied in Ch.3? are all linear equations functions? is there an instance when a linear equation is not a function? support your answer. create an equation of a nonlinear function and provide two inputs for your classmates to evaluate.
A "function" is a way of computing one number when you are given another. Linear equations with two variables define a such a functional relationship. Each of the two variables is a linear function of the other.
However, when you write an equation, linear or not, you are not always defining a function.
x + y = 14 is an equation, not a function, but can be used to derive the linear functions:
y = 14 - x and x = 14 - y
An example of nonlinear funtion is
y = 4 x^2
In Chapter 3, the similarities between functions and linear equations are that both involve mathematical relationships between variables. Both functions and linear equations can be represented by equations, and they can be graphed on a coordinate plane.
However, there are also differences between functions and linear equations. A linear equation is a specific type of function that has a constant rate of change. It can be written in the form y = mx + b, where m represents the slope and b represents the y-intercept. Linear equations describe a straight line on a graph.
Not all linear equations are functions. A function is a relationship where each input (x-value) corresponds to exactly one output (y-value). If there is any x-value that has more than one y-value, then the equation is not a function. In the context of linear equations, this occurs when the slope is zero or undefined.
Let's consider an example to illustrate this:
1) y = 2x + 3
This equation represents a linear function. For any given value of x, there is only one corresponding value of y. It can be graphed as a straight line.
2) x = 3
This equation represents a linear equation but is not a function. Since x is given, there is no variability in the output y. It represents a vertical line parallel to the y-axis and does not satisfy the definition of a function.
Now, let's create a nonlinear function:
3) y = x^2
This equation represents a nonlinear function. It is a quadratic function where the output (y) is the square of the input (x). For this function, any real value of x will have a unique value of y.
Input examples for classmates to evaluate:
- For x = 2, the corresponding y-value is 4.
- For x = -3, the corresponding y-value is 9.
These values can be obtained by substituting the given x-values into the equation y = x^2 and evaluating the expression.