What similarities and differences do you see between functions and linear equations studied in Ch. 3? Are all linear equations functions? Is there an instance in which 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.
We don't have Ch. 3 to help you.
Functions and linear equations have some similarities in that both are mathematical representations that relate one variable to another. However, there are also some differences between the two.
Similarities:
1. Both functions and linear equations involve mathematical relationships between variables.
2. Both can be graphed on a coordinate plane.
Differences:
1. A linear equation represents a straight line when graphed, while a function can have various shapes or patterns.
2. A linear equation can have at most one y-value (output) for each x-value (input). In other words, every x-value has a unique y-value. On the other hand, a function can have multiple outputs for a single input.
Not all linear equations are functions. To determine if a linear equation is a function, we apply the vertical line test. If any vertical line intersects the graph of the equation in more than one point, then the equation is not a function. This occurs when a vertical line is drawn through a vertical line segment in the graph.
For example, consider the linear equation x = 3. If we try to graph this equation, we will see that it is a vertical line that intersects the x-axis at x = 3. Since any vertical line will intersect this line at multiple points (such as (3,0) and (3,5)), the equation x = 3 is not a function.
Now, let's create an equation of a nonlinear function:
One example of a nonlinear function is a quadratic equation. Let's consider the equation y = x^2.
Inputs for your classmates to evaluate:
1. x = 2: When we substitute x = 2 into the equation, we get y = 2^2 = 4. Therefore, for x = 2, y = 4.
2. x = -3: Substituting x = -3 into the equation gives y = (-3)^2 = 9. Thus, for x = -3, y = 9.
Your classmates can evaluate these inputs by simply substituting the given values of x into the equation and calculating the corresponding values of y.