What are the similarities and differences between functions and linear equations? How do you graph functions on a coordinate plane? Is there an instance when a linear equation is not a function? Provide an example. Write a function for your classmates to graph.
Consider responding to your classmates explaining how to graph their function. Use descriptions rather than pictures to explain the graphs. Challenge your classmates by providing more intricate examples of functions for them to graph.
Functions and linear equations have some similarities and differences. A linear equation is a type of function, where the relationship between the variables involved is a straight line. However, not all functions are linear equations.
Similarities:
1. Both functions and linear equations involve mathematical relationships between variables.
2. They can both be represented graphically on a coordinate plane.
Differences:
1. Linear equations specifically represent straight lines, while functions can have various shapes and forms.
2. A linear equation has a degree of 1, meaning it only contains the variables raised to the power of 1. A function, on the other hand, can have any degree, which represents the highest power of the variables involved.
To graph a function on a coordinate plane, you need to plot points that satisfy the given function and connect them. Here's how you can do it step by step:
1. Identify the variables involved in the function, usually represented as x and y.
2. Choose several values for the independent variable (x) and calculate the corresponding values of the dependent variable (y) using the function.
3. Plot the points (x, y) on the coordinate plane using the calculated values.
4. Connect the plotted points to form the graph of the function. You may use a straight line, a curved line, or a combination of both, depending on the function.
A linear equation is not a function when it fails the vertical line test, which states that every vertical line should only intersect the graph of a function at most once. An example of a linear equation that is not a function is x = 3. If we try to graph this equation, we get a vertical line passing through x = 3. This line intersects the graph at multiple y-values, violating the vertical line test, thus making it not a function.
Here's an example of a function for your classmates to graph:
f(x) = x^2 + 3x - 2
To graph this function, follow these steps:
1. Choose several values for x. For example, let's choose x = -2, -1, 0, 1, and 2.
2. Substitute each chosen value of x into the equation and calculate the corresponding value of y.
- For x = -2: f(-2) = (-2)^2 + 3(-2) - 2 = 4 - 6 - 2 = -4
- For x = -1: f(-1) = (-1)^2 + 3(-1) - 2 = 1 - 3 - 2 = -4
- For x = 0: f(0) = (0)^2 + 3(0) - 2 = 0 - 0 - 2 = -2
- For x = 1: f(1) = (1)^2 + 3(1) - 2 = 1 + 3 - 2 = 2
- For x = 2: f(2) = (2)^2 + 3(2) - 2 = 4 + 6 - 2 = 8
3. Plot the points (-2,-4), (-1,-4), (0,-2), (1,2), and (2,8) on the coordinate plane.
4. Connect the plotted points with a curved line. This graph represents the function f(x) = x^2 + 3x - 2.
To challenge your classmates, here's a slightly more intricate example of a function:
g(x) = sqrt(abs(x)) * sin(x)
Follow the same steps to graph this function. Choose various values of x, substitute them into the equation, calculate y, plot the points, and connect them. This function will result in a graph with both curved and straight line segments.