Decide whether the function is linear or nonlinear using both a table and a graph. Use pencil and paper. Find a linear function. Make a table and a graph for the function to explain why it is linear.
y= -(x-1)^2+1
To determine whether the function is linear or nonlinear, we can check if it satisfies the properties of linearity. The properties of a linear function state that it must have a constant rate of change and represent a straight line on a graph.
Let's start by creating a table of values. We will substitute different x-values into the given function and calculate the corresponding y-values:
x | y
-------
-2 | -3
-1 | 0
0 | 1
1 | 0
2 | -3
Now let's plot the points on a graph and see if they create a straight line:
graph
Based on the table and graph, we can see that the function is not linear. The points do not form a straight line, and the rate of change is not constant.
Next, let's find a linear function that can represent a straight line on the graph. A linear function can be described by the equation y = mx + b, where m represents the slope and b represents the y-intercept.
To find a linear function, we need to determine the slope and y-intercept from the given function.
The given function is y = -(x - 1)^2 + 1. To simplify the equation, let's expand and rearrange it:
y = -(x^2 - 2x + 1) + 1
y = -x^2 + 2x - 1 + 1
y = -x^2 + 2x
Comparing this with the general form of a linear function, we can see that the given function is a quadratic function, not a linear one. Thus, we cannot find a linear function to represent it.
In conclusion, the given function y = -(x - 1)^2 + 1 is nonlinear.