Challenge 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 - 3) ^ 2 + 3
Complete the table of values for the function.
x 1 2 3 4 5
y
x 1 2 3 4 5
y -1 -3 3 -9 -17
To determine whether the function is linear or nonlinear, we can look at the table of values. If the difference in y-values is constant for each difference in x-values, then the function is linear.
In this case, looking at the table, the difference in y-values is not constant. For example, the difference between y-values for x = 2 and x = 1 is -3 - (-1) = -2, while the difference between y-values for x = 4 and x = 3 is -9 - 3 = -12. Therefore, the function is nonlinear.
To find a linear function, we need a relationship between x and y in which the difference in y-values is constant for each difference in x-values.
One example of a linear function could be: y = 2x - 1.
Here is the table of values for this linear function:
x 1 2 3 4 5
y 1 3 5 7 9
By looking at the table and graphing the linear function, we can see that the difference in y-values is constant for each difference in x-values, indicating that it is indeed a linear function.
To complete the table of values for the function y = -(x - 3)^2 + 3, substitute each value of x into the function and calculate the corresponding value of y.
Given: x = 1, 2, 3, 4, 5
Substituting x = 1:
y = -(1 - 3)^2 + 3
y = -(2)^2 + 3
y = -4 + 3
y = -1
Substituting x = 2:
y = -(2 - 3)^2 + 3
y = -(1)^2 + 3
y = -1 + 3
y = 2
Substituting x = 3:
y = -(3 - 3)^2 + 3
y = -(0)^2 + 3
y = 0 + 3
y = 3
Substituting x = 4:
y = -(4 - 3)^2 + 3
y = -(1)^2 + 3
y = -1 + 3
y = 2
Substituting x = 5:
y = -(5 - 3)^2 + 3
y = -(2)^2 + 3
y = -4 + 3
y = -1
The completed table of values for the function is:
x | y
1 | -1
2 | 2
3 | 3
4 | 2
5 | -1
Next, we will plot these points on a graph to observe the pattern and determine if the function is linear or nonlinear.
To determine whether the given function is linear or nonlinear, we need to check if the rate of change of the function is constant.
Let's start by making a table of values for the given function y = -(x - 3)^2 + 3 by substituting the given values of x.
x | y
1 | -7
2 | -2
3 | 3
4 | -2
5 | -7
Next, let's plot these points on a graph to visualize the relationship between x and y.
Now, to determine if the function is linear or nonlinear, we can examine the pattern of the points on the graph.
For a linear function, the graph would form a straight line, and each additional point would be equally spaced. However, for a nonlinear function, the graph would not be a straight line, and each additional point would not be equally spaced.
Upon plotting the points, we can observe that the graph forms a downward-opening U-shape or a parabola. This indicates that the function is nonlinear.
To find a linear function instead, we need to look for a straight line on the graph. However, in this case, the given function is nonlinear, so we cannot find a linear function using this equation.
In summary, the given function y = -(x - 3)^2 + 3 is nonlinear, as evidenced by the U-shape or parabolic graph it produces when plotted.