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. 1 y -1 2 2 3 3 4 2 5 1 Graph the function. Choose the correct graph below.
Decide whether the function is linear or nonlinear. Explain. Choose the correct answer below. OA. The function is nonlinear because the rate of change is constant and the graph of the function is a line. OB. The function is nonlinear because the rate of change is not constant and the graph of the function is not a line. OC. The function is linear because the rate of change is not constant and the graph of the function is not a line. OD. The function is linear because the rate of change is constant and the graph of the function is a line.
To complete the table of values for the function, we substitute the given values of x into the function and calculate the corresponding values of y:
1. When x = 1:
y = -(1 - 3)^2 + 3 = -(-2)^2 + 3 = -4 + 3 = -1
2. When x = 2:
y = -(2 - 3)^2 + 3 = -(-1)^2 + 3 = -1 + 3 = 2
3. When x = 3:
y = -(3 - 3)^2 + 3 = -(0)^2 + 3 = 0 + 3 = 3
4. When x = 4:
y = -(4 - 3)^2 + 3 = -(1)^2 + 3 = -1 + 3 = 2
5. When x = 5:
y = -(5 - 3)^2 + 3 = -(2)^2 + 3 = -4 + 3 = -1
The table of values for the function is:
x | y
--------
1 | -1
2 | 2
3 | 3
4 | 2
5 | -1
To graph the function, we plot the pairs (x, y) from the table on a coordinate plane:
x, y = (1, -1), (2, 2), (3, 3), (4, 2), (5, -1)
From the graph, it is clear that the points do not form a straight line. Therefore, the function is nonlinear.
The correct answer is OB. The function is nonlinear because the rate of change is not constant and the graph of the function is not a line.
To determine whether the function y = - (x - 3) ^ 2 + 3 is linear or nonlinear, let's complete the table of values and examine the graph:
Table of values:
x| y
1| -1
2| 2
3| 3
4| 2
5| 1
Now, let's plot these points on a graph:
Graph:
Note: Since graphical representation cannot be shown here, please refer to a graphing tool or draw it on paper.
After plotting the points and examining the graph, we can conclude that the function y = - (x - 3) ^ 2 + 3 is nonlinear.
Explanation:
The rate of change of a linear function is constant; however, in this case, the rate of change is not constant, as the function represents a quadratic equation. Additionally, the graph of this function is not a straight line but a curve. Therefore, the correct answer is OB. The function is nonlinear because the rate of change is not constant and the graph of the function is not a line.
To decide whether the function is linear or nonlinear, we can first complete the table of values for the given function and then graph it.
The function given is:
y = - (x - 3) ^ 2 + 3
To find the values for the table, substitute the x-values given into the function and calculate the corresponding y-values. Let's use the x-values given: 1, 2, 3, 4, and 5.
For x = 1:
y = - (1 - 3) ^ 2 + 3
y = -( -2 ) ^ 2 + 3
y = -4 + 3
y = -1
For x = 2:
y = - (2 - 3) ^ 2 + 3
y = -( -1 ) ^ 2 + 3
y = -1 + 3
y = 2
For x = 3:
y = - (3 - 3) ^ 2 + 3
y = -( 0 ) ^ 2 + 3
y = 0 + 3
y = 3
For x = 4:
y = - (4 - 3) ^ 2 + 3
y = -( 1 ) ^ 2 + 3
y = -1 + 3
y = 2
For x = 5:
y = - (5 - 3) ^ 2 + 3
y = -( 2 ) ^ 2 + 3
y = -4 + 3
y = -1
Now let's plot the points (x, y) on a graph. The x-values will be plotted on the x-axis, and the y-values will be plotted on the y-axis.
Table of values:
x | y
-------
1 | -1
2 | 2
3 | 3
4 | 2
5 | -1
Graphing the points on a coordinate plane, we get the following:
^
|
3 | o (3,3)
|
| o
| o
| o
-1 + o - - - - - - - - - - - - >
1 2 3 4 5
The points form a curve, not a straight line. This indicates that the function is nonlinear. Additionally, we can see that the rate of change of y with respect to x is not constant throughout the graph, further confirming that the function is nonlinear.
Therefore, the correct answer is:
OB. The function is nonlinear because the rate of change is not constant and the graph of the function is not a line.