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)power of 2+5
x y
-1 blank
0 blank
1 blank
2 blank
3 blank
To determine whether the function is linear or nonlinear, we can substitute different values of x into the equation and observe the corresponding values of y.
First, let's calculate the values of y for the given values of x:
When x = -1,
y = -(1-1)^2 + 5 = -0^2 + 5 = 5
When x = 0,
y = -(0-1)^2 + 5 = -(-1)^2 + 5 = -1 + 5 = 4
When x = 1,
y = -(1-1)^2 + 5 = -(0)^2 + 5 = 0 + 5 = 5
When x = 2,
y = -(2-1)^2 + 5 = -(1)^2 + 5 = -1 + 5 = 4
When x = 3,
y = -(3-1)^2 + 5 = -(2)^2 + 5 = -4 + 5 = 1
Now, let's plot these points on a graph:
x | y
-1 | 5
0 | 4
1 | 5
2 | 4
3 | 1
On the graph, we can notice that the points form a series of connected straight lines. There isn't any curve or bend in the line segments. This indicates that the function is linear.
To find a linear function that represents the data, we can calculate the slope between any two points on the line. Let's use the points (0, 4) and (1, 5) to find the slope:
Slope = (y2 - y1) / (x2 - x1)
= (5 - 4) / (1 - 0)
= 1 / 1
= 1
Now, we have the slope (m) and a point (0, 4). We can use the point-slope form of a linear equation to find the linear function:
y - y1 = m(x - x1)
y - 4 = 1(x - 0)
y - 4 = x
Therefore, the linear function that represents the given data is y = x.
The graph confirms that the function is linear, as the points are aligned in a straight line.