Determine whether the function is linear or nonlinear. The curve of the function has the points (-2, -14), (-1, -8), (0, -2) and (1, 4). If it is linear, determine the slope.
The slope of a line is defined by the change in vertical direction divided by the change in the horizontal direction. Trigonometrically, it is given by tan θ, where θ is the angle between the x-axis and the line in the anticlockwise direction.
Answer: The given function is linear in nature, and has a slope equal to 6.
Let's understand the solution step by step in detail.
The given points on the curve of the function are (-2, -14), (-1, -8), (0, -2) and (1, 4).
To check whether the function is linear or not, we check whether the curve has a constant slope or not.
Therefore, we use the slope formula m = (y2 − y1) / (x2 − x1) to check whether the slope is constant or not.
Hence, for the first two given points:
⇒ m = [(-8) - (-14)] / [-1 - (-2)] = [-8 + 14] / [-1 + 2] = 6
Here, we see that the slope is 6, from the first two points.
For the next two points:
⇒ [-2 - (-8)] / [0 - (-1)] = [-2 + 8] / [0 + 1] = 6
Here also, we see that the slope is 6.
Hence, the slope is constant for the given pairs of coordinates.
So, it is clear that the points (-2, -14), (-1, -8), (0, -2) and (1, 4) are collinear.