Question

Represent the rational function f(x) = 10 / (x - 3) by creating a table of values from -2 to 8. Determine whether the points form a smooth or a straight line.

a) Create a table of values for f(x).
b) Plot the points in the Cartesian plane.
c) Determine if the points form a smooth curve or a straight line.
d) None of the above.

Answer 1

A table of values is created for the function f(x) = 10 / (x - 3) from x = -2 to x = 8, excluding x = 3. The points are plotted on a Cartesian plane and reveal a smooth curve characteristic of a rational function, not a straight line.

Step-by-step explanation:

To represent the rational function f(x) = 10 / (x - 3), a table of values can be created for selected x-values from -2 to 8. These values should not include x = 3, since the function is undefined there (division by zero).

• x = -2, f(x) = -5
• x = -1, f(x) = -10/2 = -5
• x = 0, f(x) = 10/(-3) = -10/3
• x = 1, f(x) = 10/(-2) = -5
• x = 2, f(x) = 10/(-1) = -10
• x = 4, f(x) = 10/(1) = 10
• x = 5, f(x) = 10/(2) = 5
• x = 6, f(x) = 10/(3) ≈ 3.33
• x = 7, f(x) = 10/(4) = 2.5
• x = 8, f(x) = 10/(5) = 2

Plotting these points in the Cartesian plane and connecting them will show that they do not form a straight line, but rather a smooth curve, illustrating the characteristic shape of a rational function with a vertical asymptote at x = 3 and a horizontal asymptote as x approaches infinity.