Final answer:
To graph the given function f(x) = -3x + 1, we can plot points and connect them with a line. Then, we can shift this line vertically to graph f(x) + k for different values of k.
Step-by-step explanation:
The function given is f(x) = -3x + 1. To graph this function, we can plot a few points and connect them with a line. Let's choose some values of x and calculate the corresponding y-values:
- When x = 0, y = -3(0) + 1 = 1.
- When x = 1, y = -3(1) + 1 = -2.
Plotting these points and connecting them, we get a line that slopes downward and to the right. Now, we need to graph f(x) + k for k = -2, -4, -6. We can achieve this by shifting the original graph vertically upwards by 2 units for k = -2, 4 units for k = -4, and 6 units for k = -6. When the function f(x) = -3x + 1 is graphed, the resulting line has a slope of -3, which means it slopes downward to the right, and a y-intercept of 1, which is the point where the line crosses the y-axis. When k values of -2, -4, -6 are added to this function to create f(x) + k, the new functions become f(x) - 2, f(x) - 4, and f(x) - 6, respectively. Each of these new functions represents a parallel line to the original graph, shifted downward by 2, 4, and 6 units on the y-axis, respectively. The slope of each line remains unchanged at -3 because adding or subtracting a constant to a function affects only the y-intercept, not the slope.
If we look at the characteristic properties of linear equations in the form y = mx + b, where m is the slope and b is the y-intercept, it is clear that altering b by adding a constant k translates the line vertically but does not change the slope. Thus, option (a) is the correct answer: All three graphs are parallel to each other and to the original graph of f(x). Since these graphs are all straight lines that maintain the same slope but differ only by their y-intercept, they will neither intersect nor be perpendicular to each other, ruling out option (b).