Let f(x) = 3 sin x.
Complete parts (a) through (e) below
a. What is the y-intercept of the graph of f?
b. For what numbers x, -π ≤ x ≤ π, is the graph of f increasing?
c. What is the absolute minimum value of f?
d. For what numbers x, 0 ≤ x ≤ 2π, does f(x) = 0?
e. What are the x-intercepts of f?
a. The y-intercept of the graph of f is the value of f when x = 0. Therefore, substituting x = 0 into the equation, we have:
f(0) = 3sin(0) = 3(0) = 0.
So, the y-intercept is 0.
b. The graph of f will be increasing when the value of sin(x) is positive. In the given range of -π ≤ x ≤ π, sin(x) is positive in the first and second quadrants.
So, the graph of f is increasing for -π < x < π (excluding the endpoints).
c. The absolute minimum value of f occurs when sin(x) is at its minimum value, which is -1. Therefore, the absolute minimum value of f is:
f(x) = 3sin(x) = 3(-1) = -3.
So, the absolute minimum value of f is -3.
d. To find the values of x for which f(x) = 0, we need to solve the equation 3sin(x) = 0. This equation is satisfied when sin(x) = 0, which occurs at the x-values where the sine function equals zero, i.e., at x = 0, π, 2π, etc.
Therefore, for 0 ≤ x ≤ 2π, the values of x for which f(x) = 0 are x = 0, π, and 2π.
e. The x-intercepts of f are the values of x for which f(x) = 0. As found in part d, the x-intercepts are x = 0, π, and 2π.