Final answer:
The graph of f(x) = |(x^2 + 3x - 28) / (x - 5)| has x-intercepts at x = -7 and x = 4, with a vertical asymptote at x = 5. The y-intercept is at (0, 5.6). The domain is all real numbers except x = 5, and the range is all non-negative real numbers.
Step-by-step explanation:
To sketch the graph of the function f(x) = |(x^2 + 3x - 28) / (x - 5)|, we need to consider the absolute value, the quadratic expression, and the vertical asymptote. First, let's factor the quadratic expression to find the x-intercepts. The expression x^2 + 3x - 28 factors to (x+7)(x-4). Hence, the x-intercepts are x = -7 and x = 4.
There is a vertical asymptote at x = 5 because the denominator x - 5 equals zero and the function is undefined. The absolute value creates a mirror effect across the x-axis, so the graph will be entirely non-negative, maintaining the same shape but above the x-axis for negative values.
To find the y-intercept, we set x = 0 in the given function, resulting in y = |0^2 + 3(0) - 28| / |0 - 5| = |-28|/5 = 5.6. Therefore, the y-intercept is (0, 5.6).
The domain of the function is all real numbers except where the denominator is zero, so the domain is x ≠ 5. The range of the function is all non-negative real numbers, because of the absolute value, which is [0, ∞).