Question

Use transformations of y=(1)/(x) or y=(1)/(x²) to sketch the rational function. Label all intercepts and find the equations of all asymptotes. f(x)=(1)/((x+8)²)+9

Answer 1

To sketch f(x)=(1)/((x+8)^2)+9, shift y=1/x^2 eight units left and nine units up, resulting in a horizontal asymptote at y=9. There are no x-intercepts, a y-intercept at (0,9+1/64), and no vertical asymptotes.

Step-by-step explanation:

To sketch the rational function f(x) = \frac{1}{(x+8)^2} + 9, we first identify its transformations from the basic functions y = \frac{1}{x} or y = \frac{1}{x^2}. Let's analyze the given function step by step:

• The original function y = \frac{1}{x^2} gets shifted 8 units to the left due to the (x+8) term.
• It is then shifted 9 units up due to the +9 term at the end of the function. This means the horizontal asymptote is y=9.
• There are no x-intercepts since the function never crosses the x-axis.
• The y-intercept is found by setting x to 0, which yields f(0) = \frac{1}{64} + 9, so the y-intercept is at (0, \frac{1}{64} + 9).
• The vertical asymptote is where the denominator of the function is 0, which does not occur for this function; thus, there is no vertical asymptote.

To sketch the graph, draw a shallow parabola opening upwards, beginning at the point (-8, 9), with the parabola getting infinitely close to, but never touching, the line y=9.