Questions: f(x) = 3x + 9

f(x) = 3x + 9
Transcript text: f(x) = 3x + 9
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Solution

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Solution Steps

Step 1: Find the vertical asymptote.

The vertical asymptote occurs when the denominator is equal to zero. Set the denominator equal to zero and solve for x:

-2x - 3 = 0 -2x = 3 x = -3/2 or -1.5

Step 2: Find the horizontal asymptote.

The horizontal asymptote is determined by comparing the degrees of the numerator and denominator. Since the degrees are the same (both are degree 1), the horizontal asymptote is the ratio of the leading coefficients: y = 3/(-2) or y = -3/2 or -1.5

Step 3: Plot two points on each piece of the graph

Choose x-values to the left and right of the vertical asymptote.

  • x = -3: f(-3) = (3(-3) + 9) / (-2(-3) - 3) = 0/3 = 0. Plot the point (-3, 0).
  • x = -2: f(-2) = (3(-2) + 9) / (-2(-2) - 3) = 3/1 = 3. Plot the point (-2, 3).
  • x = 0: f(0) = (3(0) + 9) / (-2(0) - 3) = 9/-3 = -3. Plot the point (0, -3).
  • x = 1: f(1) = (3(1) + 9) / (-2(1) - 3) = 12/-5 = -2.4. Plot the point (1, -2.4).

Final Answer:

The graph of the function has a vertical asymptote at x = -1.5 and a horizontal asymptote at y = -1.5. The graph passes through the points (-3, 0), (-2, 3), (0, -3), and (1, -2.4). Use these points and the asymptotes to sketch the two branches of the hyperbola.

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