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Use technology to find points and then graph the function y, equals, left parenthesis, one half, right parenthesis, start superscript, x, minus, 1, end superscript, minus, 10, commay=(

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x−1
−10, following the instructions below.
Equation of asymptote:
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Plot at least five points with integer coordinates

1 answer

To analyze the function \( y = \left(\frac{1}{2}\right)^{x-1} - 10 \) and find its asymptote and points for graphing, we can follow these steps:

Step 1: Find the equation of the asymptote

The function is an exponential decay function, which typically has a horizontal asymptote. As \( x \) approaches infinity, \( \left(\frac{1}{2}\right)^{x-1} \) approaches 0, so we can see if there's a vertical shift affecting the asymptote.

Thus, let's find the horizontal asymptote: \[ \lim_{{x \to \infty}} \left(\frac{1}{2}\right)^{x-1} - 10 = 0 - 10 = -10 \] The equation of the horizontal asymptote is: \[ y = -10 \]

Step 2: Plot at least five points with integer coordinates

We can calculate the function values for integer values of \( x \).

  1. When \( x = 1 \): \[ y = \left(\frac{1}{2}\right)^{1-1} - 10 = 1 - 10 = -9 \] Point: \( (1, -9) \)

  2. When \( x = 2 \): \[ y = \left(\frac{1}{2}\right)^{2-1} - 10 = \frac{1}{2} - 10 = -9.5 \] Point: \( (2, -9.5) \) (not an integer, skip)

  3. When \( x = 3 \): \[ y = \left(\frac{1}{2}\right)^{3-1} - 10 = \frac{1}{4} - 10 = -9.75 \] Point: \( (3, -9.75) \) (not an integer, skip)

  4. When \( x = 4 \): \[ y = \left(\frac{1}{2}\right)^{4-1} - 10 = \frac{1}{8} - 10 = -9.875 \] Point: \( (4, -9.875) \) (not an integer, skip)

  5. When \( x = 5 \): \[ y = \left(\frac{1}{2}\right)^{5-1} - 10 = \frac{1}{16} - 10 = -9.9375 \] Point: \( (5, -9.9375) \) (not an integer, skip)

  6. When \( x = 6 \): \[ y = \left(\frac{1}{2}\right)^{6-1} - 10 = \frac{1}{32} - 10 \approx -9.96875 \] Point: \( (6, -9.96875) \) (not an integer, skip)

  7. When \( x = 7 \): \[ y = \left(\frac{1}{2}\right)^{7-1} - 10 = \frac{1}{64} - 10 \approx -9.984375 \] Point: \( (7, -9.984375) \) (not an integer, skip)

We can make calculations for negative \( x \) values:

  1. When \( x = 0 \): \[ y = \left(\frac{1}{2}\right)^{0-1} - 10 = \frac{1}{0.5} - 10 = 2 - 10 = -8 \] Point: \( (0, -8) \)

  2. When \( x = -1 \): \[ y = \left(\frac{1}{2}\right)^{-1-1} - 10 = \left(\frac{1}{2}\right)^{-2} - 10 = 4 - 10 = -6 \] Point: \( (-1, -6) \)

  3. When \( x = -2 \): \[ y = \left(\frac{1}{2}\right)^{-2-1} - 10 = \left(\frac{1}{2}\right)^{-3} - 10 = 8 - 10 = -2 \] Point: \( (-2, -2) \)

  4. When \( x = -3 \): \[ y = \left(\frac{1}{2}\right)^{-3-1} - 10 = \left(\frac{1}{2}\right)^{-4} - 10 = 16 - 10 = 6 \] Point: \( (-3, 6) \)

  5. When \( x = -4 \): \[ y = \left(\frac{1}{2}\right)^{-4-1} - 10 = \left(\frac{1}{2}\right)^{-5} - 10 = 32 - 10 = 22 \] Point: \( (-4, 22) \)

Summary of Points

Here are the points with integer coordinates we found:

  1. \( (0, -8) \)
  2. \( (-1, -6) \)
  3. \( (-2, -2) \)
  4. \( (-3, 6) \)
  5. \( (-4, 22) \)

Final Result

The equation of the horizontal asymptote is: \[ y = -10 \]

The points to plot on the graph of the function \( y = \left(\frac{1}{2}\right)^{x-1} - 10 \) are:

  1. \( (0, -8) \)
  2. \( (-1, -6) \)
  3. \( (-2, -2) \)
  4. \( (-3, 6) \)
  5. \( (-4, 22) \)

You can graph the function based on these points and observe the horizontal asymptote at \( y = -10 \).