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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, 5, commay=( 2 1 ​ ) x−1 −5, following the instructions below. Equation of asymptote: y, equalsy= Plot at least four points with integer coordinates

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To analyze and graph the function \( y = \frac{1}{2}^{(x - 1)} - 5 \), we will follow these steps:

1. Identify the Asymptote

The function is in the form of an exponential function, where the base \( \frac{1}{2} \) reflects that as \( x \) approaches infinity, \( y \) approaches \(-5\). Therefore, the horizontal asymptote is:

\[ y = -5 \]

2. Generate Points

To find points with integer coordinates, we will substitute integer values for \( x \) and calculate the corresponding \( y \).

  • For \( x = -2 \): \[ y = \frac{1}{2}^{(-2 - 1)} - 5 = \frac{1}{2}^{-3} - 5 = 8 - 5 = 3 \] Point: \((-2, 3)\)

  • For \( x = 0 \): \[ y = \frac{1}{2}^{(0 - 1)} - 5 = \frac{1}{2}^{-1} - 5 = 2 - 5 = -3 \] Point: \((0, -3)\)

  • For \( x = 1 \): \[ y = \frac{1}{2}^{(1 - 1)} - 5 = \frac{1}{2}^{0} - 5 = 1 - 5 = -4 \] Point: \((1, -4)\)

  • For \( x = 2 \): \[ y = \frac{1}{2}^{(2 - 1)} - 5 = \frac{1}{2}^{1} - 5 = \frac{1}{2} - 5 = -4.5 \] Point: \((2, -4.5)\) (This point is not an integer coordinate)

  • For \( x = 3 \): \[ y = \frac{1}{2}^{(3 - 1)} - 5 = \frac{1}{2}^{2} - 5 = \frac{1}{4} - 5 = -4.75 \] Point: \((3, -4.75)\) (Not an integer either)

  • For \( x = 4 \): \[ y = \frac{1}{2}^{(4 - 1)} - 5 = \frac{1}{2}^{3} - 5 = \frac{1}{8} - 5 = -4.875 \] Point: \((4, -4.875)\) (Not an integer)

  • Let's try \( x = -1 \): \[ y = \frac{1}{2}^{(-1 - 1)} - 5 = \frac{1}{2}^{-2} - 5 = 4 - 5 = -1 \] Point: \((-1, -1)\)

  • For \( x = 5 \): \[ y = \frac{1}{2}^{(5 - 1)} - 5 = \frac{1}{2}^{4} - 5 = \frac{1}{16} - 5 = -4.9375 \] Point: \((5, -4.9375)\) (Not an integer)

Integer Points

Here are four points with integer coordinates derived from our evaluations:

  1. \( (-2, 3) \)
  2. \( (0, -3) \)
  3. \( (1, -4) \)
  4. \( (-1, -1) \)

3. Graph the Function

You can graph the function using graphing software or a graphing calculator. The horizontal asymptote at \( y = -5 \) should be represented, along with the points plotted.

Points summary:

  • \((-2, 3)\)
  • \((-1, -1)\)
  • \((0, -3)\)
  • \((1, -4)\)

Plotting Steps:

  1. Draw the axes.
  2. Mark the horizontal line for the asymptote \( y = -5 \).
  3. Plot the points listed above on the graph.
  4. Connect the points smoothly to reflect the shape of the exponential graph.

This will accurately represent the function over the specified range.