Pregunta
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The equation of a hyperbola is given by .
Write down the equation of the new function that is formed when is transformed as follows:
Shift two units to the left
Shift 3 units up
Shift 1 unit right and 2 units down
The equation of the new hyperbola has new asymptotes at and
QUESTION 4
Sketch on the same set of axes the graphs of and
Clearly indicate all intercepts with the axes, turning point(s) and asymptote(s).

Ask by Deleon Cole. in South Africa
Jan 23,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

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After transforming the original function through shifting two units to the left, three units up, one unit right, and two units down, the new function becomes .
For the graphs of and :
  • Function :
    • Y-intercept at (0, 6)
    • X-intercepts at and
    • Turning point at (1, 0)
  • Function :
    • Y-intercept at (0, 0)
    • Horizontal asymptote at
These key points should be plotted on the same set of axes to sketch the graphs accurately.

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Extra Insights

To shift the function two units to the left, you replace with in . That gives you the new function . Shifting the function three units up will adjust the entire output by adding 3 to the function, which results in .
Next, for the shift of 1 unit right and 2 units down, you would replace with and subtract 2 from the function, yielding . Finally, adjusting the asymptotes so that the new hyperbola has asymptotes at and will involve modifying the general form to center the hyperbola accordingly.
Now, onto the sketch of and . For , start by determining the turning point via completing the square or using the vertex formula, leading to a vertex at . It opens downwards, with intercepts at and .
For , recognize that it’s an exponential function with a horizontal asymptote at , passing through the point . Plotting both functions on the same axes will highlight the intersection points, turning points, and the asymptote with clear labels for each important feature.

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