Pregunta
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QUESTION 3
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
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).
[8]

Ask by Chan Schmidt. in South Africa
Jan 23,2025

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Question 3:
After the transformations:
  1. Shift 2 units to the left:
  2. Shift 3 units up:
  3. Shift 1 unit right and 2 units down:
Question 4:
  • For :
    • Vertex:
    • X-Intercepts: and
    • Y-Intercept:
    • Graph: Downward-opening parabola symmetric about
  • For :
    • Y-Intercept:
    • Horizontal Asymptote:
    • Graph: Decreasing exponential curve approaching as increases

Solución

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The Deep Dive

To tackle the transformation of the function , we’ll make adjustments step by step according to the specified shifts.
For the first transformation, shifting two units to the left involves replacing with . Therefore, the new function is:
Next, for shifting 3 units up, you add 3 to the whole function:
For the last transformation, shifting 1 unit right (replace with ) and then 2 units down means subtracting 2 from the function:
Now, moving to the sketches of the given functions and :
The quadratic function opens downwards and can be rewritten in vertex form through completing the square, where the vertex can be found at . The intercepts are found by setting for the x-intercepts and evaluating for the y-intercept.
The exponential function shifts the standard graph to the right by 1 and up by 1. Its horizontal asymptote at can be identified as approaches infinity. Intercept points can also be determined by setting .
To sketch both functions, plot the vertex and intercepts for , where you find the maximum point and draw it accordingly, and for , identify the behavior around its asymptote. Ensure to label all intercepts and turning points for clarity!

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