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The function created from transformations of apparent function

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It seems like you are asking about the process of creating a new function by applying transformations to an "apparent function" (possibly a basic or a parent function). In mathematics, function transformations involve changing the position, size, or shape of a function's graph. Here are some common transformations:

  1. Vertical Shifts:

    • Addition: If you have a function \( f(x) \) and you create a new function \( g(x) = f(x) + k \), the graph of \( g(x) \) will be the graph of \( f(x) \) shifted vertically by \( k \) units. If \( k \) is positive, it shifts up; if negative, it shifts down.

  2. Horizontal Shifts:

    • Addition: If \( g(x) = f(x - h) \), the graph of \( g(x) \) will be shifted horizontally. If \( h \) is positive, it shifts to the right; if negative, it shifts to the left.

  3. Vertical Stretch/Compression:

    • Multiplication: If \( g(x) = af(x) \), where \( a > 1 \), the graph of \( g(x) \) is a vertical stretch of \( f(x) \). If \( 0 < a < 1 \), it compresses the graph vertically.

  4. Horizontal Stretch/Compression:

    • If \( g(x) = f(bx) \), where \( b > 1 \), the graph of \( g(x) \) is a horizontal compression of \( f(x) \). If \( 0 < b < 1 \), it stretches the graph horizontally.

  5. Reflections:

    • About the x-axis: If \( g(x) = -f(x) \), this reflects the graph of \( f(x) \) across the x-axis.
    • About the y-axis: If \( g(x) = f(-x) \), this reflects the graph of \( f(x) \) across the y-axis.

Example:

Let’s consider the function \( f(x) = x^2 \) (a parabola).

  1. Vertical Shift: \( g(x) = x^2 + 3 \) shifts the parabola up by 3 units.
  2. Horizontal Shift: \( g(x) = (x - 2)^2 \) shifts it to the right by 2 units.
  3. Vertical Stretch: \( g(x) = 2x^2 \) stretches the parabola vertically by a factor of 2.
  4. Horizontal Compression: \( g(x) = (2x)^2 \) compresses it horizontally by a factor of 2.
  5. Reflection: \( g(x) = -x^2 \) reflects the parabola across the x-axis.

By combining these transformations, you can create a wide variety of new functions based on the transformations of a base function. If you have a specific apparent function and specific transformations in mind, please provide those details!