Great choice with the cubic function \( f(x) = x^3 \). The cubic family of functions is a set of polynomial functions characterized by their degree, which here is three. Cubic functions can exhibit interesting behaviors such as inflection points, local maxima and minima, and they can cover all real values.
When you apply transformations to this base function, you typically manipulate it in a few common ways: vertical shifts, horizontal shifts, reflections, and stretches or compressions. Let's outline transformations that could be applied step-by-step to the cubic function:
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Vertical Shift: If you add or subtract a constant \( k \), the function shifts vertically.
- \( f(x) = x^3 + k \) shifts the graph up by \( k \) units if \( k > 0 \) or down by \( |k| \) units if \( k < 0 \).
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Horizontal Shift: If you replace \( x \) with \( x - h \), it shifts horizontally.
- \( f(x) = (x - h)^3 \) shifts the graph to the right by \( h \) units if \( h > 0 \) or to the left by \( |h| \) units if \( h < 0 \).
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Reflection: This occurs if you multiply the function by -1.
- \( f(x) = -x^3 \) reflects the graph over the x-axis.
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Vertical Stretch/Compression: If you multiply the entire function by a constant \( a \), this results in a stretch if \( |a| > 1 \) or a compression if \( 0 < |a| < 1 \).
- \( f(x) = a \cdot x^3 \) will stretch or compress the graph vertically.
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Horizontal Stretch/Compression: This is done by replacing \( x \) with \( bx \) where \( b \) is a positive constant.
- \( f(x) = (bx)^3 \) will compress the graph horizontally by a factor of \( 1/b \) if \( b > 1 \) or stretch it by a factor of \( b \) if \( 0 < b < 1 \).
Applying transformations can create a multitude of shapes and behaviors in the cubic family of functions.
Example Transformation Sequence
Let's say you want to apply a few transformations to the basic cubic function \( f(x) = x^3 \):
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Horizontal Shift: Shift right by 2 units
- \( g(x) = (x - 2)^3 \)
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Vertical Shift: Shift up by 3 units
- \( h(x) = (x - 2)^3 + 3 \)
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Reflection: Reflect over the x-axis
- \( j(x) = -((x - 2)^3 + 3) = -(x - 2)^3 - 3 \)
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Vertical Stretch: Stretch by a factor of 2
- \( k(x) = -2((x - 2)^3 + 3) = -2(x - 2)^3 - 6 \)
This transformation would yield a final function \( k(x) \) that represents a cubic function shifted, reflected, and stretched. Each transformation modifies the properties of the graph step-by-step from the original \( f(x) = x^3 \) to \( k(x) \) as described.
If there are specific transformations you'd like to delve into or apply, feel free to mention!