Function Transformations

Function transformations allow us to take a basic "parent" function and modify it to create a new, related function. By applying a sequence of transformations, we can shift, stretch, compress, or reflect the graph of the parent function.
Understanding these transformations is essential, as it allows us to predict the graph of a complex function based on a simpler one and to model real-world phenomena by adjusting a basic function to fit observed data. In this chapter, we will explore vertical and horizontal translations, dilations (stretches/compressions), and reflections.

A vertical translation shifts the graph of a function up or down. The transformation is defined by:$$g(x) = f(x) + k$$This transformation maps a point \((x, y)\) on the graph of \(f\) to a new point \((x, y+k)\) on the graph of \(g\).

• If \(k>0\), the graph is translated \(\boldsymbol{k}\) units up.
• If \(k<0\), the graph is translated \(\boldsymbol{|k|}\) units down.

A horizontal translation shifts the graph of a function left or right. The transformation is defined by:$$g(x) = f(x-h)$$This transformation maps a point \((x, y)\) on the graph of \(f\) to a new point \((x+h, y)\) on the graph of \(g\).

• If \(h>0\), the graph is translated \(\boldsymbol{h}\) units to the right.
• If \(h<0\), the graph is translated \(\boldsymbol{|h|}\) units to the left.

A vertical dilation stretches or compresses the graph of a function vertically. It is defined by:$$g(x) = a \cdot f(x)$$This transformation maps a point \((x, y)\) to \((x, ay)\).

• If \(|a|>1\), the graph is stretched vertically by a factor of \(a\).
• If \(0 < |a| < 1\), the graph is compressed vertically by a factor of \(a\).

A horizontal dilation stretches or compresses the graph of a function horizontally. It is defined by:$$g(x) = f(bx)$$This transformation maps a point \((x, y)\) to a new point \((\frac{x}{b}, y)\).

• If \(|b|>1\), the graph is compressed horizontally by a factor of \(\frac{1}{b}\).
• If \(0 < |b| < 1\), the graph is stretched horizontally by a factor of \(\frac{1}{b}\).

A reflection in the x-axis flips the graph of a function vertically. It is a special case of vertical dilation where \(a=-1\). It is defined by:$$g(x) = -f(x)$$This transformation maps a point \((x, y)\) to \((x, -y)\).

A reflection in the y-axis flips the graph of a function horizontally. It is a special case of horizontal dilation where \(b=-1\). It is defined by:$$g(x) = f(-x)$$This transformation maps a point \((x, y)\) to \((-x, y)\).

When multiple transformations are applied to a function, the order in which they are performed is crucial. The standard form for a transformed function is:$$ g(x) = a \cdot f(b(x-c)) + d $$To graph this function from the parent function \(f(x)\), apply the transformations in the following order:

1. Horizontal Transformations (inside the brackets):
   • Apply the horizontal stretch/compression by the factor \(\frac{1}{b}\).
   • Apply the horizontal translation (shift) by \(c\) units.
2. Vertical Transformations (outside the brackets):
   • Apply the vertical stretch/compression by the factor \(a\).
   • Apply the vertical translation (shift) by \(d\) units.

Note: Always factor out the coefficient \(b\) from the term inside the function to correctly identify the horizontal shift \(c\). For example, transform \(f(2x-6)\) as \(f(2(x-3))\). This shows a compression by \(\frac{1}{2}\) followed by a shift of 3 units right, not 6.

Describe the sequence of transformations that maps the graph of \(f(x)=\sqrt{x}\) onto the graph of \(g(x) = 3\sqrt{-x+2} - 4\).