Transforming Functions: Shifts, Stretches, and Reflections

From Parent Functions to Transformed Graphs

The core idea of transformations is simple: we start with a basic, well-understood function, which we call a parent function, and then systematically alter it. You've already built a solid library of these parent functions.

We will apply a sequence of operations to these parent functions to shift, stretch, compress, or reflect their graphs, resulting in a new, transformed function. There are four fundamental types of transformations.

1. Shifts (Translations)

Shifts, or translations, move the entire graph without changing its shape or orientation. Professor Leonard provides an excellent conceptual explanation of why these shifts work the way they do.

As the video explained, we can summarize the rules for shifts as follows:

• Vertical Shift: Given a function $y=f(x)$, the graph of $y=f(x)+k$ is the graph of $f(x)$ shifted:

  • Up by $k$ units if $k>0$.
  • Down by $|k|$ units if $k<0$.

• Horizontal Shift: Given a function $y=f(x)$, the graph of $y=f(x-h)$ is the graph of $f(x)$ shifted:

  • Right by $h$ units if $h>0$. (e.g., $f(x-3)$ shifts right by 3).
  • Left by $|h|$ units if $h<0$. (e.g., $f(x+2)$ which is $f(x-(-2))$ shifts left by 2).

2. Stretches, Compressions, and Reflections

These transformations change the shape or orientation of the graph.

Vertical Transformations: y = a * f(x)

Multiplying the entire function by a constant a affects the y-coordinates of every point on the graph.

• Vertical Stretch/Compression:
  • If $|a|>1$, the graph is stretched vertically by a factor of $a$. The graph appears narrower.
  • If $0<|a|<1$, the graph is compressed vertically by a factor of $a$. The graph appears wider.
• Reflection across the x-axis:
  • If $a<0$, the graph is reflected across the x-axis. This is because all positive y-values become negative and vice-versa.

Horizontal Transformations: y = f(b * x)

Multiplying the input x by a constant b affects the x-coordinates. Like horizontal shifts, these are often counter-intuitive.

• Horizontal Stretch/Compression:
  • If $|b|>1$, the graph is compressed horizontally by a factor of $1/|b|$.
  • If $0<|b|<1$, the graph is stretched horizontally by a factor of $1/|b|$.
• Reflection across the y-axis:
  • If $b<0$, the graph is reflected across the y-axis.

Professor Leonard also explains these concepts clearly.

A Systematic Approach to Combining Transformations

Real-world functions often involve multiple transformations. To handle this complexity, we need a systematic approach. A transformed function can generally be written in the form:

$g(x)=a\cdot f(b(x-h))+k$

Here, f(x) is the parent function, and a, b, h, k are the parameters that control the transformations. The following resource provides a fantastic, structured method for analyzing these combined transformations.

The most powerful tool from this reading is the point-mapping formula. If a point (x, y) is on the graph of the parent function y = f(x), the corresponding point (X, Y) on the transformed graph g(x) = a f(b(x - h)) + k is given by:

This formula provides a systematic, almost algorithmic way to transform a graph. You identify a few key points on the parent graph, apply this mapping to find their new locations, and then sketch the transformed graph through the new points.

Worked Example: Transforming the Natural Logarithm

Let's apply this method to graph the function $g(x)=3-\ln (2-x)$ starting from the parent function $f(x)=\ln (x)$.

1. Rewrite in Standard Form:
First, we must rearrange $g(x)$ to match the standard form $af(b(x-h))+k$.

$g(x)=3-\ln (2-x)=-\ln (-x+2)+3=-1\cdot \ln (-1(x-2))+3$

2. Identify Parameters and Parent Function:

• Parent Function: $f(x)=\ln (x)$
• Parameters: $a=-1$, $b=-1$, $h=2$, $k=3$

This tells us we have:

• A reflection across the x-axis ($a=-1$).
• A reflection across the y-axis ($b=-1$).
• A horizontal shift right by 2 ($h=2$).
• A vertical shift up by 3 ($k=3$).

3. Identify Key Features of the Parent Function:
The graph of $f(x)=\ln (x)$ has:

• A key point at $(1,0)$.
• Another point at $(e,1)$ (since $\ln (e)=1$).
• A vertical asymptote at $x=0$.

4. Apply the Point-Mapping Formula:
The mapping is: $(X,Y)=\left(\frac{x}{-1}+2,-1\cdot y+3\right)=(-x+2,-y+3)$.

• Transform the point $(1,0)$:
  $X=-1+2=1$
  $Y=-0+3=3$
  The new point is $(1,3)$.

• Transform the point $(e,1)$:
  $X=-e+2\approx -2.718+2=-0.718$
  $Y=-1+3=2$
  The new point is $(2-e,2)$.

• Transform the asymptote $x=0$:
  Since the asymptote is a vertical line defined by an x-value, we apply the horizontal part of the mapping to it.
  New asymptote: $X=-0+2=2$. The new vertical asymptote is $x=2$.

5. Sketch the Graph:
Plot the new asymptote at $x=2$. Plot the transformed points $(1,3)$ and $(2-e,2)$. Then, draw the logarithmic curve through these points, approaching the asymptote. The double reflection means the graph will be in the top-left quadrant relative to its starting point.

This systematic process allows you to construct a complex graph with confidence, piece by piece.

Conclusion

In this lesson, we developed a comprehensive toolkit for transforming functions. By understanding how shifts, stretches, and reflections are encoded in a function's equation, you can now visualize and sketch complex graphs from simple parent functions.

Key Takeaways:

• Transformations include shifts (translations), stretches/compressions, and reflections.
• Changes outside the function f(...) cause vertical transformations.
• Changes inside the function f(...) cause horizontal transformations, which often behave opposite to intuition.
• Any combination of these transformations can be represented by the general form $g(x)=a\cdot f(b(x-h))+k$.
• The most reliable way to graph a transformed function is to use the point-mapping formula $(X,Y)=(\frac{x}{b}+h,a\cdot y+k)$ on key points from the parent graph.

In our next lesson, we will continue exploring ways to build new functions from old ones. We will move beyond transformations to investigate how functions can be combined using arithmetic operations (addition, subtraction, multiplication, division) and the powerful concept of function composition.