Lesson 3.4: Transformations of Graphs

Introduction

In this lesson, we will explore the topic of transformations of graphs. Understanding how to manipulate and transform functions is crucial in mathematics, as it helps us to visualize and analyze changes to graphs through translations, stretches, and reflections. By the end of this lesson, you, students, should be able to:

Understand how to perform translations, stretches, and reflections using the general forms of transformations.
Combine multiple transformations and apply them in the correct order.
Relate transformed equations to their respective graphs.
Apply a single transformation to a graph and describe its effect.
Execute a sequence of transformations correctly.

To kick things off, let’s first define what we mean by transformations of functions and how they apply to the graphs of those functions.

Understanding Transformations

Transformations involve changing the position or shape of the graph of a function without altering its basic form. The four primary types of transformations are:

Translations: Shifting the graph left, right, up, or down.
Stretches: Enlarge or narrow the graph with respect to the x or y-axis.
Reflections: Flipping the graph over a line, usually the x-axis or y-axis.
Combinations: Applying two or more transformations in sequence.

Understanding these transformations will provide a strong foundation for further exploring functions and their geometric representations.

Translations

Vertical Translations: $y = f(x) + a$

Vertical translations involve shifting the graph up or down. If we take a function $f(x)$ and add a constant $a$, the graph shifts vertically.

If $a > 0$, the graph shifts up.
If $a < 0$, the graph shifts down.

Example 1: Let’s consider the function $f(x) = x^2$. The graph is a parabola that opens upwards and has its vertex at the origin (0, 0).

If we now consider $g(x) = f(x) + 3 = x^2 + 3$, this transformation shifts the graph up by 3 units. The vertex of the new graph $g(x)$ will be at (0, 3).

Horizontal Translations: $y = f(x + a)$

Horizontal translations shift the graph left or right. Adding a constant inside the function affects its x-values:

If $a > 0$, the graph shifts left.
If $a < 0$, the graph shifts right.

Example 2: Again, using the basic function $f(x) = x^2$, consider the transformation $g(x) = f(x + 2) = (x + 2)^2$. This translates the graph of $f(x)$ 2 units to the left. Now the vertex of the graph is at (-2, 0).

Combined Translations

When combining vertical and horizontal translations, we need to apply the transformations in the correct sequence.

Example 3: For the function $f(x) = x^2$, let’s apply both a vertical and a horizontal translation:

Shift left by 2 units (transform to $f(x + 2)$) → $g(x) = (x + 2)^2$.
Shift up by 3 units (transform to $g(x) + 3$) → $h(x) = (x + 2)^2 + 3$.

In this case, the vertex shifts from (0, 0) to (-2, 3).

Stretches

Vertical Stretch: $y = af(x)$

A vertical stretch involves multiplying the function by a constant factor $a$. This transformation changes the y-values:

If $a > 1$, the graph stretches away from the x-axis.
If $0 < a < 1$, the graph compresses towards the x-axis.
If $a < 0$, the graph also reflects across the x-axis.

Example 4: Let’s modify $f(x) = x^2$. If we compute $g(x) = 2f(x) = 2x^2$, the graph stretches vertically by a factor of 2. The vertex remains at (0, 0), but any point on this graph will now be twice as high as it was previously.

Horizontal Stretch: $y = f(bx)$

A horizontal stretch (or compression) occurs when the input to the function is multiplied by a factor $b$:

If $b > 1$, the graph compresses horizontally.
If $0 < b < 1$, the graph stretches horizontally.
If $b < 0$, the graph reflects across the y-axis.

Example 5: Let’s take the function $f(x) = x^2$ and transform it to $g(x) = f(0.5x) = (0.5x)^2$. In this case, the graph stretches horizontally by a factor of 2, resulting in a wider parabola than the original function.

Reflections

Reflections flip the graph over a specific axis:

Reflection over the x-axis: $y = -f(x)$

To reflect a function over the x-axis, we multiply the function by -1.

Example 6: For the function $f(x) = x^2$, the reflection would yield $g(x) = -f(x) = -x^2$. This flips the graph over the x-axis, turning the parabola that opens upwards into one that opens downwards.

Reflection over the y-axis: $y = f(-x)$

For reflections over the y-axis, we replace $x$ with $-x$ in the function.

Example 7: Using $f(x) = x^2$, this gives us $g(x) = f(-x) = (-x)^2$. Since the square function is even, the reflection doesn't change the graph— it remains the same; however, for $h(x) = -f(-x) = -(-x)^2$, we would see a reflection over both the x and y-axes resulting in a graph that opens downwards but maintains its symmetry.

Combining Transformations

Transformations can be combined to achieve the desired effect on the graph. The order matters when dealing with multiple transformations, so we must apply them carefully.

Order of Operations

Start with the original function $f(x)$.
Apply horizontal translations first, followed by horizontal stretches/compressions.
Next, apply reflections, and finally, vertical transformations (stretches and translations).

Example 8: Let’s apply a combination of transformations to $f(x) = x^2$:

First, shift left by 3: $y = f(x + 3)$ → $(x + 3)^2$.
Then, stretch vertically by a factor of 2: $y = 2f(x + 3)$ → $2(x + 3)^2$.
Finally, reflect over the x-axis: $y = -2f(x + 3)$ → $-2(x + 3)^2$. This results in a graph that is reflected, stretched, and translated.

Conclusion

In this lesson, we have discussed the fundamental transformations of functions, including translations, stretches, and reflections. You have learned how to manipulate the equations of functions to observe the corresponding changes in the graphs. Understanding these transformations not only aids in the visual comprehension of functions but also enhances your problem-solving skills in algebra and calculus.

Study Notes

Transformations of Graphs involve moving or changing the shape of graphs without changing their fundamental characteristics.
Vertical Translation: $y = f(x) + a$ shifts the graph up or down.
Horizontal Translation: $y = f(x + a)$ shifts the graph left or right.
Vertical Stretch/Compression: $y = af(x)$ stretches or compresses the graph along the y-axis.
Horizontal Stretch/Compression: $y = f(bx)$ stretches or compresses the graph along the x-axis.
Reflection: $y = -f(x)$ reflects over the x-axis and $y = f(-x)$ reflects over the y-axis.
Combining Transformations requires careful application of the transformations in a specific order.
Always remember that transformations can drastically change the appearance of a graph, though the basic relationships within the function remain intact.