# How is the graph of mc034-1.jpg related to its parent function, mc034-2.jpg? A guide to understanding transformations of functions

In this article, we will explore how the graph of mc034-1.jpg is related to its parent function, mc034-2.jpg. We will also learn some general rules for transforming functions and how to apply them to different types of functions.

## What are parent functions and transformations?

A parent function is a basic function that can be used as a starting point for more complex functions. For example, the parent function for linear functions is y = x, and the parent function for quadratic functions is y = x^2.

A transformation is a change in the shape, position, or size of a function’s graph. There are four main types of transformations:

• Translation: This is a shift of the graph horizontally or vertically by adding or subtracting a constant to the input or output of the function.
• Reflection: This is a flip of the graph over the x-axis or y-axis by multiplying the input or output of the function by -1.
• Stretching: This is a change in the steepness of the graph by multiplying the input or output of the function by a factor greater than 1.
• Shrinking: This is a change in the flatness of the graph by multiplying the input or output of the function by a factor between 0 and 1.

## How to transform a function using algebra?

To transform a function using algebra, we can use the following general rules:

• To translate the graph left or right by h units, replace x with (x – h) in the function.
• To translate the graph up or down by k units, add or subtract k to the function.
• To reflect the graph over the x-axis, multiply the function by -1.
• To reflect the graph over the y-axis, replace x with -x in the function.
• To stretch or shrink the graph horizontally by a factor of b, replace x with (x / b) in the function.
• To stretch or shrink the graph vertically by a factor of a, multiply the function by a.

The graph of mc034-1.jpg is given by y = -(x + 3)^2 + 4, and its parent function is mc034-2.jpg, which is y = x^2. To see how they are related, we can compare their equations using the rules above.

First, we notice that there is a negative sign in front of the function, which means that there is a reflection over the x-axis. This flips the shape of the parabola from opening upward to opening downward.

Next, we notice that there is a +3 inside the parentheses with x, which means that there is a translation left by 3 units. This shifts the vertex of the parabola from (0, 0) to (-3, 0).

Finally, we notice that there is a +4 outside the parentheses, which means that there is a translation up by 4 units. This shifts the vertex of the parabola from (-3, 0) to (-3, 4).

Therefore, we can say that the graph of mc034-1.jpg is obtained from its parent function by reflecting over the x-axis, translating left by 3 units, and translating up by 4 units.

## Conclusion

In this article, we learned how to transform functions using algebra and how to relate the graph of mc034-1.jpg to its parent function, mc034-2.jpg. We hope this article was helpful and informative for you. If you have any questions or feedback, please leave them in the comments below. Thank you for reading!