For students who have taken Algebra I and Geometry, graphing linear equations is an important skill to apply. Unfortunately, some functions are more difficult to graph than others, and the quadratic function y = x^2 falls under this category. In this article, I will guide you through some different methods for graphing the parabola y = x^2, so that you can master this important function.

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One method for transforming graphs of functions is to use the translation of coordinate points in the graph. In this example, we apply a horizontal and vertical shift for moving the graph of y = x^2. By shifting the original graph of y = x^2 left three spaces and up by two, we get the new graph of:

f(x) = (x + 3)^2 – 2

Another way to transform our graphs of functions is by altering the steepness of the graph by stretching or compressing it. Vertical stretches or compressions of a graph can be made by using values greater than one or less than one, respectively, for the input variable x. When the value is greater than one, the graph shrinks vertically, and when the value is less than one, it stretches vertically.

In this case, we multiply the original function by 2, resulting in the new function of:

g(x) = 2(x^2)

A horizontal stretch results by taking a value less than one for the output variable y, which makes the graph wider, or greater than one, which makes the graph narrower. Here, we multiply the original function by 0.5, resulting in the new function of:

h(x) = (x^2) / 2

Transformations of graphs result from reflecting or “flipping” graphs across the x and/or y axes. In this example, we reflect our original graph of y = x^2 across the y-axis, which results in a new function where only the input variable x is inverted (flipped), which results in the new function:

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k(x) = – (x)^2

Now, let’s flip our original graph of y = x^2 across the x-axis, which resulting in a new function where only the output variable y is inverted (flipped):

l(x) = (-x)^2

Finally, in this example, we reflect our original graph of y = x^2 across both the x and y axes, which results in reverting the original graph of y = x^2 back to its original form, which is the new function of:

m(x) = – (-x)^2

I hope this brief overview of transformations of functions proves useful for your future mathematical needs. Graphing functions is an important skill for all students of mathematics because it enables them to visualize and understand the relationship between different variables, so if you are still struggling with this topic, I recommend seeking out additional resources and practicing with different functions.

Feel free to share this article with other students who may be struggling with this topic or leave a comment below if you have any other questions,

The Math Dude

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FAQ

Q: What are the different types of transformations?

A: Translation, reflection and stretching/compressing

Q: What does it mean to translate a graph?

A: Moving the graph left, right, up, or down along the x or y-axis

Q: What does it mean to reflect a graph?

A: Flipping the graph across the x or y axis, or both

Q: What does it mean to stretch or compress a graph?

A: Making the graph wider or narrower by altering the values of x or y

Q: What are the steps to graph a function?

A: Find the vertex, plot the axis of symmetry, find the x and y-intercepts, and use the transformations learned to graph additional points