# Graphing Quadratic Equations Using Transformations

A quadratic equation is any polynomial equation of the second degree. The standard form of such an equation is: 0 = ax^{2} + bx + c. In this equation, a, b, and c must all be real numbers and a cannot be zero.

When we graph a quadratic equation, we get a parabola. There are numerous ways to graph quadratic equations including factoring and using the axis of symmetry, but we'll focus on transformations in this article. Let's get started!

## The procedure for graphing quadratic equations using transformations

When we talk about graphing using transformations, we're talking about taking the basic parabola (y = x^{2}) and modifying it based on another quadratic equation. The basic parabola is called the "parent graph" and we can figure out how to transform it based on the values of a and c of the quadratic's standard form.

Recall that all quadratic equations can be written in the form ax^{2} + bx + c. The graph of y = ax^{2} is the same as y = x^{2} except that the graph is stretched vertically by a factor of a. If a is negative, the parabola also opens downward instead of upward and if a is between 1 and -1 it undergoes vertical compression or "squish." Similarly, the graph y = x^{2} + c looks like y = x^{2} except that it's shifted c units up (if c is positive) or down (if negative).

Seeing these rules in action can make them easier to understand, so let's look at a sample problem. Let's graph the function y = 2x^{2} - 5 using transformation.

If we're going to graph a quadratic equation using transformation, the first thing we have to do is graph the parent function, y = x^{2}.

Next, we look at our equation to figure out our a and c values. The a value is 2. It's positive, so our parabola will still open upward. Therefore, the only transformation we have to make is stretching the graph vertically by a factor of 2:

Our c value is -5, so we also have to shift the graph down by 5 units:

With that, we successfully sketched the graph of y = 2x^{2} - 5. The upside of this method is that we don't need to do any algebra, instead approximating what the graph should look like. This is also the biggest downside, as the result doesn't give us any specific points to work with. Working neatly is essential for this reason.

## Practice Problems

a. Sketch the following quadratic equation using transformation: y = -1/2 (x - 3)^{2} + 2

Whenever we graph a quadratic equation using transformation, the first thing we have to do is sketch the parent graph:y = x^{2}

You might be thinking that we have to convert our equation to the standard ax^{2} + bx + c format, but we can see that a = -1/2 and c is 2 the way it is. The only extra step is that we have to replace the x in y = x^{2} with (x - 3), which has the effect of shifting the image 3 units to the right:

Our a value is -1/2. Since we're working with a negative a value, the parabola will flip upside down. Likewise, the graph undergoes a vertical compression (or "squish") by a factor of 2:

Finally, our c value is 2. We need to shift the entire graph two units up:

We have sketched y = -1/2 (x - 3)^{2} + 2 successfully!

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## Graphing quadratic equations using transformation

Sketching quadratic equations using transformation is less math-intensive than most other methodologies, but the entire process can also feel very abstract. If your student isn't sure what "vertically stretch by 5 units" means or when to flip the parabola, a 1-on-1 math tutor can provide fresh explanations and examples until they feel confident enough to move on. Likewise, a tutor can create a learning environment where mistakes are encouraged as part of the educational process. If you would like to learn more about the benefits of tutoring or how easy it is to get started with Varsity Tutors, contact our Educational Directors today.