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By rearranging a quadratic equation, you can end up with an infinite number of ways to express the same thing. Learn about the three main forms of a quadratic and the pros and cons of each.
Any time you throw something into the air, it's going to follow a parabolic path. From throwing your wrapper into the trash to throwing a 50-yard touchdown to launching a bird that seems a little bit angry, we see parabolas all over the place. It makes sense then that we want to be able to graph them because the graphs can help us answer questions, like will my bird hit its target?
But as is often the case in math, there is more than one way to go about it. In this case, by simply rearranging the parts of the quadratic equation, we can end up with an infinite number of ways to express the same thing. While most of the ways to write the quadratic equation are redundant and useless, there are three forms that actually have unique uses. These three main forms that we graph parabolas from are called standard form, intercept form and vertex form. Each form will give you slightly different information and have its own unique advantages and disadvantages. In this video, we'll go through both for all the different forms.
Let's begin with standard form, y = ax^2 + bx + c. There it is in general form, and here are a few specific examples of what one might look like: y = x^2 + x + 1 and y = -4x^2 - 5x + 9.
To be completely honest, the main reason this one makes the cut as a useful form is because it's the easiest and most basic to write. While the other forms will require some fancy rearranging with algebra tricks, like factoring or completing the square, most quadratics will be in standard form straight from the beginning. This means that you can dive right into the problem from the get-go, while the other forms will often make you do work before you can even begin. Once we get past that, though, standard form doesn't have too much to offer. Perhaps, its most useful trait is that the a value tells you whether the parabola is concave up (positive a value) or concave down (negative a value), but it turns out that all the forms are going to have this ability.
The second trait of standard form has to do with the y-intercept of the parabola. Since the y-intercept is where x=0, substituting this in shows us that the a and b terms drop out, leaving us with only the c value. Therefore, the c value is always the y-intercept. This is kind of cool, but substituting x=0 into the other forms to find the y-intercept is pretty easy too. The last thing you can do with standard form is calculate the axis of symmetry with the formula x = -b / 2a. Once again, while this is kind of cool, finding the axis of symmetry is possible and actually easier with the other forms.
The next form we'll go over is intercept form, y = a(x - p)(x - q). This is the general form, and here are a few specific examples: y = -(x - 1)(x + 5) and y = 3(x + 5)(x + 9).
While it is true that every once in awhile you'll be given a problem that's already in intercept form, it will often be the case that you'll have to first factor the standard-form equation to make it look like intercept form. Although this can sometimes be a headache, there are advantages to doing the work. The a value will, again, tell you whether the parabola is concave up or down, and if you want to find the y-intercept, you can simply substitute in x=0 and quickly evaluate a(-p)(-q).
Where intercept form gets its name and passes standard form in usefulness, is in its ability to not just tell you where the y-intercept is but also where the x-intercepts are. Because the x-intercepts are where y=0, substituting in either p or q will give you a zero in your product, turning the entire equation into zero. Therefore, p and q are the two x-intercepts, or roots, of your quadratic. Be careful with the signs on your roots, though. Because the general equation has a -p and -q, an (x - 5) would actually mean a root at x=5, while an (x + 5) would mean a root at x= -5.
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Lastly, because parabolas are symmetrical, the axis of symmetry must lie directly in between the two roots. This means you can find it on your graph by working your way into the middle or algebraically by calculating the average between the two points: x = (p + q)/2.
And finally we come to vertex form: y = a(x - h)^2 + k. This is the general form, and these are some specific examples: y = 9(x + 5)^2 - 1 and y = -(x - 3)^2 - 1.
This time, getting your quadratic into this form requires you to complete the square, which is possibly the hardest algebraic trick of them all. But if you can, you are going to be rewarded for your hard work. First off, the a value still tells us whether it's concave up or down, and the y-intercept is still easily found by substituting in x= 0 and evaluating. But now, just like intercept form gave us the intercepts, vertex form will give us the vertex of our parabola straight from the equation: h is going to become the x-coordinate, and k will become the y-coordinate, of our vertex. Now, we can easily tell where the axis of symmetry is simply by remembering that it goes right through the middle of the graph where the vertex is. Therefore, the axis of symmetry is just the line x = h.
To review, depending on how you organize it, a quadratic equation can be written in three different forms: standard, intercept and vertex. No matter the form, a positive a value indicates a concave-up parabola, while a negative a value means concave down.
Standard form is the most basic and easy to come up with but has limited helpfulness. Intercept form is what you get if you're willing to factor the quadratic first, and in addition to all that standard form tells you, it also gives you the x-intercepts (or roots) of the parabola. Vertex form is what you get if you complete the square on the standard-form equation first, and in addition to all that standard form tells you, it also gives you the coordinates of the vertex.
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