When students learn the quadratic function, they might be confused and intimidated, but this scary-looking thing can be conquered easily if you understand how the function actually corresponds to the graph.
The quadratic function is still a function, and it’s imperative to review what a function is before anything else. Written in its simple form as y=x2, this expression means that the input x is squared to produce output y. The xy-plane essentially keeps track of what the output is for every possible input (if you want to know the output for an input of 2, simply check how high the curve is when it’s above the x value of 2). You probably also heard of the term “parabola” before: it’s the shape of the quadratic function’s graph.
But usually, you are not given the simple y=x2. Luckily, there are only three important forms of the quadratic function. Lets see what they are through an example.
The vertex form: y=a(x-b)2+d
I made this example graph in Desmos, an online graphing calculator, using the vertex form. It’s arguably the best way to move the parabola into precise locations away from the origin (or else the graph would be very useless). The vertex form might look daunting, but the additional constants a, b, and d are there to modify the input and create many different parabolas, like I mentioned before.
So what do a, b, and d do? Let's look at the function for this example: y=2(x-4)2-2. Here, a=2, b=4, d=-2. Notice how it’s counterintuitive where although b seems to be negative, but it’s actually positive, and if it were to have a positive sign in front of it (ex. y=2(x+4)2-2), b would actually be negative (-4). The exact reason why we flip the sign for b are found in lessons on the transformations of functions.
Notice that d=-2 moves the function down by 2 units, and b=4 moves the function right by 4 units. If d is positive, then the function moves up, and if b is negative, the function moves left. The vertex form effectively move the vertex of the parabola (ie. the tip) to anywhere on the xy-plane using b and d.
The standard form: y=ax2+bx+c
Also known as the general form, this form is not as useful. Here the b is not the all-mighty b in the vertex form. To get the standard form, you simply do some algebra and expand the parenthesis in the vertex form. The standard form for the above example is y=2x2-16x+30.
There is one new constant here: c, and this actually helps you a little bit by telling you the y-intercept of the parabola (where the curve crosses the y-axis). Here, c=30, and thus the y-intercept is 30, which is outside of the image above.
The x-intercept form: y=a(x-m)(x-n)
Also known as the factored form, this is what your teacher means when they tell you to factor a quadratic function. You might be perplexed why you spent weeks learning factorization: this is the preparation for learning quadratic functions.
The x-intercepts are where the graph crosses the x-axis, but what do they have to do with factors? In fact, x-intercepts are the factors of a quadratic function! Here the new constants m and n tell you exactly where the x-intercepts are on the x-axis. The equation for this example is y=2(x-3)(x-5), and m=3, n=5. Like b from the vertex form, you have to flip the signs when finding m and n. Also, which one you call m and which you call n don’t matter.
If I give you y=2(x-3)(x-5) rather than the graph, you can immediately mark 3 and 5 on the x-axis, and if you are good at factoring, you can also factor the standard form y=2x2-16x+30 to get the x-intercept form.
The constant a
In all three forms, there is the constant called a. This important number controls the shape of the graph. Let's take the vertex form of the example and change a to see what happens; except for a, the function stays the same: y=a(x-4)2-2
You can see when a is bigger, the parabola rises up more with the same x-value. The constant a “stretches” the parabola; when a=1, an increase of 2 in x produces an increase of 4 in y, and when a=2, the increase in y doubles to 8.
This is the quadratic function. It’s the introductory step to functions where the input is raised to a power, and if you understand the quadratic function, you would understand its successors (ex. polynomial functions) with ease.
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