Коли дві квадратичні функції співпадають?

There are two ways to tell if quadratic functions are identical:

1. If their equations, $y = f(x)$, are equivalent
2. If their graphs share 3 points in common

Both of these are pretty straightforward to work with (yay!). We'll look at an example of each below.

Approach 1: Are the equations equivalent?

This will probably require you to mess around with (i.e., rearrange) at least one of the equations to see if you can get it to look like the other.

The equation of a quadratic function in $x$ will look like $y = ax^2+bx+c$: a quadratic equation in two variables. Let's first review how to tell if two quadratic equations in a single variable are equivalent.

So, are these two quadratic equations equivalent?

The bottom one appears to be in its simplest form, so let's see if we can take the top one and rearrange it to look the same.

Now, we need to factor the quadratic expression on the left, $x^2 - 2x - 8$.

We can see that, once we simply and factor in the second equation, the quadratic expressions on the left are the same.

Approach 2: Do the functions/graphs share 3 points in common?

Let's use the same two quadratic expressions as we did before, just so we can show that both approaches can be used to determine if they are identical.

If two quadratic functions are the same, they will share at least 3 distinct points on their graphs. To determine if quadratic expressions share points without actually graphing them, simply pick three values for x and plug them into each expression. If each x produces the same value for y in both expressions, then we can say they are equivalent.

Let's try it out.

We're going to pick values for x that are easy to work with (i.e., small):

x=0 x=1 x=4 (this one makes the second expression equal to 0)

Okay! This means that both graphs share the point (0,-8). One down, two to go :)

Yes! This means that both graphs share the point (1,-9). We only need one more point in common to show that the graphs are exactly the same.

Woohoo! This means that both graphs share the point (4,0). This is their third point in common. Note that any other points we select would also be shared by both graphs, but we only need 3 to say that they are identical.

Two quadratic functions are identical if they coincide at 3 distinct points. This is a good, and accurate definition, but without graphing the entire function, how else can we verify this is true?

One method for this is to plug in 3 separate values for x and see if they come up with the same answer. For example, we have two quadratic functions: $$y_1=x^2+2x+3, y_2=(x-2)^2+6x-1$$Let's decide if these are identical or not. We can choose different values for x and see what the functions return. Let's use $x=0, x=1, x=2$. Solving for $x=0$, either with a calculator or by had gives us $y_1=3, y_2=3$. That's great, but not enough to prove they are identical. We need 2 more points. This time, when $x=1$ we get $y_1=6, y_2=6$. This is looking good. Just one more point will do it. Last, when $x=2$ we get $y_1=11, y_2=11$. This is enough to prove the functions are identical.

(As a side note, $x=2$ was a smart choice for this problem because then the first term in $y_2$ would become zero since $(2-2)^2=0$. Utilize tricks like this whenever possible to make your calculations faster and easier).

Another method to use would be putting both quadratic functions into standard form. For quadratic functions, this form is:$$y=ax^2+bx+c$$ Let's use the same functions from the first example, $y_1$ and $y_2$. $y_1$ is already in standard form, so the next step would be to simplify $y_2$ and put it in standard form.$$y_2=(x-2)^2+6x-1$$$$y_2=x^2-4x+4+6x-1$$$$y_2=x^2+2x+3$$Putting $y_2$ into standard form shows us that it is the same as $y_1$.

Two quadratic functions are identical if their equations are the same, or if they coincide at three distinct points. This is true because there is at most one quadratic through any three distinct points.