Write the equation of the parabola in vertex form. vertex (2,4), point (3,1) f(x)=

Answer 1

That's parabola, hence we can say the standard form is f(x) = ax² + bx + c

The vertex is (3,1) and we know that the X-component of this parabola is found by (-b/2a).

Hence, -b/2a = 2 ==> -b = 4a

Let's plug (3,1) and (2,4) to the function

For (3,1) ==> 1 = 9a + 3b + c

For (2,4) ==> 4 = 4a + 2b + c

From these two equations when subtracted, we get 5a + b = -3 and we know, b = -4a.

Then, let's plug 'b' as -4a

5a - 4a = -3

a = -3 ==> a = -3

Then, b = -4 x -3 = 12

To find "c", let's plug a & b into the function at any known value.

-27 + 36 + c = 1

Then, c = -8

Then, the function is f(x) = -3x² +12x -8



Answer 2

You are looking for an equation in the form f(x)=a*(x-h)²+k

where the point (h, k) is the vertex. Here, the added constant k can be thought of as the height or distance of the vertex from the x axis, similarly to how in a linear equation given as f(x)=mx+b, b provides the height of the y intercept, which is the distance from the x axis at x=0, so f(x)=b

In vertex form, if you set x=h, then you’ll get f(x)=a*(h-h)²+k=k, the positive constant representing the “height”

However, this height may not be the height at the intercept, since instead of calculating at x=0 to cancel out the variable portion of the equation, you calculate x=h. Therefore, the variable h will be the x coordinate of the vertex, which is the distance of the vertex from the y intercept.

Using the provided vertex, we plug in the x coordinate of the vertex for h and the y coordinate of the vertex for k to obtain f(x)=a*(x-2)²+4.

From here, you can simply plug in the point provided, where x=3 and y=1=f(x). Therefore,

1=a*(3-2)²+4

-3=a*(1)^{2. Subtracting 4 from both sides}

a=-3 Simplifying

plugging into vertex form of equation, we obtain the answer

f(x)= -3*(x-2)²+4