Solution: Yes, our friend is correct. Explanation: See solution.
Practice makes perfect
Let's first determine if every quadratic can be written in standard form by using the definition of a quadratic function. In a quadratic function, the highest degree is 2, so it can only have a total of 3 terms — the 2^(nd) degree term, the 1^(st) degree term, and the 0^(th) degree term.
x^2, x^1, and x^0
A quadratic function can be any combination of these 3 terms. Each term can be multiplied by any real number constant. Let's allow our constants to be a, b, and c.
y=ax^2+bx+c
By the definition of a quadratic function, we can see that any quadratic can be written in standard form. Now, let's determine if any quadratic can be written in vertex form.
y=a(x-h)^2+k
The graph of quadratic function is a parabola. All parabolas have a vertex, due to the shape of a parabola. The constants h and k can be any real number, and they can represent any vertex. Even if the vertex is (0,0), vertex form still holds.
y=a(x-h)^2+k ⇒ y=a(x-0)^2+0 ⇔ y=ax^2
Using h and k, we can place the parabola anywhere on the graph. Using the constant a, we can make the parabola any possible width as well as change the direction it opens. Therefore, our friend is correct — vertex form can make any possible parabola, since we have control of the direction, width, and placement of the parabola.
Alternative Solution
Converting Standard Form to Vertex Form
Another way to show that every quadratic can be written in vertex form is to show that standard form can be transformed to vertex form.
Now, notice that - b2a and 4ac+b^24a are constants. We can replace them with simpler constants h and k.
y=a(x-( - b/2a)^2)+ 4ac+b^2/4a ⇒
y=a(x- h)^2+ k
As we can see, any equation in standard form can be converted to vertex form. Since any quadratic function can be written in standard form, this also means that any quadratic can be written in vertex form.
