Answer:
- vertex form: f(x) = a(x -h)^2 +k
- standard form: f(x) = ax^2 +bx +c
Step-by-step explanation:
Often, you're given one form and asked to write the equation in the other form. Here, we can show the relationship between the two forms.
In <u>vertex form</u>, the vertex of the function (h, k) is obvious in the way the function expression is written:
f(x) = a(x -h)^2 +k . . . . . . . for vertex (h, k) and vertical scale factor "a"
If we "simplify" this form, we get ...
f(x) = a(x^2 -2hx +h^2) +k
f(x) = ax^2 -2ah + (ah^2 +k) . . . . . "standard form" from vertex form
Comparing this to standard form, we can see the relations between the coefficients are ...
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In <u>standard form</u>, terms are written in descending order of the exponent of the variable.
f(x) = ax^2 +bx +c
Generally, coefficients are named in alphabetical order, starting with "a" for the leading coefficient (the coefficient of the highest-degree term).
We can use the relations shown above to find the vertex from from these coefficients.
b = -2ah
h = -b/(2a) . . . . . divide by the coefficient of h
And the other coefficient of the vertex is ...
k = c - ah^2 . . . . subtract ah^2 from the equation for c
k = c - b^2/(4a)
Then ...
f(x) = a(x +b/(2a))^2 +(c -b^2/(4a)) . . . . . "vertex form" from standard form
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You may notice that the key relationship is that between "b" and "h". It is useful to remember it:
h = -b/(2a)
