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Write an equation for the quadratic function
f whose graph passes through the points
(6,0),(7,0), and
(5,6).

f(x)=

Write an equation for the quadratic function $f$ whose graph passes through the points $(6,0),(7,0)$ , and $(5,6)$ . $\newline$ $f(x)=$

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Write a quadratic function from its x-intercepts and another point

Full solution

Q. Write an equation for the quadratic function

f

whose graph passes through the points

(6,0),(7,0)

, and

(5,6)

.

\newline

f(x)=

Identify x-intercepts: Identify the x-intercepts from the given points. The x-intercepts are the x-values where the y-values are zero. Here, the x-intercepts are at $(6,0)$ and $(7,0)$ .
Set up quadratic form: Set up the general form of the quadratic equation using the x-intercepts. The form is $f(x) = a(x - p)(x - q)$ , where $p$ and $q$ are the x-intercepts. Thus, $f(x) = a(x - 6)(x - 7)$ .
Use third point: Use the third point $(5,6)$ to find the value of ' $a$ '. Substitute $x = 5$ and $f(x) = 6$ into the equation: $6 = a(5 - 6)(5 - 7)$ . Simplify the right side: $6 = a(-1)(-2) = 2a$ .
Solve for 'a': Solve for 'a'. From $6 = 2a$ , divide both sides by $2$ to get $a = 3$ .
Substitute 'a' back: Substitute 'a' back into the equation. The final equation of the quadratic function is $f(x) = 3(x - 6)(x - 7)$ .

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