Three points on the graph of the function $f(x)$ are $(0,0)$ $(1,1)$ and $(2,4)$ which represents $f(x)$

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Algebra 2

Find the vertex of a parabola

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Q. Three points on the graph of the function

f(x)

are

(0,0)

(1,1)

and

(2,4)

which represents

f(x)

Determine Function Form: Determine the general form of the function. $\newline$ Since we have $3$ points, we can assume that the function $f(x)$ is a quadratic function of the form $f(x) = ax^2 + bx + c$ .
Set Up Equations: Use the given points to set up a system of equations. $\newline$ Substitute the $x$ and $y$ values from the points into the quadratic equation to get three equations: $\newline$ $1$ . For point $(0,0)$ : $0 = a(0)^2 + b(0) + c$ , which simplifies to $c = 0$ . $\newline$ $2$ . For point $(1,1)$ : $1 = a(1)^2 + b(1) + c$ , which simplifies to $a + b + c = 1$ . $\newline$ $3$ . For point $(2,4)$ : $4 = a(2)^2 + b(2) + c$ , which simplifies to $y$ $0$ .
Solve Equations: Solve the system of equations. $\newline$ From equation $1$ , we have $c = 0$ . $\newline$ Substitute $c = 0$ into equations $2$ and $3$ to get: $\newline$ $a + b = 1$ (simplified from $a + b + 0 = 1$ ) $\newline$ $4a + 2b = 4$ (simplified from $4a + 2b + 0 = 4$ )
Find $a$ and $b$ : Solve for $a$ and $b$ . We can multiply the first equation by $2$ to eliminate $b$ : $2a + 2b = 2$ Now we have a system of two equations: $2a + 2b = 2$ $4a + 2b = 4$ Subtract the first equation from the second to solve for $a$ : $b$ $0$ $b$ $1$ $b$ $2$
Substitute Values: Substitute the value of $a$ into one of the equations to find $b$ . Using $a + b = 1$ and substituting $a = 1$ , we get: $1 + b = 1$ $b = 0$
Write Final Function: Write the final function $f(x)$ . Now that we have $a = 1$ , $b = 0$ , and $c = 0$ , the function $f(x)$ is: $f(x) = 1x^2 + 0x + 0$ Simplify to get: $f(x) = x^2$