Question

the table represents a quadratic function. write an equation of the function in standard form. x 1 2 3 4 f(x) 2 3 2 -1 the function is f(x)=□.

Explanation:

Step1: Recall the standard - form of a quadratic function

The standard - form of a quadratic function is $f(x)=ax^{2}+bx + c$.

Step2: Substitute the given points into the function

When $x = 1$, $f(1)=a(1)^{2}+b(1)+c=a + b + c=2$.
When $x = 2$, $f(2)=a(2)^{2}+b(2)+c=4a + 2b + c=3$.
When $x = 3$, $f(3)=a(3)^{2}+b(3)+c=9a+3b + c=2$.

Step3: Solve the system of equations

Subtract the first equation from the second equation:
$(4a + 2b + c)-(a + b + c)=3 - 2$, which simplifies to $3a + b=1$.
Subtract the second equation from the third equation:
$(9a+3b + c)-(4a + 2b + c)=2 - 3$, which simplifies to $5a + b=-1$.

Step4: Solve for $a$ and $b$

Subtract the equation $3a + b=1$ from $5a + b=-1$:
$(5a + b)-(3a + b)=-1 - 1$,
$2a=-2$, so $a=-1$.
Substitute $a=-1$ into $3a + b=1$:
$3(-1)+b=1$,
$-3 + b=1$, so $b = 4$.

Step5: Solve for $c$

Substitute $a=-1$ and $b = 4$ into $a + b + c=2$:
$-1+4 + c=2$,
$3 + c=2$, so $c=-1$.

Answer:

$f(x)=-x^{2}+4x - 1$