Three points on the graph of the function $f(x)$ are $(0,1)$ $(1,4)$ and $(2,9)$ which represents $f(x)$

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Find the vertex of a parabola

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Q. Three points on the graph of the function

f(x)

are

(0,1)

(1,4)

and

(2,9)

which represents

f(x)

Identify Pattern: Identify the pattern in the given points. $\newline$ We have the points $(0,1)$ , $(1,4)$ , and $(2,9)$ . Let's look for a pattern in the $y$ -values as the $x$ -values increase. $\newline$ From $(0,1)$ to $(1,4)$ , the $y$ -value increases by $3$ . $\newline$ From $(1,4)$ to $(2,9)$ , the $y$ -value increases by $(1,4)$ $2$ . $\newline$ This suggests that the function might be quadratic, as the differences between consecutive $y$ -values are not constant but increase by a constant amount (the second differences are constant).
General Form: Use the general form of a quadratic function to find $a$ , $b$ , and $c$ . The general form of a quadratic function is $f(x) = ax^2 + bx + c$ . We will use the given points to create a system of equations to solve for $a$ , $b$ , and $c$ .
Create Equations: Create equations using the given points and the general form of a quadratic function. $\newline$ Using the point $(0,1)$ , we get the equation: $\newline$ $1 = a(0)^2 + b(0) + c$ $\newline$ This simplifies to: $\newline$ $c = 1$ $\newline$ Using the point $(1,4)$ , we get the equation: $\newline$ $4 = a(1)^2 + b(1) + c$ $\newline$ This simplifies to: $\newline$ $a + b + c = 4$ $\newline$ Using the point $(2,9)$ , we get the equation: $\newline$ $9 = a(2)^2 + b(2) + c$ $\newline$ This simplifies to: $\newline$ $4a + 2b + c = 9$
Substitute Values: Substitute the value of $c$ into the other two equations. $\newline$ We know that $c = 1$ , so we can substitute this into the other two equations to get: $\newline$ $a + b + 1 = 4$ $\newline$ $4a + 2b + 1 = 9$ $\newline$ Now we simplify these equations to: $\newline$ $a + b = 3$ $\newline$ $4a + 2b = 8$
Solve Equations: Solve the system of equations for $a$ and $b$ . We have two equations: $a + b = 3$ $4a + 2b = 8$ We can multiply the first equation by $2$ to help eliminate one of the variables: $2a + 2b = 6$ Now we subtract this new equation from the second equation: $(4a + 2b) - (2a + 2b) = 8 - 6$ $2a = 2$ Dividing both sides by $2$ gives us: $a = 1$ Now we substitute $a = 1$ into the first equation: $b$ $1$ $b$ $2$ $b$ $3$
Final Function: Write the final function using the values of $a$ , $b$ , and $c$ . We have found that $a = 1$ , $b = 2$ , and $c = 1$ . Therefore, the function that passes through the points $(0,1)$ , $(1,4)$ , and $(2,9)$ is: $f(x) = 1x^2 + 2x + 1$