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Three points on the graph of the function $f(x)$ are $(0,6)$ $(1,9)$ and $(2,12)$ 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

(0,6)

(1,9)

and

(2,12)

which represents

f(x)

Identify Pattern: Identify the pattern in the $y$ -values of the given points. $\newline$ We have the points $(0,6)$ , $(1,9)$ , and $(2,12)$ . We notice that as $x$ increases by $1$ , $y$ increases by $3$ .
Check Linearity: Determine if the points lie on a line. $\newline$ Since the change in $y$ is consistent as $x$ increases by $1$ , we suspect that these points might lie on a straight line. A linear function has the form $f(x) = mx + b$ , where $m$ is the slope and $b$ is the $y$ -intercept.
Calculate Slope: Calculate the slope $m$ of the line. $\newline$ Using the points $(0,6)$ and $(1,9)$ , we calculate the slope $m$ as follows: $\newline$ $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 6}{1 - 0} = \frac{3}{1} = 3$
Use Y-Intercept: Use the y-intercept $b$ from one of the points. $\newline$ Since the point $0,6$ is on the graph of the function, when $x = 0$ , $y = 6$ . This gives us the y-intercept $b = 6$ .
Write Equation: Write the equation of the line using the slope and y-intercept. $\newline$ The function $f(x)$ is then $f(x) = 3x + 6$ .
Verify Function: Verify the function with the third point $(2,12)$ . Substitute $x = 2$ into the function $f(x) = 3x + 6$ to check if $y = 12$ . $f(2) = 3(2) + 6 = 6 + 6 = 12$ Since this matches the $y$ -value of the third point, our function is correct.

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