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Three points on the graph of the function $f(x)$ are $(0,3)$ $(1,6)$ and $(2,9)$ which represents $f(x)$

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Roots of integers

Full solution

Q. Three points on the graph of the function

f(x)

are

(0,3)

(1,6)

and

(2,9)

which represents

f(x)

Given Points Analysis: We are given three points on the graph of the function $f(x)$ : $(0,3)$ , $(1,6)$ , and $(2,9)$ . To find the function $f(x)$ , we need to determine if there is a pattern or relationship between the $x$ -values and the $y$ -values that these points represent.
Identifying Linear Relationship: Looking at the $x$ -values ( $0$ , $1$ , $2$ ) and the corresponding $y$ -values ( $3$ , $6$ , $9$ ), we can see that as the $x$ -value increases by $1$ , the $y$ -value increases by $3$ . This suggests a linear relationship between $x$ and $y$ .
Calculating Slope: To confirm the linear relationship, we can calculate the slope $m$ of the line using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ . Using the points $(0,3)$ and $(1,6)$ , we get $m = \frac{6 - 3}{1 - 0} = \frac{3}{1} = 3$ .
Writing Equation in Slope-Intercept Form: Since the slope is consistent across the points given, we can write the slope-intercept form of the line as $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. We already know the slope $m$ is $3$ , so the equation becomes $y = 3x + b$ .
Finding Y-Intercept: To find the y-intercept $b$ , we can use any of the given points. Let's use the point $(0,3)$ . Plugging the values into the equation $y = 3x + b$ , we get $3 = 3(0) + b$ , which simplifies to $3 = b$ .
Writing Final Function: Now that we have both the slope and the y-intercept, we can write the function $f(x)$ as $f(x) = 3x + 3$ .

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