Full solution

Q. Three points on the graph of the function

f(x)

are

(0,6)

(1,8)

and

(2,10)

which represents

f(x)

Identify Pattern: Identify the pattern in the given points. $\newline$ We have the points $(0,6)$ , $(1,8)$ , and $(2,10)$ . Let's look at the $x$ and $y$ values to see if there is a linear relationship. $\newline$ For $x = 0$ , $y = 6$ . $\newline$ For $x = 1$ , $y = 8$ . $\newline$ For $x = 2$ , $(1,8)$ $0$ . $\newline$ We can see that as $x$ increases by $(1,8)$ $2$ , $y$ increases by $(1,8)$ $4$ .
Determine Linearity: Determine if the points lie on a straight line. Since the difference in $y$ is consistent as $x$ increases by $1$ , we can suspect that these points lie on a straight line. This suggests a linear function of 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 $(1,8)$ , we can calculate the slope $m$ as follows: $\newline$ $m = \frac{y_2 - y_1}{x_2 - x_1}$ $\newline$ $m = \frac{8 - 6}{1 - 0}$ $\newline$ $m = \frac{2}{1}$ $\newline$ $m = 2$
Find Y-Intercept: Use the y-intercept from one of the points. $\newline$ Since the point $(0,6)$ is given, we know that when $x = 0$ , $y = 6$ . This means the y-intercept $b$ is $6$ .
Write Line Equation: Write the equation of the line. $\newline$ Using the slope $m = 2$ and the y-intercept $b = 6$ , we can write the equation of the line as: $\newline$ $f(x) = 2x + 6$
Verify with Third Point: Verify the equation with the third point $(2,10)$ . Substitute $x = 2$ into the equation $f(x) = 2x + 6$ to see if $y$ equals $10$ . $f(2) = 2(2) + 6$ $f(2) = 4 + 6$ $f(2) = 10$ Since the third point $(2,10)$ satisfies the equation, our function is correct.