Question

How do you use function notation to write the equation of the line with the slope of 2 and y-intercept of $ \left( 0,\dfrac{-6}{7} \right)$ ?

Answer 1

Hint: In this question, we need to write the equation of a line whose slope and intercept are given. We need to write the equation using function notation. For this, we will first use the slope-intercept form of a line given by y = mx+c to find the equation of a line in (x,y) point form where m is the slope of the line and c is the y-intercept. After that, we will use f(x) in place of y to find the equation in function notation.

Complete step by step answer:
Here we are given the slope of the line as 2. The line forms y intercept at the point $ \left( 0,\dfrac{-6}{7} \right)$ . So we can say the line cuts the x axis at the point $ \left( 0,\dfrac{-6}{7} \right)$ . The length of the intercept will be the distance from origin to the point $ \left( 0,\dfrac{-6}{7} \right)$ i.e. it will be $ \dfrac{-6}{7}$ . So we can say that the value of y intercept is $ \dfrac{-6}{7}$ .
Now we know that an equation of a line in slope-intercept form is given by y = mx+c where m is the slope of the line and c is the y-intercept. So let us use it to find the required equation of a line. We are given the slope of the line as 2, therefore the value of m will be equal to 2. Also, the value of y-intercept is given as $ \dfrac{-6}{7}$ therefore, the value of c will be equal to $ \dfrac{-6}{7}$.
We have m = 2 and c = $ \dfrac{-6}{7}$ .
Putting both these values in the equation of the line in slope intercept form we have $ y=2x+\left( \dfrac{-6}{7} \right)\Rightarrow y=2x-\dfrac{6}{7}$ which is the equation of line.
But we require the equation of a line in function notation. We know that y is the dependent variable that depends on x so it can be written as a function of x. Therefore we can write y as f(x). Hence our equation becomes $ f\left( x \right)=2x-\dfrac{6}{7}$ .
Therefore the required equation of line is $ f\left( x \right)=2x-\dfrac{6}{7}$ .

Note:
Students should take care of the signs while solving this type of question. Note that the given point (y-intercept) is of the form (0,y), so it cuts the y-axis at $ \dfrac{-6}{7}$ which will be the value of c in y = mx+c. Students should keep in mind the equation of a line in slope-intercept form. Note that every y should be changed into the form of f(x) and the right side should be a function of x only.