Question

17. Find the equation of the line that is tangent to the graph of F(x) and parallel to the given

line. Function: F(x) = 2x^2 Line: 4x + y + 3 = 0

Answer 1

The equation of the line is
`y = -4\cdot x -2`.

Step-by-step explanation:

Let the function be
`f(x) = 2\cdot x^(2)` and the line be
`4\cdot x + y +3 = 0`. First, we transform the equation of the line into explicit form:

`y = -4\cdot x + 3` (1)

By Differential Calculus, the slope at any point of the function is represented by its first derivative, that is:

`f'(x) = 4\cdot x` (2)

If the line tangent to the function must be parallel to
`y = -4\cdot x + 3`, then
`f'(x) = -4`. In consequence, we clear
`x` in (2):

`4\cdot x = -4`

`x = -1`

Then, we evaluate the function at the result found above to determine the associated value of
`y = f(x)`:

`f(-1) = 2\cdot (-1)^(2)`

`y = f(-1) = 2`

By Analytical Geometry we know that an equation of the line can be formed by knowing both slope (
`m`) and y-intercept (
`b`). If we know that
`x = -1`,
`y = 2` and
`m = -4`, then the y-intercept of the equation of the line is:

`b = y -m\cdot x`

`b = 2-(-4)\cdot (-1)`

`b = 2 -4`

`b = -2`

Based on information found previously and the equation of the line, we have the equation of the line that is tanget to the graph of
`f(x)`, so that it is parallel to
`4\cdot x + y +3 = 0` is:

`y = -4\cdot x -2`

The equation of the line is
`y = -4\cdot x -2`.