Question

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

Answer 1

Answer:

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

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 $y = -4\cdot x -2$.