Find the coordinates of the focus and equation of the directrix for the parabola given by y2 = −4x. The general formula for this parabola is y2 = 4px. Therefore, the value of p is . The coordinates of the focus are . The equation of the directrix is .

The coordinates of the focus are anf the equation of the directrix is , respectively.

Step-by-step explanation:

Let be , which represents a parabola with a horizontal axis of symmetry. To find the location of the focus and to determine the equation of the directrix we must find the distance between focus and vertex (), dimensionless, a value than can be extracted from the following definition:

(Eq. 1)

Where:

- Independent variable, dimensionless.

- Dependent variable, dimensionless.

, - Coordinates of the vertex, dimensionless.

By direct comparison, we get the following coincidences:

(Eq. 2)

From (Eq. 2) we get that distance between focus and vertex is:

Which is consistent with the fact parabola has an absolute maximum.

Now, the location of the focus is:

(Eq. 3)

If know that , and , coordinates of the focus is:

And the equation of the directrix is represented by:

(Eq. 4)

()

The coordinates of the focus are anf the equation of the directrix is , respectively.

(f - g)(x) = 5x^3 - 3

step-by-step explanation:

rearrange these functions vertically, with like terms in separate columns:

f(x) = 5x^3 - 2

g(x) =         +1

now subtract, column by column:

(f - g)(x) = 5x^3 - 3

step-by-step explanation:

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