Question

Two points on the graph of the linear function f are (0,5) and (3,8). Write a function g whose graph is a reflection in the x-axis of the graph of f.

Answer 1

Explanation:

As first step we must obtain the equation of the line. According to Analytical Geometry, we can get a equation of the line by knowing two different points. A linear function is represented by the following expression:

`y = m\cdot x + b`

Where:

`x` - Independent variable, dimensionless.

`y` - Dependent variable, dimensionless.

`m` - Slope, dimensionless.

`b` - y-Intercept, dimensionless.

If we know that
`(x_(1), y_(1)) = (0, 5)` and
`(x_(2), y_(2)) = (3, 8)`, the following system of linear equations is formed:

`5 = 0\cdot m + b`

`b = 5` (Eq. 1)

`8 = 3\cdot m + b`

`3\cdot m + b = 8` (Eq. 2)

From (Eq. 1) in (Eq. 2) we get the value of
`m`:

`3\cdot m + 5 = 8`

`3\cdot m = 3`

`m = 1`

The equation of the line that contains the points (0, 5) and (3, 8) is
`f(x) = x + 5`.

As next step we must apply a reflection in the x-axis, whose operation is defined as follows:

`(x', y') \longrightarrow (x, -y)`

That is:

`x' = x`

`y' = -y`

Then, the function
`g(x)` is
`g(x) = -x-5`. We plot each function and include the result in the attachment below.