Question

How do you write `f(x)= |-18x-17|` as piecewise functions?

Answer 1

Equation 1: `y = 18x + 17, x ≥ -17/18`
Equation 2: `y = -18x - 17, x <-17/18`

Explanation:

When writing piece-wise equations for absolute value functions, you will have two equations, one to the left of the vertex and the other to the right of the vertex. So, it makes sense that we find the location of the vertex.

There is no vertical transformation, so the vertex lies on the x-axis.

`0 = |-18x - 17|`

`0 = -18x - 17`

`x = -17/18`

We now know the x-intercept of the graph: `(-17/18, 0)`. All that is left to do is to find another point that the function passes through. It makes sense that we find the `y` intercept.

`y = |-18(0) - 17|`

`y = 17->(0, 17)`

Now that we know two points, we can find the equation of the line to the right of the x intercept.

Start by finding the slope.

`m = (y_2 - y_1)/(x_2 - x_1)`

`m = (17 - 0)/(0 - (-17/18))`

`m = 17/(17/18)`

`m = 18`

By point slope form:

`y - y_1 = m(x - x_1)`

`y - 17 = 18(x - 0)`

`y - 17 = 18x`

`y = 18x + 17`

We now have our first piece-wise equation: `y = 18x + 17, x ≥ -17/18`.

Finding the equation of the second is extremely simple. All you must do is multiply the entire right-hand side of the equation by `-1`.

`y = -18x - 17`, would be the equation of our second piece-wise equation. This would be for `x < -17/18`.

Hopefully this helps!