Question

How do you graph `q(x)=-4abs(x-2)-1` using transformations?

Answer 1

See below.

Explanation:

Let's say that the parent function is `f(x) = absx`.

graph{abs x [-10, 10, -5, 5]}

To obtain the function `q(x)`, we must perform a series of transformations. First, we can vertically stretch `f(x)` by a factor of `4`.

`4 * f(x) = 4 * abs x = 4 abs x`

As a reminder, here are some rules for horizontal and vertical stretches and shrinks for `f(x)`:

• A vertical stretch by a factor of `a` is denoted `a * f(x)`
• A vertical shrink by a factor of `1/a` is denoted `1/a * f(x)`
• A horizontal stretch by a factor of `a` is denoted `f(1/a * x)`
• A horizontal shrink by a factor of `1/a` is denoted `f(a * x)`

Let's call this `g(x)`: `g(x) =4absx`.

graph{4 abs x [-10, 10, -5, 5]}

We can now reflect `g(x)` over the `x`-axis. A reflection of `g(x)` over the `x`-axis is represented by `-g(x)`, while a reflection over the `y`-axis would be represented by `g(-x)`.

`-g(x) = - 4 abs x`

Let's call this `h(x)`: `h(x) = -4 abs x`.

graph{-4 abs x [-10, 10, -5, 5]}

We can now horizontally and vertically translate (or shift) `h(x)`. The rules for these translations are below:

• A vertical shift `a` units up is denoted `h(x) + a`

• A vertical shift `a` units down is denoted `h(x) - a`

• A horizontal shift `a` units right is denoted `h(x-a)`

• A horizontal shift `a` units left is denoted `h(x+a)`

In this case, we are shifting `h(x)` `1` unit down and `2` units right.

`h(x-2) -1 = -4 abs (x-2) - 1`

graph{-4 abs x -1 [-10, 10, -5, 5]}

graph{-4 abs (x -2) - 1 [-10, 10, -5, 5]}

This is the final function `p(x) = -4 abs (x-2) - 1`.