MHF4U - 1.4
Unit 1 - Lesson 2
Example 1: State the transformations for the following functions from their parent function, sketch the shape of the graph and state the domain and range
a) $f(x)=\frac{-4}{3(x-10)}-5$

Step 1 ：First, we identify the parent function. The parent function here is \(f(x)=\frac{1}{x}\).

Step 2 ：The transformations applied to the parent function are: a vertical stretch by a factor of 4, a reflection in the x-axis, a horizontal shift 10 units to the right, and a vertical shift 5 units down. So the transformed function is \(f(x)=\frac{-4}{3(x-10)}-5\).

Step 3 ：To sketch the graph, we start with the graph of the parent function \(f(x)=\frac{1}{x}\), then apply the transformations in the order: vertical stretch, reflection, horizontal shift, and vertical shift.

Step 4 ：The domain of the function is all real numbers except for the value that makes the denominator zero. In this case, the denominator is zero when \(x=10\). So the domain of \(f(x)\) is \(x \neq 10\), or \(x \in \boxed{(-\infty, 10) \cup (10, \infty)}\) in interval notation.

Step 5 ：The range of the function is all real numbers except for the value that the function approaches as \(x\) approaches the value excluded from the domain. In this case, as \(x\) approaches 10, the function approaches \(-5\). So the range of \(f(x)\) is \(y \neq -5\), or \(y \in \boxed{(-\infty, -5) \cup (-5, \infty)}\) in interval notation.