5.3 Graphing Radical Functions
- Graph radical functions, including square root and cube root functions.
- Identify the effect of transformations on the key features of the graphs of radical functions.
Assignment
- All vocabulary copied into notes
- radical function
- p261 11–17, 20–33 (pdf)
- Answers to odd questions / even questions
Additional Resources
Transforming Radical Functions
See section 1.2. Seriously. There is nothing new here. Just keep in mind that radicals with odd indices, like cube root, can have negative numbers in their domain. Square roots are restricted to non-negative numbers.
Rewrite a Radial Function
Simplifying a radical functions can make spotting transformation easier.
\[\begin{align} f(x) &= \sqrt{4x+16} + 7 \\ &= \sqrt{4(x + 4)} + 7 \\ f(x) &= 2\sqrt{x+4} + 7 \end{align}\]It’s usually a good idea to factor so that $x$ has a coefficient of $1$. Horizontal scale can move key points, like vertices, making it difficult to see horizontal translations. In the original function above, the function was moved left $16$, but then scaled horizontally by $4$, meaning it ended up only being moved $4$ units to the left. It’s easier to spot this in the factored version.
The two functions above are identical, blurring the lines between horizontal and vertical scaling.
\[\begin{align} f(x) &= \sqrt{9x} \\ g(x) &= 3\sqrt{x} \end{align}\]Vertical is typically the easier of the two to process, so rewriting in that form is desired.