Transformations

Example 1

Describe $\displaystyle g(x)=-2(x-3)^2+1$ relative to $\displaystyle f(x)=x^2$.

Shift right $\displaystyle 3$, reflect across the x-axis, stretch vertically by $\displaystyle 2$, and move up $\displaystyle 1$. The vertex is $\displaystyle (3,1)$.

Example 2

Write the graph of $\displaystyle y=|x|$ shifted left $\displaystyle 2$ and down $\displaystyle 5$.

A left shift replaces $\displaystyle x$ with $\displaystyle x+2$ and a downward shift subtracts $\displaystyle 5$: $\displaystyle y=|x+2|-5$.

Problem 1.2.1

Describe $\displaystyle y=3\sqrt{x+1}-2$ relative to $\displaystyle y=\sqrt{x}$.

Problem 1.2.2

Transform $\displaystyle y=x^2$ by reflecting across the x-axis and moving down $\displaystyle 4$.

Problem 1.2.3

Compare the parent graph and transformed graph $\displaystyle y=2(x-1)^2-3$. State its vertex and vertical scale.

Show worked solution

The vertex moves to $\displaystyle (1,-3)$ and the vertical scale factor is $\displaystyle 2$. The graph opens upward and is narrower than $\displaystyle y=x^2$ at the same output height above its vertex.