M151Q Precalculus · Unit 1: Functions, graphs, models · on Exam 1
MFG 2.3 Transformations of graphs
Study notes for Transformations of graphs in M151Q Precalculus at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- Starting from a basic graph $y = f(x)$: $f(x) + k$ shifts up $k$; $f(x - h)$ shifts right $h$ (note the sign); $af(x)$ stretches vertically by $a$ (compresses if $0
- Order matters: apply stretches and reflections before shifts. $g(x) = -2(x+3)^2 + 4$: shift left 3, stretch by 2, reflect over the $x$-axis, shift up 4. Vertex $(-3, 4)$.
- Read the transformation from a graph by locating where the "anchor" point of the parent (vertex, corner, origin) landed and how the heights scaled.
- Transformations move the domain and range along: $\sqrt{x-2} - 5$ has domain $[2,\infty)$ and range $[-5,\infty)$.
Key formulas
Vertical
$$y = f(x) + k\ (\text{up } k),\qquad y = a f(x)\ (\text{stretch by } a),\qquad y = -f(x)\ (\text{flip over } x\text{-axis})$$
Horizontal
$$y = f(x - h)\ (\text{right } h),\qquad y = f(-x)\ (\text{flip over } y\text{-axis})$$
General form
$$y = a\,f(x - h) + k$$
Worked example
Write the formula for the graph of $y = \sqrt x$ shifted right 2 and down 5, reflected across the $x$-axis. State its domain and range.
Show the solution
$y = -\sqrt{x - 2} - 5$. Domain $[2,\infty)$ (inside the root must be $\ge 0$). The root part is $\le 0$ after the reflection, so the range is $(-\infty, -5]$.
Common mistakes
- Reading $f(x - 2)$ as a shift left. Inside the parentheses, the sign is reversed.
- Shifting before stretching when the formula stretches first (a·f(x) + k).
- Forgetting that a reflection flips the range.
Exam tipName each transformation in the order the formula applies it, then move one anchor point step by step. Write the final anchor point on your graph.
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