Transformations and Piecewise Functions

Transformations of functions

A transformation of $f(x)$ changes its graph in a predictable way. Here is the whole toolkit in one place:

• $f(x - h)$ shifts the graph horizontally by $h$ (to the right when $h$ is positive).
• $f(x) + k$ shifts the graph vertically by $k$.
• $-f(x)$ reflects the graph over the $x$-axis.
• $f(-x)$ reflects the graph over the $y$-axis.
• $a\cdot f(x)$ scales all $y$-values vertically by a factor of $|a|$. If $a$ is negative, the graph is also reflected over the $x$-axis.

Anything applied to $x$ inside the function changes the graph horizontally; anything applied to the function's output changes it vertically. The horizontal changes feel backwards: $f(x+4)$ moves the graph left 4, because the input must be 4 smaller to get the same output. The vertical changes behave as you would expect.

For a specific numerical evaluation, simplify the input first, evaluate $f$, and then apply whatever is outside $f$. When several horizontal changes combine in the same expression, rewrite that inside expression clearly before deciding the shift or reflection, instead of assuming a fixed order applies to every combination.

Applying a transformation step by step

Worked example. Let $f(x) = x^2$ and $g(x) = -2f(x + 1) + 3$. Find $g(1)$.

Apply the inside shift first, then the outside operations in order.

$g(1) = -2f(1+1) + 3 = -2f(2) + 3$

$= -2(4) + 3$

$= -8 + 3 = -5$

Two common slips show up in the answer choices. One is evaluating $f$ at the original input and skipping the inside change: computing $f(1)$ instead of $f(2)$ above. The other is dropping the negative sign or the multiplier that sits outside $f$. Writing out each stage on its own line prevents both.

Compare three look-alikes with $f(x) = x^2$ at $x = 3$. $f(3) + 1 = 10$ adds outside the function; $f(3 + 1) = 16$ adds inside. They are different transformations with different answers. Likewise $f(2x)$ at $x = 3$ means $f(6)$, so you double the input before squaring. And $f(-x)$ at $x = -5$ means $f(5)$, so the negatives cancel before squaring.

Building an equation from a description

A transformation question can also run in reverse. Instead of transforming a given equation, you are told what happened to a graph and asked to build the resulting equation, or to match a description to the correct graph among several options.

Translate each phrase into the rule above, one at a time:

• "shifted left 4" becomes $x \to x + 4$ inside.
• "shifted down 1" becomes $-1$ at the end.
• "reflected over the $x$-axis" puts a negative on the whole function.
• "reflected over the $y$-axis" replaces $x$ with $-x$ inside.

Starting from $y = x^2$, shifting left 4, down 1, and reflecting over the $x$-axis gives $y = -(x+4)^2 - 1$. Note that the reflection negates the entire squared expression, and the shift down is applied after, so the $-1$ is not negated. Similarly, a graph of $y = |x|$ shifted right 3 and up 2 is $y = |x - 3| + 2$.

When both a shift and a reflection act on the input, the order matters, so read the description carefully. Reflecting $y = \sqrt{x}$ over the $y$-axis and then shifting right 2 gives $y = \sqrt{-(x-2)} = \sqrt{2 - x}$, whereas shifting right first and reflecting second would give a different function.

Piecewise functions

A piecewise function uses different rules for different parts of its domain. Evaluating a piecewise function at a specific input means first identifying which condition that input satisfies, then applying only the rule attached to that condition. Using the wrong piece is the most common error on this topic.

Evaluating a piecewise function

Worked example. A piecewise function is $p(x) = 2x + 1$ if $x < 2$, and $p(x) = x^2 - 3$ if $x \ge 2$. Find $p(2) + p(-1)$.

Check which condition each input satisfies before applying a rule.

$p(2)$: since $2 \ge 2$, use the second piece: $2^2 - 3 = 1$.

$p(-1)$: since $-1 < 2$, use the first piece: $2(-1) + 1 = -1$.

$p(2) + p(-1) = 1 + (-1) = 0$.

Boundary values deserve special care. Look at the inequality symbol: $\le$ and $\ge$ include the boundary, while $<$ and $>$ do not. With three or more pieces, test the input against each interval in order, like $-2 \le x < 3$. Each input gets exactly one rule.

Finally, a piecewise function can be asked to "solve $p(x) = k$ for a number $k$." Solve with each piece separately, and then keep only the solutions that satisfy that piece's own condition. A solution that lies outside the piece's interval must be rejected.

Where this appears as a model

Costs that change structure past a threshold are piecewise. A shipping cost of $5 for the first pound plus $2 for each additional pound is a piecewise function; a 4-pound package costs $5 + 2(3) = \$11$. Phone plans, taxi fares, and tax brackets all work the same way: decide which range your input is in, then use that range's rule only.

Strategy summary

• Inside the parentheses affects $x$ (horizontal, backwards). Outside affects the output (vertical).
• Do the inside substitution first, evaluate $f$, then apply what is outside.
• Translate each phrase of a description into one algebraic change, then assemble.
• For a piecewise function, test the condition first, check the boundary symbol, and use only that one rule.