Functions: Analysis and Transformation

Why Functions Are the Language of Precalculus

Every topic in this course — polynomials, exponentials, trigonometry, polar coordinates — is a function. Not a formula to memorize, but a relationship between quantities that follows a rule. Before we study any of those specific families, we need to sharpen the tools that apply to all of them: identifying shape, predicting behavior from an equation, and describing how one function relates to another through transformations.

If Algebra II was about learning to solve equations in each family, Precalculus is about seeing the families as a unified landscape. Transformations are the map.

The Parent Function Gallery

A parent function is the simplest member of a function family — the version with no shifts, stretches, or reflections. Every other function in the family is a transformation of the parent.

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Every function you encounter in Precalculus descends from one of these parents. The first analytical question is always: which parent am I looking at?

Vertical and Horizontal Shifts

A vertical shift moves the graph up or down. A horizontal shift moves it left or right.

For a parent function

$$f(x)$$

:

• $$f(x)+k$$

  shifts up by

  $$k$$

  units (down if

  $$k<0$$

  )

• $$f(x-h)$$

  shifts right by

  $$h$$

  units (left if

  $$h<0$$

  )

The horizontal shift is counterintuitive:

$$f(x-3)$$

shifts right, not left. Think of it this way — the function now needs

$$x=3$$

to produce what the parent produced at

$$x=0$$

.

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Reflections

A reflection flips the graph across an axis.

• $$-f(x)$$

  reflects across the x-axis (all y-values change sign)

• $$f(-x)$$

  reflects across the y-axis (all x-values change sign)

For

$$f(x)=\sqrt{x}$$

:

• $$-\sqrt{x}$$

  opens downward instead of upward

• $$\sqrt{-x}$$

  extends to the left instead of the right

❓

Concept Check

Start with f(x) = x^3. Describe the graph of g(x) = -(x - 1)^3 + 2 by identifying each transformation applied to the parent, in order.

▸

Starting from f(x) = x^3: (1) horizontal shift right 1 unit (x - 1), (2) reflection across the x-axis (the negative sign), (3) vertical shift up 2 units (+ 2). The inflection point moves from (0, 0) to (1, 2), and the S-curve is flipped upside down.

Stretches and Compressions

A vertical stretch or compression multiplies all y-values by a constant

$$a$$

:

• $$a\cdot f(x)$$

  with

  $$|a|>1$$

  : vertical stretch (graph gets taller)

• $$a\cdot f(x)$$

  with

  $$0<|a|<1$$

  : vertical compression (graph gets shorter)

A horizontal stretch or compression replaces

$$x$$

with

$$bx$$

:

• $$f(bx)$$

  with

  $$|b|>1$$

  : horizontal compression (graph gets narrower)

• $$f(bx)$$

  with

  $$0<|b|<1$$

  : horizontal stretch (graph gets wider)

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The General Transformation Formula

All transformations combine into one template:

$$g(x)=a\cdot f\left(b(x-h)\right)+k$$

Reading a transformed equation means identifying these four parameters and describing each effect. Writing a transformed equation means choosing parameters to match a described or graphed function.

Piecewise Functions

A piecewise function uses different rules on different intervals of the domain. The function definition specifies which rule applies where.

$$f(x)=\{\begin{array}{ll}x+3& \text{if }x<1\\ 5& \text{if }1\le x\le 4\\ 2x-3& \text{if }x>4\end{array}$$

To evaluate a piecewise function, first determine which interval the input falls into, then apply the corresponding rule.

• $$f(-2)=(-2)+3=1$$

  (using the first rule, since

  $$-2<1$$

  )

• $$f(3)=5$$

  (using the second rule, since

  $$1\le 3\le 4$$

  )

• $$f(6)=2(6)-3=9$$

  (using the third rule, since

  $$6>4$$

  )

Piecewise functions appear everywhere in applied mathematics: tax brackets, shipping rates, speed limits that change along a highway. The mathematical challenge is reading the conditions carefully and checking boundaries.

❓

Concept Check

A parking garage charges $5 for the first hour, $3 for each additional hour up to 5 hours total, and a flat $20 for any stay longer than 5 hours. Write a piecewise function C(t) for cost as a function of time t (in hours), where t is greater than 0.

▸

C(t) = 5 if 0 is less than t ≤ 1; C(t) = 5 + 3(t - 1) = 3t + 2 if 1 is less than t ≤ 5; C(t) = 20 if t is greater than 5. Note the boundary values: at t = 1, the cost is $5; at t = 5, the cost is 3(5) + 2 = $17, and any time beyond 5 hours costs a flat $20.

Domain and Range Revisited

In 8th grade, domain and range were lists of values from a table. In Precalculus, they are intervals.

Transformations shift the domain and range predictably. If the parent square root has domain

$$[0,\infty )$$

, then

$$g(x)=\sqrt{x-3}+1$$

has domain

$$[3,\infty )$$

and range

$$[1,\infty )$$

. The shift right 3 moves the domain start from 0 to 3; the shift up 1 moves the range start from 0 to 1.

Building Toward Calculus

Every topic in Precalculus will ask you to do what this unit has practiced: recognize a function's family, read its transformations, identify its domain and range, and describe its behavior. The polynomial functions in the next unit are more complex than quadratics, but the analytical approach is the same — start with the parent, track the transformations, and let the structure guide your understanding.