6.4 Transformations and Interpreting Functions in Context

6.4 Transformations and Interpreting Functions in Context

The last Table 11 Functions skill is interpreting functions in a context — reading f(3) = 18 as “after 3 hours the tank holds 18 gallons,” not as a floating algebra trick. Higher Skills Insight bands reward fluency using function notation to describe transformations: writing f(x − 4) + 2 for a shift, rather than only sliding a sketch — the 276–300 absolute-value item is exactly that, in function notation. This section ties those together, with piecewise rules at the level AAF actually uses.

/practice/accuplacer-advanced-algebraPractice questions with detailed explanations

The five transformations you must name

Start with a parent graph y = f(x). New graphs are written by composing simple changes with f.

| Notation | Geometric effect | Example with f(x) = |x| | |---|---|---| | f(x) + k | Vertical shift up k (down if k < 0) | |x| + 3 moves the V up 3 | | f(x − h) | Horizontal shift right h (left if h < 0) | |x − 4| moves the V right 4 | | a f(x) | Vertical stretch if |a| > 1; vertical shrink if 0 < |a| < 1; x-axis reflection if a < 0 | 3|x| is steeper; −|x| opens down | | f(−x) | Reflection across the y-axis | |−x| = |x| (this parent is already symmetric) | | −f(x) | Reflection across the x-axis | −|x| is an upside-down V |

Two facts cause most errors.

Horizontal substitutions move the opposite direction from the sign you see. f(x − 3) is a shift right 3, because x − 3 = 0 when x = 3: the old input 0 now happens at 3. f(x + 3) = f(x − (−3)) is a shift left 3. Vertical changes + k move the way the sign looks: +2 is up, −2 is down.

Inside versus outside. Anything outside f, such as 2f(x) or f(x)+5, changes outputs (y). Anything inside the argument, such as f(2x) or f(x−5), changes inputs (x). f(2x) is a horizontal shrink by 2, not a vertical stretch. AAF is more likely to test f(x−h) and af(x) than the f(2x) shrink, but the inside/outside split still saves you.

Combined: g(x) = 2f(x − 3) + 1 is right 3, vertical stretch 2, up 1, in that mental order: horizontal first (inside), then stretch, then vertical shift (outside). If f is |x|, the vertex moves from (0, 0) to (3, 1) and the arms get steeper. If f is x^2, the vertex of 2(x − 3)^2 + 1 is likewise (3, 1) — the same transformation language as vertex form.

f(−x) versus −f(x): for f(x) = x^3 + 1, f(−x) = −x^3 + 1 (y-axis reflection) and −f(x) = −x^3 − 1 (x-axis reflection). They are not the same function. Describing “flip over the x-axis” in notation is −f(x), not f(−x). That sentence is the higher-band transformation fluency item.

A vertical shrink example: (1/2)f(x) halves every output. If f(2) = 10, then (1/2)f(2) = 5. A horizontal counterpart f(x/2) would wait until the input is twice as large to produce the old output; do not mix those.

Worked substitution: g(x) = 3(x + 8)

Context items often hide a transformation inside a formula that looks too simple. Let g(x) = 3(x + 8). This is a linear function, equivalently g(x) = 3x + 24. To evaluate, substitute the entire input into (x + 8) first.

g(4) = 3(4 + 8) = 3(12) = 36.

Traps:

• 3(4) + 8 = 20 treats the 8 as added after multiplying, as if the rule were 3x + 8.
• 3(4) alone is 12 — dropping the inner +8 entirely.
• 3(12) + 8 = 44 substitutes correctly, then adds 8 again as if g were 3(x+8)+8.
• Distribution 3(4) + 3(8) = 12 + 24 = 36 is valid, but only if you distribute to both terms inside.

If a story says a lab records temperature g(t) = 3(t + 8) degrees after t minutes past dawn, then at t = 4 the temperature is 36, not 20. The +8 is inside the input to the scale factor 3: eight minutes of warm-up are built into the clock. Interpreting in context means attaching units and the correct substitution, not inventing a new formula.

Another original context: a parking garage charges C(h) = 2(h − 1) + 5 dollars for h hours after the first hour is bundled into a $5 gate fee. C(4) = 2(3) + 5 = 11. Writing 2(4 − 1) + 5 is the function; writing 2(4) − 1 + 5 = 12 drops the parentheses and misreads the story. C(1) = 5, the gate fee with no extra hours billed.

Piecewise functions at AAF level

A piecewise function uses different formulas on different parts of the domain. AAF pieces are short: two linear pieces, or a linear piece glued to a constant, or the definition of absolute value.

Let p(x) = x + 2 when x < 0, and p(x) = 2x when x ≥ 0.

Then p(−3) = −3 + 2 = −1 (left piece) and p(4) = 8 (right piece). p(0) = 0 because 0 uses x ≥ 0. The graph is two rays that meet at the origin in this particular example; they need not meet. If the pieces disagree at a boundary, the inequality (< versus ≤) tells you which value is defined. AAF will not usually ask you to debate a single missing point, but it will ask you to pick the correct piece.

Absolute value is piecewise: |x| = −x for x < 0 and x for x ≥ 0. Transformations of |x| are still piecewise after you shift: |x − 4| + 1 uses 5 − x to the left of 4 and x − 3 to the right of 4. The V-graph in Graphs of Functions, Vertical Line Test, Maxima and Minima is this piecewise rule drawn.

For a shipping-style context: a mailer charges $4 for weights up to 2 lb and $4 + 1.5(w − 2) for w > 2. That is piecewise. S(2) = 4 and S(6) = 4 + 1.5(4) = 10. Do not apply the second piece at w = 2 if the story says “over 2 lb.” Do not apply 1.5w to the whole weight; the 1.5 multiplies only the extra pounds.

Interpreting f in a sentence

When h(t) is height in meters after t seconds:

That table is the same f(x+2) versus f(x)+2 distinction from Section 6.1, now with units. If a water tank has volume V(t) = 40 − 3t gallons after t minutes of draining, V(5) = 25 gallons remain, V(t) = 10 solves 40 − 3t = 10 so t = 10 minutes, and the tank is empty when V(t) = 0, t = 40/3 minutes. Domain in context is not all reals: t ≥ 0 and V(t) ≥ 0, so 0 ≤ t ≤ 40/3. Context restricts the algebraic domain.

Average rate of change on [1, 5] is (V(5) − V(1))/(5 − 1) = (25 − 37)/4 = −3 gallons per minute, which matches the slope of this linear model. AAF may not use the phrase “average rate of change,” but it will ask how fast a quantity changes between two inputs, or what V(8) means in gallons, not as a bare 16.

If a quadratic model h(t) = −5t^2 + 20t + 2 gives height of a thrown object, the vertex time t = −b/(2a) = 2 seconds is when height is maximum: h(2) = −20 + 40 + 2 = 22 meters. Interpreting that 22 as “the object travels 22 meters horizontally” is the wrong quantity. Function notation names one output of one rule; read the units on h.

After you can shift, reflect, substitute g(4) = 3(4+8), and read h(3) = 18 in words, move to mixed CAT practice. The Functions slice is the heaviest Table 10 weight on AAF — typically up to four of the twenty items — and it reuses the vertex skill you already built for quadratics.