Look only at additions/subtractions outside the function: g(x) = f(x) + k shifts the graph vertically by k units. If k > 0 the whole graph moves up k units; if k < 0 it moves down |k| units. Example: f(x) + 3 → up 3; f(x) − 2 → down 2. Quick reminders from the CED essentials: - Additive vertical: g(x) = f(x) + k → vertical translation by k (1.12.A.1). - Multiplicative vertical (g(x) = a·f(x)) changes height (vertical dilation); if a < 0 you also reflect across the x-axis (1.12.A.3). - Horizontal shifts (inside the input, f(x − h)) move left/right, not up/down (1.12.A.2). - Combined transforms stack: do horizontal dilations/shifts inside first, then vertical dilations/reflections, then vertical translations last to see the net up/down move (use algebra to rewrite if needed) (1.12.A.5). On the AP exam you’ll be asked to identify these quickly from expressions and graphs—practice problems in the Topic 1.12 study guide help (https://library.fiveable.me/ap-pre-calculus/unit-1/transformations-functions/study-guide/6S5lhzaXAYrpVwQz). For extra practice across Unit 1, use the unit page (https://library.fiveable.me/ap-pre-calculus/unit-1) and the 1000+ practice problems (https://library.fiveable.me/practice/ap-pre-calculus).

f(x) + k shifts the graph vertically by k: every y-value becomes y + k, so g(x) = f(x)+k is a vertical translation up k (if k>0) or down k (if k<0). This is an additive transformation that changes the range but not x-coordinates (CED 1.12.A.1). f(x + k) shifts the graph horizontally, but the direction is opposite the sign of k: g(x) = f(x + k) is a horizontal translation by −k units (so +k moves left, −k moves right). This is an additive transformation that changes the x-values (domain) but preserves y-values relative to the parent shape (CED 1.12.A.2). Quick tips: vertical changes (outside f) affect outputs/range; horizontal changes (inside f) affect inputs/domain and have that sign flip. Both can combine with dilations/reflections (CED 1.12.A.3–A.5). For a focused review, see the Topic 1.12 study guide (https://library.fiveable.me/ap-pre-calculus/unit-1/transformations-functions/study-guide/6S5lhzaXAYrpVwQz) and practice problems (https://library.fiveable.me/practice/ap-pre-calculus). On the AP exam, expect to describe these translations and how they change domain/range or produce combined transformations.

Think of the input as being “pre-shifted.” For g(x) = f(x + 2) you’re feeding f an input that’s 2 bigger than x. Put u = x + 2 so g(x) = f(u). Solve for x: x = u − 2. A point on f at (u, f(u)) becomes a point on g at (u − 2, f(u)). So every x-coordinate moves left 2 units. Quick example: if f(0) = 3, then g(−2) = f(−2 + 2) = f(0) = 3. The y-value that used to occur at x = 0 now occurs at x = −2 → left shift. This matches the CED rule (g(x) = f(x + h) gives a horizontal translation by −h) in 1.12.A.2. For more practice and visuals on horizontal vs vertical translations and combined transformations, check the Topic 1.12 study guide (https://library.fiveable.me/ap-pre-calculus/unit-1/transformations-functions/study-guide/6S5lhzaXAYrpVwQz) and try problems on Fiveable’s practice page (https://library.fiveable.me/practice/ap-pre-calculus).

Start with the parent f(x). Build g(x) by applying transformations in this order: inside (horizontal) then outside (vertical). Use these rules from the CED (Topic 1.12): - Horizontal: replace x by (x − h) to shift right h; by (x + h) to shift left h. Replacing x by b·x gives a horizontal compression by factor 1/b (if |b|>1) or stretch by factor 1/|b| (if 0<|b|<1). If b<0 you also reflect across the y-axis. - Vertical: multiply f by a to stretch (|a|>1) or compress (0<|a|<1) vertically; if a<0 reflect across the x-axis. Add k outside to shift up (k>0) or down (k<0). General formula: g(x) = a·f(b(x − h)) + k. Example: parent f(x)=x^2. If you want vertical stretch by 3, horizontal compression by 2, shift right 1 and up 4: g(x) = 3·f(2(x − 1)) + 4 = 3·[2(x − 1)]^2 + 4 = 12(x − 1)^2 + 4. Remember to check domain/range changes and note reflections when a or b are negative. For a quick AP-aligned refresher, see the Topic 1.12 study guide (https://library.fiveable.me/ap-pre-calculus/unit-1/transformations-functions/study-guide/6S5lhzaXAYrpVwQz). For extra practice problems, go to (https://library.fiveable.me/practice/ap-pre-calculus).

Use the sign that’s outside the function (multiply the output) to reflect across the x-axis, and the sign inside the input (replace x with −x) to reflect across the y-axis. - Reflection over the x-axis: g(x) = a·f(x) with a < 0. Typical shorthand: g(x) = −f(x). This flips all y-values: if f(2)=3, then −f(2)=−3. (CED 1.12.A.3) - Reflection over the y-axis: g(x) = f(bx) with b < 0. Typical shorthand: g(x) = f(−x). This flips the x-coordinates: the point (2,3) on f becomes (−2,3) on f(−x). (CED 1.12.A.4) Keep ordering in mind: horizontal changes happen inside the parentheses (affect x), vertical changes outside (affect y). So −f(−x) does both: reflect over y then x (equivalently a 180° rotation about the origin for most graphs). These transformation rules and practice problems are in the Topic 1.12 study guide (https://library.fiveable.me/ap-pre-calculus/unit-1/transformations-functions/study-guide/6S5lhzaXAYrpVwQz). For extra practice, try the AP Precalculus practice set (https://library.fiveable.me/practice/ap-pre-calculus).

Think of transformations as changes to the input x (inside f) and output y (outside f). A reliable step-by-step you can always use: 1. Write the transformed function in the standard form g(x) = a·f(b(x − h)) + k. Label: horizontal shift h (right if h>0), horizontal scale factor = 1/|b| (compression if |b|>1), vertical scale factor = |a| (stretch if |a|>1), vertical shift k (up if k>0). Signs: a<0 flips across x-axis; b<0 flips across y-axis. 2. To find where each parent point (x0,y0) goes, use these exact mapping formulas (safe and unambiguous): - New x = x0 / b + h (if b<0, that introduces the y-axis reflection because x0/b changes sign) - New y = a·y0 + k (if a<0, that includes the x-axis reflection) 3. If you prefer stepwise graphing, apply transformations in this order to points: - Apply horizontal operation that changes x (solve for x as above), which accounts for horizontal shift and dilation/reflection. - Apply f to get y (conceptual). - Apply vertical dilation/reflection (multiply by a). - Apply vertical translation (+k). 4. Always test with 3–5 key points (zeros, peaks, intercepts) and check domain/range changes. On the AP exam they expect you to identify additive (translations) and multiplicative (dilations/reflections) changes and show mapping or algebraic rewriting (CED 1.12.A). Need more practice? Use the Topic 1.12 study guide (https://library.fiveable.me/ap-pre-calculus/unit-1/transformations-functions/study-guide/6S5lhzaXAYrpVwQz) and extra practice problems (https://library.fiveable.me/practice/ap-pre-calculus).

Quick rule: a factor outside f(x) changes the graph vertically; a factor attached to x (inside) changes it horizontally. - Vertical dilation: g(x) = a·f(x). If |a| > 1 it’s a vertical stretch by factor a; if 0 < |a| < 1 it’s a vertical compression (shrink). If a < 0 you also get a reflection across the x-axis. (CED: 1.12.A.3) - Horizontal dilation: g(x) = f(bx). This changes x-scale. If |b| > 1 the graph compresses horizontally (gets narrower); if 0 < |b| < 1 the graph stretches horizontally (gets wider). Equivalently, f(bx) = f(x/(1/b)), so a horizontal “stretch by factor k” is written f(x/k). If b < 0 you also get a reflection across the y-axis. (CED: 1.12.A.4) Spot them fast: coefficient outside = vertical; coefficient inside (with x) = horizontal. Be ready to combine transformations—order matters when interpreting formulas on the AP (free-response problems use these skills). For extra practice and examples see the Topic 1.12 study guide (https://library.fiveable.me/ap-pre-calculus/unit-1/transformations-functions/study-guide/6S5lhzaXAYrpVwQz) and lots of practice problems (https://library.fiveable.me/practice/ap-pre-calculus).

Think of g(x) = f(2x) as “you’re feeding f a faster input.” For g to give the same output f gives at x = a, you need 2x = a, so x = a/2. That means every feature (peak, zero, etc.) of f that sits at x = a moves to x = a/2 in g—all x-values are halved. Halving x-values makes the graph narrower (a horizontal compression by factor 1/2). Quick numeric check: f(x) = x^2. Then g(x) = f(2x) = (2x)^2 = 4x^2. The parabola is “skinnier”: the point that was at x = 1 for f is at x = 0.5 for g. If b > 1 in g(x) = f(bx), the graph compresses horizontally by 1/b; if 0 < b < 1 it stretches (gets wider). This is the horizontal dilation idea in the CED (1.12.A.4). For more practice and AP-style problems on transformations, check the Topic 1.12 study guide (https://library.fiveable.me/ap-pre-calculus/unit-1/transformations-functions/study-guide/6S5lhzaXAYrpVwQz) and the AP Precalculus practice bank (https://library.fiveable.me/practice/ap-pre-calculus).

Short answer: it depends on the type of transformation. - Vertical translations g(x)=f(x)+k: domain unchanged; range is shifted by +k (every y-value moves up/down by k). - Horizontal translations g(x)=f(x−h): range unchanged; domain shifts by +h (every x where f is defined is moved right by h). - Vertical dilations/reflections g(x)=a·f(x) (a≠0): domain unchanged; range is multiplied by a. If a<0 you also reflect across the x-axis, so signs of range values flip. - Horizontal dilations/reflections g(x)=f(bx) (b≠0): range unchanged; domain is scaled by 1/b (points x in the parent become x/b in the new graph). If b<0 there’s also a reflection across the y-axis. - Combinations: apply these rules in order (horizontal shifts/stretches affect domain; vertical ones affect range). Always check for new restrictions (e.g., vertical shifts can create/remove values like y=0 for sqrt or reciprocal parent functions). On the AP exam they assume “domain = real x for which f(x) is real,” so always test transformed inputs/outputs for invalid values (holes, asymptotes). For a quick refresher see the Topic 1.12 study guide (https://library.fiveable.me/ap-pre-calculus/unit-1/transformations-functions/study-guide/6S5lhzaXAYrpVwQz) and try practice problems (https://library.fiveable.me/practice/ap-pre-calculus).

Start with the parent parabola f(x)=x^2. A reflection across the x-axis is a vertical multiplicative transformation by a = −1, and a shift 3 units right is the horizontal additive transformation x → x−3. Combine them: g(x) = −f(x−3) = −(x−3)^2. Why that works (CED terms): g(x)=a·f(x−h) with a=−1 (vertical dilation + reflection over x-axis, EK 1.12.A.3) and h=3 (horizontal translation right by 3, EK 1.12.A.2). The domain stays all real numbers; the range changes from [0, ∞) to (−∞, 0]. This kind of construction is exactly the skill tested in Topic 1.12—practice making combined additive/multiplicative transformations (see the Topic 1.12 study guide for more examples: (https://library.fiveable.me/ap-pre-calculus/unit-1/transformations-functions/study-guide/6S5lhzaXAYrpVwQz)). For extra practice, check the unit practice set (https://library.fiveable.me/practice/ap-pre-calculus).

Do them in this order: horizontal shift, vertical dilation + reflection, then vertical shift. Specifically: 1. Start with y = f(x). 2. Shift right 1 (replace x with x − 1) → this moves the graph horizontally by +1 (horizontal translation). 3. Apply −2 to the output: multiply y-values by 2 (vertical dilation by factor 2) and because it’s negative, reflect across the x-axis. 4. Finally add 3: translate the whole graph up 3 units (vertical translation). So graphically: f(x) → f(x−1) (right 1) → −2·f(x−1) (vertical stretch 2 + reflect over x-axis) → −2f(x−1)+3 (up 3). Note domain/range can change after these, and this is exactly the kind of combined additive/multiplicative transformation the CED covers (1.12.A.5). For a quick review, check the Topic 1.12 study guide (https://library.fiveable.me/ap-pre-calculus/unit-1/transformations-functions/study-guide/6S5lhzaXAYrpVwQz). For extra practice problems, see the unit practice set (https://library.fiveable.me/practice/ap-pre-calculus).

Use this combined form: g(x) = a · f( b(x − h) ) + k What each piece does (AP CED terms): - +k (outside) = vertical translation up k (or down if k < 0). - (x − h) inside = horizontal translation right h (left if h < 0). - a (outside, a ≠ 0) = vertical dilation by factor a; if a < 0 there’s also a reflection across the x-axis. - b (inside, b ≠ 0) = horizontal dilation/compression: it scales the x-input. Practically, if |b|>1 the graph compresses horizontally (closer to the y-axis); if 0<|b|<1 it stretches horizontally. If b<0 there’s also a reflection across the y-axis. Order note (why form matters): inside changes (horizontal shift & horizontal scaling) act on x first; the outside a and +k then scale/shift vertically. This matches the CED essentials on additive and multiplicative transformations (Topic 1.12). For a quick refresher, see the Topic 1.12 study guide (https://library.fiveable.me/ap-pre-calculus/unit-1/transformations-functions/study-guide/6S5lhzaXAYrpVwQz) and drill practice at (https://library.fiveable.me/practice/ap-pre-calculus).

Because a horizontal dilation rescales x-values, you look at what input x in g(x) = f(bx) gives the same output as f at some original input. If f(u) has a point at (u, f(u)), then g(x) = f(bx) equals that same output when bx = u, so x = u / b. Every original x-value u is moved to u/b. That means distances along the x-axis are multiplied by 1/b: points get closer when b>1 (a horizontal compression) and farther when 0https://library.fiveable.me/ap-pre-calculus/unit-1/transformations-functions/study-guide/6S5lhzaXAYrpVwQz). For extra practice problems, try the AP Precalculus practice set (https://library.fiveable.me/practice/ap-pre-calculus).

Think of transformations in the order the graph actually changes, not the order they appear in the formula. A safe, consistent rule: 1. Work the inside of f( ): do horizontal translation first (write bx + c as b(x − h) so you see h), then apply the horizontal dilation/reflection from b (f(bx) compresses by 1/b; b < 0 also reflects across the y-axis). 2. Next apply vertical changes: do vertical dilation/reflection (multiply the output by a; a < 0 reflects across the x-axis), then the vertical translation (add k). So for g(x) = a·f(b(x − h)) + k: start with f(x), shift right h, compress/stretch by factor 1/|b| (and reflect if b<0), stretch/compress vertically by |a| (and reflect if a<0), then shift vertically by k. Rewrite inside (factor bx + c) to spot h, and practice with examples—this is exactly Topic 1.12 in the CED (translations, dilations, reflections). For more worked examples, check the Topic 1.12 study guide (https://library.fiveable.me/ap-pre-calculus/unit-1/transformations-functions/study-guide/6S5lhzaXAYrpVwQz) and try practice problems (https://library.fiveable.me/practice/ap-pre-calculus).

Short version: if the change is inside the function’s parentheses, it affects the input (x) → horizontal changes; if it’s outside, it affects the output (f(x)) → vertical changes. Quick rules you can memorize (CED terms): - g(x) = f(x) + k → additive, vertical translation up k (range shifts). - g(x) = f(x − h) → additive inside, horizontal translation right h (shifts domain). - g(x) = a·f(x) (a ≠ 0) → multiplicative, vertical dilation by factor a; a < 0 also reflects across x-axis. - g(x) = f(bx) (b ≠ 0) → multiplicative inside, horizontal dilation by factor 1/b; b < 0 also reflects across y-axis. Examples: f(x)+3 moves whole graph up 3 (output changes). f(x+3) moves it left 3 (input changed). 2·f(x) is a vertical stretch; f(2x) is a horizontal compression. Order tip: do horizontal shifts/stretches before vertical ones when writing function forms. Practice recognizing these on the exam—Topic 1.12 (transformations) appears on multiple-choice and FRQs. For a quick refresher and practice, see the Topic 1.12 study guide (https://library.fiveable.me/ap-pre-calculus/unit-1/transformations-functions/study-guide/6S5lhzaXAYrpVwQz) and lots of practice problems (https://library.fiveable.me/practice/ap-pre-calculus).