Transformations of graphs
The graph of y = (x − 3)² + 1 is the graph of y = x² moved 3 to the right and 1 up. The vertex goes from (0, 0) to (3, 1).
Guess first: does x − 3 move the curve left or right?
Start in the lab before the rules. It opens on g(x) = (x − 3)² + 1, drawn over the dashed parent y = x². Before you look closely, guess: does the x − 3 push the curve left or right? Now find the red key point. It sits at (3, 1). Right, then. Set k to −2 and the whole curve drops. Set a to −2 and it flips and gets narrower. Hold on to the first surprise, a minus sign that moved the graph to the right, because the rest of the unit explains it.
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Four parent graphs to know by shape
A handful of basic graphs are worth knowing by shape: y = x², the parabola; y = |x|, a V with its corner at the origin; y = √x, half of a sideways parabola that starts at the origin; and y = x³, which rises through the origin with a flat moment there. OpenStax calls these toolkit functions. Most graphs you will meet are one of them moved, flipped or stretched, and four rules describe all of those changes. If you know the parent and the rules, you can sketch the new graph in a few seconds without a table.
Add 20 outside and the roof vent graph rises by 20
Adding a constant outside the function moves the graph up or down. If g(x) = f(x) + k, every output grows by k, so the graph shifts up k units when k > 0 and down when k < 0. OpenStax gives an example from a green building whose roof vents open and close during the day: if V(t) is the area of open vents, in square feet, t hours after midnight, and the manager opens 20 more square feet all day and night, the new function is S(t) = V(t) + 20, the same graph lifted by 20. Only the outputs change; the inputs stay the same.
Why x − 3 moves the graph right, not left
Changing the input moves the graph sideways, and here the direction surprises people. The graph of g(x) = f(x − h) is the graph of f moved h units to the right. To see why, ask where g gets the output that f had at 0: g(x) = f(0) when x − h = 0, that is, at x = h. Everything happens h units later. So (x − 3)² is shifted right 3, and |x + 2| = |x − (−2)| is shifted left 2. A minus sign inside the brackets means right, a plus sign means left.
A minus sign outside flips up and down; inside, left and right
A minus sign in front, g(x) = −f(x), flips every output, so the graph is reflected over the x-axis: −√x hangs below the axis instead of rising above it. A minus sign on the input, g(x) = f(−x), reflects the graph over the y-axis: √(−x) starts at the origin and goes to the left. Multiplying the outputs by a positive number a stretches the graph vertically when a > 1 and compresses it when 0 < a < 1, so 2|x| is a steeper V and (1/2)x² a wider parabola.
Follow one point and the rest of the graph comes with it
For g(x) = a f(x − h) + k, a reliable way to sketch is to follow a few points of the parent. A point (x, y) on f goes to (x + h, a·y + k) on g. The point (4, 2) is on y = √x, so on y = √(x − 1) + 3 it becomes (5, 5). The corner (0, 0) of |x| on y = 2|x − 3| + 1 becomes (3, 1). Go back to the lab at the top, pick a parent, move a, h and k, and watch the red key point; the panel lists the moves in order and shows where two other points land.
- How does the graph of g(x) = x² + 3 compare with y = x²?
- Up 3
- Down 3
- Right 3
- Left 3
- How does g(x) = (x − 4)² compare with y = x²?
- Right 4
- Left 4
- Up 4
- Down 4
- How does g(x) = |x + 2| compare with y = |x|?
- Left 2
- Right 2
- Up 2
- Down 2
- What does the graph of g(x) = −√x look like compared with y = √x?
- Reflected over the x-axis
- Reflected over the y-axis
- Moved down 1
- Stretched by 2
- What is the vertex of g(x) = (x + 1)² − 5?
- (−1, −5)
- (1, −5)
- (−1, 5)
- (1, 5)
- Where does the corner of y = 2|x − 3| + 1 sit?
- (3, 1)
- (−3, 1)
- (3, 2)
- (1, 3)
- How does y = (1/2)x² compare with y = x²?
- Compressed vertically (wider)
- Stretched vertically (narrower)
- Moved down 1/2
- Reflected
- (4, 2) is on y = √x. Which point is on y = √(x − 1) + 3?
- (5, 5)
- (3, 5)
- (5, −1)
- (4, 5)
- Which change does g(x) = f(−x) make?
- Reflection over the y-axis
- Reflection over the x-axis
- Shift left 1
- No change
- g(x) is y = x³ moved down 2. What is g(2)?
- Reading y = (x + 2)² as "move right 2". x + 2 = x − (−2), so h = −2 and the graph moves left 2. The vertex is at (−2, 0). Check: (−2 + 2)² = 0.
- Treating f(x + 2) and f(x) + 2 as the same graph. For f(x) = x², f(x + 2) = (x + 2)² moves left 2, while f(x) + 2 = x² + 2 moves up 2. At x = 1 they give 9 and 3.
- Sending (2, 4) on y = x² to (5, 3) on y = (x + 3)² − 1. Here h = −3, so x moves left: (2 − 3, 4 − 1) = (−1, 3). Check: (−1 + 3)² − 1 = 3.
- f(x) + k: up k (down if k < 0).
- f(x − h): right h; f(x + h): left h.
- −f(x): reflect over the x-axis; f(−x): over the y-axis.
- a·f(x): stretch if a > 1, compress if 0 < a < 1. A point (x, y) goes to (x + h, a·y + k).
Sources for this course (8)
- Curriculum: 교육부 고시 제2022-33호 [별책 8] 수학과 교육과정 (ncic.re.kr/inv/org/download.do?year=2022&seq=10003559&orgType=ogi4, PDF read 2026-10-19: 중학교 1~3학년군 [9수02-06] 그래프 해석(증가와 감소), [9수02-14] 함수의 개념·함숫값, 용어 함수·함숫값·기울기·평행이동, 고려 사항 「한 양이 변함에 따라 다른 양이 하나씩 정해지는 두 양 사이의 대응 관계를 이용하여 도입」; 고등학교 공통수학2 [10공수2-01-06] 평행이동, [10공수2-01-07] 원점·x축·y축·직선 y = x에 대한 대칭이동, [10공수2-03-01] 함수의 개념(두 집합 사이의 대응 관계), [10공수2-03-02] 합성함수, [10공수2-03-03] 역함수, 용어 정의역·치역·공역·대응·일대일대응·일대일함수·합성함수·역함수, 기호 f: X → Y, g ∘ f, (g ∘ f)(x), y = g(f(x)), f⁻¹, y = f⁻¹(x), 교수·학습 용어 조건제시법; 미적분Ⅰ 용어 구간·닫힌구간·열린구간 [a, b], (a, b), 평균변화율, [12미적Ⅰ-02-07] 증가와 감소, 극댓값·극솟값)
- Textbook: OpenStax, College Algebra 2e, 3.1 Functions and Function Notation (relation, domain and range of ordered pairs, {(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)}, the odd/even relation that is not a function, coffee menu prices, vertical line test, horizontal line test, one-to-one: openstax.org/books/college-algebra-2e/pages/3-1-functions-and-function-notation)
- Textbook: OpenStax, College Algebra 2e, 3.2 Domain and Range (no even root of a negative number, no division by 0, interval notation with brackets and parentheses, (0, 100] for "more than 0 and at most $100", {(2, 10), (3, 10), (4, 20), (5, 30), (6, 40)}: .../pages/3-2-domain-and-range)
- Textbook: OpenStax, College Algebra 2e, 3.3 Rates of Change and Behavior of Graphs (average rate of change, increasing and decreasing intervals, local maxima and minima: .../pages/3-3-rates-of-change-and-behavior-of-graphs)
- Textbook: OpenStax, College Algebra 2e, 3.4 Composition of Functions (f(g(x)); f(x) = 2x + 1 and g(x) = 3 − x give f(g(x)) = 7 − 2x and g(f(x)) = −2x + 2, so composition is not commutative; c(s) calories and s(t) sit-ups; f(x) = 1/x and g(x) = x − 3: .../pages/3-4-composition-of-functions)
- Textbook: OpenStax, College Algebra 2e, 3.5 Transformation of Functions (g(x) = f(x) + k vertical shift, f(x − h) horizontal shift, −f(x) and f(−x) reflections, af(x) stretches and compressions, roof vent example S(t) = V(t) + 20 (open vent area in square feet plus 20): .../pages/3-5-transformation-of-functions)
- Textbook: OpenStax, College Algebra 2e, 3.7 Inverse Functions (f⁻¹ is not an exponent and not 1/f, C = (5/9)(F − 32), 75 °F is about 24 °C, 26 °C is about 79 °F, x² restricted to [0, ∞), reflection about y = x: .../pages/3-7-inverse-functions)
- History block only: MacTutor, University of St Andrews, The function concept (Leibniz, August 1673, the word "function"; Johann Bernoulli's letter of 2 September 1694; Euler, Introductio in analysin infinitorum 1748 and Institutiones calculi differentialis 1755; Dirichlet 1837; Goursat 1923 "y = f(x)": mathshistory.st-andrews.ac.uk/HistTopics/Functions)
Checked: 2026-10-18