The parabola
Square the numbers from −3 to 3 and you get 9, 4, 1, 0, 1, 4, 9. Plotted, they trace a U shape called a parabola.
Set a to 2, then 1/2, then −1
Start with the lab, not the definitions. It shows y = x² next to a dashed copy that stays put. Set a to 2. The curve gets narrower, and at x = 2 it already reaches 8. Now try 1/2. Then −1. One of them spreads the cup out; the other turns it upside down. Keep those three pictures in mind, because everything further down explains them.
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Why −2 and 2 land at the same height
Lines come from equations in which x appears only to the first power. Put x² into the equation and the graph bends. The simplest case is y = x². Make a table: x = −3 gives 9, x = −2 gives 4, x = −1 gives 1, x = 0 gives 0, and then the same outputs come back in reverse for x = 1, 2 and 3. Two different inputs, such as −2 and 2, share one output, because a negative number squared is positive. Join the points with a smooth curve and you get a U that is lowest at the origin and climbs faster and faster on both sides.
| x | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y = x² | 9 | 4 | 1 | 0 | 1 | 4 | 9 |
A function that can be written as f(x) = ax² + bx + c, with a not equal to 0, is a quadratic function, and OpenStax calls this the general form. Its graph is always a parabola. If a were 0, the x² term would vanish and you would be back to a line, which is why the condition matters. The numbers b and c may be zero: y = x², y = 3x² − 5 and y = −x² + 4x are all quadratic functions. Setting x = 0 leaves only c, so c is the y-intercept, the place where the parabola crosses the y-axis.
Fold the curve along x = 1 and the halves match
Every parabola has one turning point, the vertex. When the parabola opens upward the vertex is its lowest point, and its y-coordinate is the minimum value of the function. When it opens downward the vertex is the highest point and gives the maximum value. Through the vertex runs a vertical line called the axis of symmetry. Fold the graph along that line and the two halves land on each other. This saves work. Once you know one point on the curve, you get a second point for free by reflecting it across the axis. A basketball shot seen from the side draws close to the same arc, and the top of that arc is its vertex.
Why a = 2 is narrow and a = −1 is upside down
The sign of a decides the direction. If a > 0 the parabola opens up, like a cup; if a < 0 it opens down, like an upside-down cup. The size of a decides the width. With y = 2x², every output is twice that of y = x², so the curve climbs twice as fast and looks narrower: at x = 2 it is already at 8. With y = (1/2)x², the outputs are halved and the curve is wider: at x = 2 it is only at 2. A larger |a| means a narrower parabola, and a smaller |a| a wider one. That is what the lab at the top showed. With a = 2 the curve hugged the y-axis, with 1/2 it spread out, and −1 flipped the cup.
- For f(x) = x², what is f(−4)?
- For y = 3x², what is y when x = −2?
- Which parabola opens downward?
- y = −2x² + 1
- y = 2x² − 1
- y = x² − 9
- y = (1/2)x²
- Which parabola is the narrowest?
- y = 3x²
- y = x²
- y = (1/2)x²
- y = −(1/3)x²
- What is the y-intercept of y = x² − 4x + 7? Give the y-value.
- A parabola has axis of symmetry x = 3 and passes through (1, 5). Which other point must be on it?
- (5, 5)
- (1, −5)
- (3, 5)
- (−1, 5)
- What is the vertex of y = x² + 2?
- (0, 2)
- (2, 0)
- (0, −2)
- (−2, 0)
- Which of these is a quadratic function?
- y = 5 − x²
- y = 4x − 3
- y = 7
- y = 2/x
- What is the maximum value of y = −x² + 4?
- On y = x² − 1, what is y at x = −2? (It matches the value at x = 2.)
| Parabola | The U-shaped graph of a quadratic function |
| Vertex | The turning point: lowest if a > 0, highest if a < 0 |
| Axis of symmetry | The vertical line through the vertex |
| y-intercept of ax² + bx + c | (0, c) |
- Writing −3² = −9 for the point at x = −3 on y = x². −3² means −(3²). The input is the whole number −3, so you need (−3)² = 9, and the point is (−3, 9). Put brackets around a negative input before squaring it.
- Calling y = −2x² wider than y = x² because −2 is smaller than 1. Width depends on |a|, not on a. |−2| = 2 is bigger than 1, so y = −2x² is the narrower one. The minus sign only turns it upside down.
- Reading the y-intercept of y = (x − 1)² − 2 as −2. c is the y-intercept only when the function is written as ax² + bx + c. Here, put x = 0 instead: (0 − 1)² − 2 = 1 − 2 = −1, so the curve crosses the y-axis at (0, −1), as in the figure.
- f(x) = ax² + bx + c with a ≠ 0 is a quadratic function; its graph is a parabola.
- a > 0 opens up, a < 0 opens down; larger |a| is narrower.
- The vertex is the lowest or highest point; the axis of symmetry passes through it.
- The y-intercept is (0, c).
Sources for this course (6)
- 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 [9수02-19] 다항식의 곱셈과 인수분해(해설의 곱셈 공식 m(a + b), (a + b)², (a − b)², (a + b)(a − b), (x + a)(x + b), (ax + b)(cx + d)), [9수02-20] 이차방정식, [9수02-21] 이차함수의 개념, [9수02-22] 이차함수의 그래프(최댓값·최솟값은 x의 범위가 실수 전체인 경우만), 용어 인수·인수분해·완전제곱식·이차방정식·중근·근의 공식·이차함수·포물선·축·꼭짓점·최댓값·최솟값·평행이동, 「이차방정식은 해가 실수인 경우만」, 「근과 계수와의 관계는 다루지 않는다」, 「구한 해가 문제 상황에 적합한지 확인」; 고등학교 공통수학1 [10공수1-02-02] 실근과 허근·판별식, [10공수1-02-03] 근과 계수의 관계, [10공수1-02-06] 이차함수의 최대·최소와 실생활(제한된 범위, 공학 도구))
- Textbook: OpenStax, College Algebra 2e , 5.1 Quadratic Functions (parabola, vertex, axis of symmetry x = −b/(2a), general form ax² + bx + c, standard form a(x − h)² + k also called vertex form, f(x) = 2x² + 4x − 4 = 2(x + 1)² − 6, f(x) = −3(x + 2)² + 4, vertex of 2x² − 6x + 7 by −b/(2a) (the completing-the-square example x² − 6x + 7 in the course is our own), satellite dish cross-sections, 80 ft fence: maximum area 800 sq ft at L = 20, newspaper revenue maximum at $31.80 and $2,528,100, ball H(t) = −16t² + 80t + 40: top at 2.5 s, 140 ft, ground after about 5.458 s, rock H(t) = −16t² + 96t + 112: openstax.org/books/college-algebra-2e/pages/5-1-quadratic-functions)
- Textbook: OpenStax, Algebra and Trigonometry 2e, 2.5 Quadratic Equations (zero-product property, (x − 2)(x + 3) = x² + x − 6, square root property, completing the square, quadratic formula, discriminant table: openstax.org/books/algebra-and-trigonometry-2e/pages/2-5-quadratic-equations)
- Reference value: NIST CODATA standard acceleration of gravity 9.80665 m/s², about 32.174 ft/s² (physics.nist.gov/cgi-bin/cuu/Value?gn)
- Reference value: NIST, "Putting the Best Foot Forward" (1 foot = 0.3048 m exactly since 1959; nist.gov/pml/us-surveyfoot, checked 2026-10-19)
- History block only: MacTutor, University of St Andrews, Quadratic, cubic and quartic equations (Babylonians about 1800 BC, algorithmic completing the square with positive answers; Euclid about 300 BC geometric; Brahmagupta (598-665) admits negative quantities; al-Khwarizmi (c 800) six types, x² + 10x = 39; Abraham bar Hiyya, Liber embadorum 1145: mathshistory.st-andrews.ac.uk/HistTopics/Quadratic_etc_equations).
Checked: 2026-10-18