Quadratic Graphs

1Summary 重點摘要

y = ax² + bx + c draws a parabola (U shape).

2Vocabulary cards Vocabulary

3Step by step Steps

1. 1. What is a quadratic functionA quadratic function has x² as its highest power, in the standard form y = ax² + bx + c where a, b, c are constants and a ≠ 0 (with a = 0 the x² term vanishes and the graph becomes a straight line). Examples: y = x² (a = 1, b = 0, c = 0), y = x² + 2x + 1 (a = 1, b = 2, c = 1), y = −2x² + 5 (a = −2, b = 0, c = 5). The curve appears in real life: the flight of a basketball and the jet of a fountain both trace approximate parabolas — the most intuitive physical picture of a quadratic, and the reason it appears so widely in physics and engineering.
2. 2. The shape of a parabolaThe graph of a quadratic is called a parabola, and its shape is governed by a. A positive a opens the parabola upward (U shape), as in y = x²; a negative a opens it downward (inverted U), as in y = −x². The larger |a|, the narrower the curve: y = 3x² is narrower than y = x², and y = 0.5x² is wider. Quick reading: look only at the coefficient of x² — its sign fixes the direction of opening and its size the width. The values of b and c shift the parabola's position (left-right and up-down) but never its direction. Check the sign of a first and the overall shape is settled at once.
3. 3. The vertexThe vertex is the highest or lowest point of the parabola — the turning point of the graph. When the parabola opens upward (a positive), the vertex is the global minimum; when it opens downward (a negative), the vertex is the global maximum. The x-coordinate of the vertex is x = −b ÷ 2a. For y = x² − 4x + 1: x = −(−4) ÷ (2 × 1) = 2, and substituting gives y = 4 − 8 + 1 = −3, so the vertex is (2, −3); since a = 1 is positive it is a minimum, and the function's least value is −3. The classic slip is the sign in −b: the whole of b flips, so b = −4 gives −b = 4, and missing this turns the graph sideways.
4. 4. Roots (crossing the x-axis)The roots are the values of x that make y = 0 — the x-coordinates where the parabola meets the x-axis. Three relations are possible: the curve lies wholly above (or below) the axis — no real roots; it touches the axis at exactly one point — one (repeated) root; it crosses at two points — two roots. For y = x² − 4: setting x² − 4 = 0 gives x = 2 or x = −2, two roots, and the graph crosses the x-axis at (2, 0) and (−2, 0). Roots and vertex agree: the midpoint of the roots is the vertex's x-coordinate; above, (2 + (−2)) ÷ 2 = 0, and the vertex indeed sits at x = 0 — a useful check on the working.
5. 5. The y-interceptThe y-intercept is the y-coordinate where the parabola crosses the y-axis, found by putting x = 0. In y = ax² + bx + c, substituting x = 0 gives y = c, so the y-intercept always equals the constant term, and the curve passes through (0, c). For y = x² + 2x + 1 the intercept is 1 — the point (0, 1); for y = 2x² − 5x − 3 it is −3 — the point (0, −3). This is the easiest of the three features to find: read the constant term. The intercept may be positive or negative regardless of the opening; mark it first when sketching, add the vertex, and the parabola's position is largely fixed.
6. 6. Drawing a quadratic graphSketching a parabola takes five steps. First find the vertex: compute x = −b ÷ 2a, then substitute for y. Second find the y-intercept: read the constant c. Third choose several x-values (say two integers either side of the vertex), compute their y-values and tabulate. Fourth plot all the points. Fifth join them with a smooth curve — a parabola is curved, so never connect the dots with a straightedge. For y = x² − 2x − 3: the vertex is at x = 1, y = −4, that is (1, −4); the y-intercept is −3; computing x = −1, 2, 3 gives y = 0, −3, 0; five points trace an upward parabola whose roots −1 and 3 emerge naturally.
7. 7. Common mistakesFour errors recur in quadratic graphs. First, the vertex formula sign: for y = x² − 4x + 1 the vertex sits at x = −(−4) ÷ 2 = 2; dropping the double negative yields x = −2 and flips the whole graph sideways. Second, drawing the parabola as a polyline: the joins must be smoothly curved; ruling straight segments loses shape marks. Third, arithmetic on substitution: with a negative x, the square keeps its sign — at x = −2, x² = 4, not −4. Fourth, confusing roots with the y-intercept: roots lie on the x-axis (y = 0) while the intercept lies on the y-axis (x = 0), found by different moves. Self-check when finished: the vertex on the axis of symmetry, flanking points symmetric, and the y-intercept equal to c.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Quadratic: y = ax² + bx + c
• Draws a parabola (U shape)
• a positive up, negative down
• Roots = x values when y = 0
• Vertex is highest or lowest point