الرياضيات (بالإنجليزية) الصف الثالث الإعدادي، الأسبوع 2: The quadratic function f(x) = ax² + bx + c: the parabola, the y-intercept and the vertex at x = −b/2a

The quadratic function

الأسبوع 2 · الدالة التربيعية: الصورة القياسية والرأس ومحور التماثل وصورة الرأس والتمثيل البياني والقيمة العظمى أو الصغرى · 3 دروس

The quadratic function f(x) = ax² + bx + c: the parabola, the y-intercept and the vertex at x = −b/2a

Mathematics Prep 3 Term 1, Unit 2, Lesson 1

GoalRecognise a quadratic function, find its y-intercept, and find the vertex of its parabola.

New words

quadratic function · f(x) = ax² + bx + c with a ≠ 0; its domain is all real numbersparabola · the curve that is the graph of a quadratic functionvertex · the turning point of the parabola, on its axis of symmetryaxis of symmetry · the vertical line through the vertex that cuts the parabola into two mirror halves

Explanation

A quadratic function has the form f(x) = ax² + bx + c, with a ≠ 0 (if a = 0 we get a straight line). Its graph is a parabola with a vertical line of symmetry. The y-intercept is where x = 0, so it is f(0) = c, the point (0, c). Where is the vertex? Solve f(x) = c: ax² + bx = 0, so x(ax + b) = 0, giving x = 0 and x = −b/a. These two points have the same height, so by symmetry the axis lies exactly halfway between them, at x = −b/2a. The vertex is the point (−b/2a, f(−b/2a)): first find its x-coordinate, then put it into f to get the y-coordinate.

Worked examples

Find the vertex and the y-intercept of f(x) = x² − 6x + 5. Here a = 1, b = −6, c = 5. The x-coordinate of the vertex is −b/2a = 6/2 = 3. Then f(3) = 9 − 18 + 5 = −4, so the vertex is (3, −4). The y-intercept is f(0) = 5. Check the symmetry: f(6) = 36 − 36 + 5 = 5, and 3 is halfway between 0 and 6. ✓

Find the vertex of f(x) = −2x² + 8x − 3. Here a = −2, b = 8, c = −3. The x-coordinate is −8/(2 × (−2)) = −8/(−4) = 2. Then f(2) = −8 + 16 − 3 = 5. The vertex is (2, 5) and the y-intercept is (0, −3).

For which values of m is f(x) = (m − 4)x² + 5x − 1 a quadratic function? It needs a ≠ 0, so m − 4 ≠ 0 and m ≠ 4. For m = 4 the rule is 5x − 1, a straight line.

Class activity

Equal heights. For f(x) = x² − 6x + 5 find f(1) and f(5), then f(2) and f(4). What do you notice about each pair, and which x is exactly in the middle of each pair?

Practice

Is f(x) = x(x − 5) + 3 a quadratic function? Give a, b, c.

Find the vertex of f(x) = x² + 4x + 1.

What is the y-intercept of f(x) = 2x² − 7x + 9?

The point (2, k) lies on f(x) = x² − 6x + 5. Find k.

The point (4, 1) is on the parabola of f(x) = x² − 6x + 5? Decide.