الأسبوع 2 · الدالة التربيعية: الصورة القياسية والرأس ومحور التماثل وصورة الرأس والتمثيل البياني والقيمة العظمى أو الصغرى · 3 دروس
3
Vertex form f(x) = a(x − h)² + k: the vertex (h, k), and the path of a thrown ball
Mathematics Prep 3 Term 1, Unit 2, Lesson 1
GoalRead the vertex, the axis and the maximum or minimum from the vertex form, and use it in a ball problem.
New words
vertex form · f(x) = a(x − h)² + k, with vertex (h, k)standard form · f(x) = ax² + bx + cheight function · a rule h(t) that gives the height of an object after t secondstrajectory · the curved path of a thrown object
Explanation
The same parabola can be written as f(x) = a(x − h)² + k. A square is never negative, so (x − h)² is smallest (zero) when x = h, and there f(x) = k. This shows at once that the vertex is (h, k) and the axis is x = h. The same rule for a decides the opening. Watch the signs: in (x + 2)² + 4 we have x + 2 = x − (−2), so h = −2 and the vertex is (−2, 4), not (2, 4). To change the standard form into the vertex form, find the vertex first; then h = −b/2a, k = f(h), and a is unchanged. A thrown ball is a nice use: its height is a quadratic function of the time t, and the greatest height is the value k at the vertex. Time cannot be negative, so use only the t-values that make sense.
Function
Vertex (h, k)
Axis
Opens
Max / min value
2(x − 3)² + 1
(3, 1)
x = 3
up
min 1
(x + 5)² − 2
(−5, −2)
x = −5
up
min −2
7 − (x − 4)²
(4, 7)
x = 4
down
max 7
Worked examples
State the vertex, axis and extreme value of f(x) = 7 − (x − 4)². Write it as −(x − 4)² + 7: a = −1, h = 4, k = 7. The vertex is (4, 7), the axis is x = 4, and since a < 0 the maximum value is 7.
Write f(x) = x² − 6x + 5 in vertex form. The vertex is (3, −4) and a = 1, so f(x) = (x − 3)² − 4. Check: (x − 3)² − 4 = x² − 6x + 9 − 4 = x² − 6x + 5 ✓.
A footballer kicks a ball. Its height in metres after t seconds is h(t) = −5(t − 2)² + 20, for 0 ≤ t ≤ 4. The vertex is (2, 20): the ball is highest, 20 m, after 2 seconds. At t = 0: h = −20 + 20 = 0, so it starts on the ground. At t = 4: h = −20 + 20 = 0, so it lands after 4 seconds.
Class activity
Sign check. Expand (x + 2)² + 4 and (x − 2)² + 4 and compare with the vertex you expect. Which sign of h goes with each bracket?
Practice
1
Find the vertex and the extreme value of f(x) = 3(x − 1)² + 5.
Vertex (1, 5); a = 3 > 0, so the minimum value is 5.
2
Find the vertex of f(x) = (x + 4)² − 3.
x + 4 = x − (−4), so h = −4 and the vertex is (−4, −3).
