5
The function y = ax² + bx + c
Textbook: pp. 14–17
GoalFind the vertex and axis of symmetry of y = ax² + bx + c and relate the graph to shifts by completing the square.
New words
completing the square · to‘la kvadratni ajratishtranslation (shift) · parallel ko‘chirish (siljitish)coordinates of the vertex · parabola uchining koordinatalariequation of a parabola · parabolaning tenglamasi
Explanation
Every function y = ax² + bx + c can be written, by completing the square, as y = a(x − x₀)² + y₀, where x₀ = −b : (2a) and y₀ = y(x₀). So its graph is the parabola y = ax² shifted along Ox by x₀ (to the right if x₀ > 0, to the left if x₀ < 0) and along Oy by y₀ (up if y₀ > 0, down if y₀ < 0). The vertex of the parabola is the point (x₀, y₀), and the axis of symmetry is the line x = x₀, parallel to Oy. The branches point up when a > 0 and down when a < 0. The equality y = ax² + bx + c is called the equation of the parabola.
Worked examples
y = x² + 4x + 1: x₀ = −4 : 2 = −2, y₀ = 4 − 8 + 1 = −3. The vertex is (−2, −3). With a full square: y = (x + 2)² − 3.
The parabola with vertex (1, −2) through (3, 6): y = a(x − 1)² − 2, 6 = a · 4 − 2, so a = 2. The equation is y = 2(x − 1)² − 2 = 2x² − 4x. Check: 2 · 9 − 12 = 6.
Class activity
“Find the vertex”: the teacher hands out formulas y = ax² + bx + c; teams quickly compute x₀, then y₀, and mark the vertex on the plane. The fastest correct team wins.
Practice
1
State the vertex of the parabola y = 2(x − 3)² + 1.
(3, 1)
2
Find the coordinates of the vertex of y = x² − 6x + 5.
(3, −4)
3
Write the equation of the axis of symmetry of y = −x² + 8x − 3.
x = 4
4
Why does the function take its greatest or smallest value at x₀ = −b : (2a)?
In y = a(x − x₀)² + y₀ we have (x − x₀)² ≥ 0, equal to zero only at x = x₀, so y₀ is the extreme value.