☰ Contents · Mathematics

Linear and quadratic models

Lessons 27 · 1 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 10, Parts 1 and 2, 1st edition. EXTREMUM PRESS, Tashkent, 2017
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Linear and quadratic models

Textbook, Part 2: pp. 12–22
GoalModel real situations with linear and quadratic functions: compute values, find the parabola’s vertex and its intersections with the axes, and recover a formula from a graph.
New words
slope · burchak koeffitsiyentivertex of a parabola · parabola uchiaxis of symmetry · simmetriya o‘qiquadratic function · kvadrat funksiya
Explanation

f(x) = ax + b is a linear function whose graph is a line with slope a; for instance, fee = fixed charge + hourly rate · time is linear. y = ax² + bx + c (a ≠ 0) is a quadratic function whose graph is a parabola. It meets the y-axis at (0, c), and its intersections with the x-axis are the roots of ax² + bx + c = 0. The axis of symmetry is x = −b/(2a) and the vertex lies on it: to get its ordinate, substitute x = −b/(2a) into the formula. If a > 0 the branches go up and the vertex gives the least value; if a < 0 they go down and the vertex gives the greatest value. If the roots x₁, x₂ are known, the formula can be written y = a(x − x₁)(x − x₂); we find a from another point of the parabola.

Worked examples
A rectangular plot next to a wall is fenced with 60 m of netting (the wall is the fourth side). If the side perpendicular to the wall is x, the area is S(x) = x(60 − 2x) = −2x² + 60x. a = −2 < 0, the vertex is at x = −60/(2 · (−2)) = 15: S(15) = 15 · 30 = 450 m², the greatest area. S(x) = 0 at x = 0 and x = 30, where the parabola meets the x-axis.
A parabola meets the x-axis at x = −1 and x = 5 and its least value is −18. It has the form y = a(x + 1)(x − 5); the axis is x = (−1 + 5)/2 = 2, and at the vertex y(2) = a · 3 · (−3) = −9a = −18, so a = 2. Hence y = 2(x + 1)(x − 5) = 2x² − 8x − 10.
Class activity

Make a table for a mobile plan: monthly fee C(g) = 8000g + 40 000 so‘m (g — extra GB). Compute the values for g = 0, 1, 2, 3, 4, plot the points and check that they lie on one line.

Practice
1
Write the equation of the axis of symmetry of y = ax² + bx + c.
2
Find the least value of y = x² − 6x + 5.
3
A phone battery starts at 100% and loses 8% per hour: B(t) = 100 − 8t. After how many hours is 36% left?
4
Why does a parabola with a > 0 have its least value at the vertex?