Theory of Intelligence
11TH•Mathematics•Theory Of Quadratic Functions•Question 2

Step-by-Step Solution

Find the maximum or minimum point by sketching the following quadratic functions. Also find their domain and range:

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Step-by-Step Solution

Step 1: Identify Parameters and Vertex

Rewriting in standard descending order f(x)=ax2+bx+cf(x) = ax^2 + bx + c:

a=−1,b=−1,c=6a = -1, \quad b = -1, \quad c = 6

Since a=−1<0a = -1 < 0, the parabola opens downwards, which means the vertex is an absolute maximum point.

The xx-coordinate of the vertex hh:

h=−b2a=−−12(−1)=−12=−0.5h = -\frac{b}{2a} = -\frac{-1}{2(-1)} = -\frac{1}{2} = -0.5

The yy-coordinate kk:

k=f(−12)=6−(−12)−(−12)2=6+12−14=24+2−14=254=6.25k = f\left(-\frac{1}{2}\right) = 6 - \left(-\frac{1}{2}\right) - \left(-\frac{1}{2}\right)^2 = 6 + \frac{1}{2} - \frac{1}{4} = \frac{24 + 2 - 1}{4} = \frac{25}{4} = 6.25

The vertex is located at (−12,254)\left(-\frac{1}{2}, \frac{25}{4}\right).

Step 2: Intercepts & Axis of Symmetry

  • yy-intercept: f(0)=6  ⟹  (0,6)f(0) = 6 \implies (0, 6).
  • xx-intercepts: Factor −x2−x+6=−(x2+x−6)=−(x+3)(x−2)=0  ⟹  x=−3-x^2 - x + 6 = -(x^2 + x - 6) = -(x + 3)(x - 2) = 0 \implies x = -3 and x=2x = 2.
  • Axis of Symmetry: x=−12=−0.5x = -\frac{1}{2} = -0.5

Step 3: Domain and Range

  • Domain: Defined for all real numbers: Domain=R=(−∞,∞)\text{Domain} = \mathbb{R} = (-\infty, \infty)
  • Range: The highest peak on the curve is y=6.25=254y = 6.25 = \frac{25}{4}, and the curve descends downwards to −∞-\infty: Range=(−∞,254]=(−∞,6.25]\text{Range} = \left(-\infty, \frac{25}{4}\right] = (-\infty, 6.25]

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