Page 125 - Algebra

Remember
• f(x) = ax2 + bx + c = 0, where a ≠ 0 is a parabola. In a parabola,
−𝑏
• Line of symmetry is x = 2𝑎
−𝑏 −𝑏
• Coordinates of the vertex (2𝑎, f(2𝑎))
• a < 0, the vertex is the maximum point
• a > 0, the vertex is the minimum point
• Vertex Form of the Parabola f(x) = a(x – h)2 + k, h and k are the vertex
• Factored form of a parabola
f(x) = a(x – b)(x – c)
The x-intercept or the solution of the equations are x = b, and x = c
The graph of
Worked Example
The equation of a parabola is given by f(x) = 3(x + 3)(x + 5). Find the minimum or the maximum value of the function:
Solution:
f(x) = 3(x + 3)(x + 5)
Solve the equation
f(x) = 3(x2 + 5x + 3x + 15)
f(x) = 3(x2 + 8x + 15)
Write the equation in the form of f(x) = a(x – h)2 + k, where k is the minimum or the maximum value of the vertex.
f(x) = 3(x2 + 8x + 15)
f(x)=3(x2 +8x)+45
Now complete the square by adding and subtracting 48
f(x)=3(x2 +8x)+45+48–48
f(x) = 3(x2 + 8x + 16) + 45– 48
f(x) = 3(x + 4)2 – 3
Compare it with f(x) = a(x – h)2 + k
a = 3, h = – 4, k = – 3
As a > 0, the function has a minimum value.
The minimum value of the function is – 3
Page 124 of 177
Algebra I & II