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Quadratic Function

A quadratic function is a function that can be written in the form f(x)=ax2+bx+cf(x) = ax^2 + bx + c, where aa, bb, and cc are constants and a≠0a \neq 0. Its graph is always a U-shaped curve called a parabola.

A quadratic function is a polynomial function of degree 2, expressed in standard form as f(x)=ax2+bx+cf(x) = ax^2 + bx + c, where aa, bb, and cc are real numbers and a≠0a \neq 0. The restriction a≠0a \neq 0 is essential — without the x2x^2 term, the function would be linear, not quadratic. The graph of every quadratic function is a parabola that opens upward when a>0a > 0 and downward when a<0a < 0. The parabola has a vertex, which represents the minimum or maximum value of the function.

Key Formula

f(x)=ax2+bx+cf(x) = ax^2 + bx + c
Where:
  • aa = the coefficient of x², which determines whether the parabola opens up (a > 0) or down (a < 0) and how wide or narrow it is
  • bb = the coefficient of x, which affects the horizontal position of the vertex
  • cc = the constant term, which is the y-intercept of the parabola

Worked Example

Problem: For the quadratic function f(x)=2x2−8x+6f(x) = 2x^2 - 8x + 6, find the vertex and determine whether the parabola opens upward or downward.
Step 1: Identify the coefficients from the standard form f(x)=ax2+bx+cf(x) = ax^2 + bx + c.
a=2,b=−8,c=6a = 2, \quad b = -8, \quad c = 6
Step 2: Find the x-coordinate of the vertex using the formula x=−b2ax = -\frac{b}{2a}.
x=−−82(2)=84=2x = -\frac{-8}{2(2)} = \frac{8}{4} = 2
Step 3: Substitute x=2x = 2 back into the function to find the y-coordinate of the vertex.
f(2)=2(2)2−8(2)+6=8−16+6=−2f(2) = 2(2)^2 - 8(2) + 6 = 8 - 16 + 6 = -2
Step 4: Since a=2>0a = 2 > 0, the parabola opens upward, meaning the vertex is the minimum point.
Answer: The vertex is at (2,−2)(2, -2), and the parabola opens upward.

Visualization

Why It Matters

Quadratic functions model many real-world situations where something rises and then falls (or vice versa). The path of a thrown ball, the profit of a business as a function of price, and the area of a rectangle with a fixed perimeter are all described by quadratic functions. Finding the vertex tells you the maximum height, the optimal price, or the greatest possible area.

Common Mistakes

Mistake: Forgetting the negative sign in the vertex formula and computing x=b2ax = \frac{b}{2a} instead of x=−b2ax = -\frac{b}{2a}.
Correction: The formula has a negative sign in the numerator. For f(x)=2x2−8x+6f(x) = 2x^2 - 8x + 6, you need x=−−84=2x = -\frac{-8}{4} = 2, not x=−84=−2x = \frac{-8}{4} = -2.
Mistake: Assuming every quadratic function has two x-intercepts.
Correction: A quadratic function can have two, one, or zero x-intercepts depending on whether the discriminant b2−4acb^2 - 4ac is positive, zero, or negative.

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