Quadratic Functions

General Information

Quadratic functions are parabolas, functions that are the first power in one variable and the second power in the other variable. Usually, the parabola will be the first power in y and the second power in x. There are three major forms in which quadratic functions will appear.

• Standard Form: y = ax² + bx + c
• Vertex Form: y = a(x - h)² + k; (h, k) is the vertex of the parabola
• Factored Form: y = a(x - z₁)(x - z₂); z₁ and z₂ are the zeroes/roots of the function

Although all quadratic functions can be converted to each of these forms, if the function does not have real roots, it is usually not expressed in factored form.

Standard Form

Below are the features of standard form of a quadratic function.

• The axis of symmetry has equation x = -^b⁄_2a.
• The x coordinate of the vertex is -^b⁄_2a.
• The value c is the y intercept of the function - (0, c) is a point on the function.
• If c = 0, the function has two real roots, one of which is 0.
• If a > 0, the function opens upward; if a < 0, the function opens downward.
• If a and c have different signs, the function will have 1 real root if b is 0, and the function will have 2 real roots if b is anything other than 0

Vertex Form

Below are the features of vertex form of a quadratic function.

• (h, k) is the vertex (lowest point or highest point) of the function.
• If k = 0, the function has one unique root of multiplicity 2; in other words, the vertex lies on the x axis.
• If a and k have the same sign, the function has no real zeroes/roots; in other words, the function will never touch the x axis.
• If a and k have different signs, the function has two real, distinct zeroes/roots; in other words, the function will intersect the x axis twice.

Factored Form

Below are the features of factored form of a quadratic function.

• The zeroes/roots of the function are easily found.
• The axis of symmetry is the average of the two zeroes/roots.

Example Problems

1. Characterize the roots - 2 real, distinct or 1 real, double or 2 imaginary, complex conjugate - of the following quadratic function:
   
   • y = 7(x - 11)² - 1
   
   
2. Which of the following parabolas has a vertex in the third quadrant and has two real solutions?
   
   1. y = 10(x - 7)² + 6
   2. y = 10(x + 7)² + 6
   3. y = -10(x + 7)² + 6
   4. y = 10(x + 7)² - 6


3. Characterize the roots - 2 real, distinct or 1 real, double or 2 imaginary, complex conjugate - of the following quadratic function:
   
   • y = 8(x - 14)² + 9
   
   
4. Which of the following parabolas has a vertex in the first quadrant and has no real solutions?
   
   1. y = 2(x + 1)² - 6
   2. y = -2(x - 1)² + 6
   3. y = 2(x - 1)² + 6
   4. y = 2(x + 1)² + 6


5. Find the roots and vertex coordinates of the following quadratic function.
   • y = x² - 4x - 60