Quadratic functions

Quadratic

Functions

Quadratic function: function whose equation is a polynomial of the second degree, i.e. has the form , where a ≠ 0

Quadratic equations discriminant is the number if : two solutions , if : one solution (also called two coinciding

solutions) if : solutions (in fact, the solutions are non-real

complex numbers) the solutions of the equation are the zeroes of the

corresponding quadratic function

sign of a determines whether parabola has opening upwards ( ) or downwards ( )

sign of d determines the number of x-interceptsif : x-axis and parabola intersect in two pointsif : x-axis and parabola have one common point,

horizontal axis is tangent to parabolaif : x-axis and parabola do not intersect

c is the y-intercept of the parabola

Graph of a quadratic function: parabola

is the highest/lowest point of the parabola y-value of the vertex is the maximum/minimum

value of the corresponding quadratic function x-coordinate of the vertex is the x-value whose

corresponding y-value is maximum/minimum value

x-coordinate of the vertex is y-coordinate of the vertex is found by plugging

its x-coordinate into the equation

Vertex of a parabola

if , where and are the solutions of the corresponding quadratic equation

if cannot be factorized

Factoring polynomials of the second degree

Quadratic inequalities (also with other inequality signs): sketch the graph of the left hand side and draw conclusions from the graph

Setting up the equation of a parabola if three points are given

Solving simple systems of two equations and two unknowns involving squares of the unknowns.

Applications and word problems involving quadratic functions, more specifically: setting up an equation for one or more quadratic functions given a description in words and using these equations to solve a problem by calculating a function value, solving a quadratic equation, solving a quadratic inequality, finding the maximum or minimum value, …

Example 1. Find the zeroes of the function with equation .

Example 2. For which values of b do the solutions of the equation coincide?

Example 3. Find the numbers b and c in the equation of the quadratic function f such that f reaches its maximal value 5 at x = 3.

Example 4. Factor the polynomial if possible. Example 5. Solve for x: .

Examples

Example 6. Find the equation of the parabola going through the points , and .

Example 7. Solve the system Example 8. For a certain good, the relation

between the number q of items sold and the price p per item is given by . Express the total revenue r in terms of p and find the price for which the revenue is maximal.

Examples

Example 9. A company charges 200 EUR for each leather bag. In order to motivate shop keepers to buy greater lots of bags, the company decides to implement the following pricing strategy: for each bag in excess of 150 bags, the price per bag will be reduced by 1 EUR for all bags in the lot (not only for the supplementary ones). For example, if the shop keeper decides to buy a lot of 155 bags, he receives the reduction five times and it applies to each of the 155 bags. Hence, each bag in the lot costs 195 EUR then. What is the maximum revenue?