Required knowledge and skills: quadratic functions
Overview
1. Quadratic function: function whose equation is a polynomial of the second degree, i.e. has the form
y  ax 2  bx  c , where a  0
2. Quadratic equations ax 2  bx  c  0
a. discriminant is the number d  b 2  4ac
b. if d  0 : two solutions x1 
b d
b d
, x2 
2a
2a
b
2a
d. if d  0 : solutions (in fact, the solutions are non-real complex numbers)
e. the solutions of the equation are the zeroes of the corresponding quadratic function
3. Graph of a quadratic function: parabola
a. sign of a determines whether parabola has opening upwards ( a  0 ) or downwards ( a  0 )
b. sign of d determines the number of x-intercepts
i. if d  0 : x-axis and parabola intersect in two points
ii. if d  0 : x-axis and parabola have one common point, horizontal axis is tangent to parabola
iii. if d  0 : x-axis and parabola do not intersect
c. c is the y-intercept of the parabola
4. Vertex of a parabola
a. is the highest/lowest point of the parabola
b. y-value of the vertex is the maximum/minimum value of the corresponding quadratic function
c. x-coordinate of the vertex is the x-value whose corresponding y-value is maximum/minimum
value
b
d. x-coordinate of the vertex is
2a
e. y-coordinate of the vertex is found by plugging its x-coordinate into the equation
5. Factoring polynomials of the second degree
c. if d  0 : one solution (also called two coinciding solutions) x1  x 2 
a. if d  0 : ax 2  bx  c  a( x  x1 )( x  x 2 ) , where x1 and x 2 are the solutions of the
corresponding quadratic equation
b. if d  0 : ax 2  bx  c cannot be factorized
6. Quadratic inequalities ax 2  bx  c  0 (also with other inequality signs): sketch the graph of the left
hand side and draw conclusions from the graph
7. Setting up the equation of a parabola if three points are given
8. Solving simple systems of two equations and two unknowns involving squares of the unknowns.
9. 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, …
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Examples
Example 1. Find the zeroes of the function with equation  2 x 2  3 x  5  0 .
Example 2. For which values of b do the solutions of the equation 2 x 2  bx  5  0 coincide?
Example 3. Find the numbers b and c in the equation f ( x)   x 2  bx  c of the quadratic function f such
that f reaches its maximal value 5 at x = 3.
Example 4. Factor the polynomial 4 x 2  8 x  5 if possible.
Example 5. Solve for x:  x 2  3x  30  x( x  1) .
Example 6. Find the equation of the parabola going through the points P(1, 10) , Q(1, 3) and R(2, 5.5) .
10 x  3 y 2  48
Example 7. Solve the system 
x  3 y  0
Example 8. For a certain good, the relation between the number q of items sold and the price p per item is
given by q  120  0.4 p . Express the total revenue r in terms of p and find the price for which the revenue
is maximal.
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?
Solutions to the examples
Example 1.  1 and 2.5
Example 2. b  2 10 or b  2 10
Example 3. b  6 and c  4
Example 4. 4( x  0.5)( x  2.5) or (2 x  1)(2 x  5)
Example 5. x  3 or x  5
Example 6. y  2 x 2  3.5 x  4.5
Example 7. ( x  6 and y  2 ) or ( x  24 and y  8 )
Example 8. r  0.4 p 2  120 p , r is maximal if p  150
Example 9.30 625 EUR
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