Required knowledge and skills: linear functions
Overview
1. Linear function: function whose equation is linear, i.e. has the form y  mx  b , where m  0
2. Graph of a linear function: straight line
a. drawing the graph when the equation is given
b. setting up the equation when the graph is given (see also item 5)
3. Interpretation of the coefficient m in the equation y  mx  b
a. slope of the line
b. rate of change: if x increases by 1 unit, then y changes by m units
y y 2  y1
c. difference quotient: m 
, where P1 ( x1 , y1 ) and P2 ( x 2 , y 2 ) are points on the line

x x 2  x1
d. sign of m determines whether line is increasing or decreasing
e. absolute value of m determines how steep the line is
f. parallel lines have equal slopes
g. slopes of perpendicular lines have product equal to –1
4. Interpretation of the coefficient b in the equation y  mx  b
a. b is y-intercept
b. b is function value of 0
c. b  0 iff origin is on the line
5. Setting up the equation of a line
a. if slope m and one point P ( x 0 , y 0 ) are given: y  y 0  m( x  x 0 )
b. if two points are given
6. Linear equations:
a. solving linear equations
b. solving other equations leading to a linear equation
c. (graphical) interpretation of the solution of a linear equation: solution of equation mx  b  0
i. is x-intercept of the corresponding line
ii. is the zero of the corresponding linear function
7. Linear inequalities
a. solving linear inequalities
b. graphical interpretation of linear inequalities
i. solutions of mx  b  0 form interval where corresponding line is above the horizontal axis
ii. solutions of m1 x  b1  m2 x  b2 form interval where the line corresponding to left hand side
is above the line corresponding to right hand side
8. Implicit equation of a line: ax  by  c  0 or, equivalently, ax  by  d , (where a and b are not
simultaneously 0); horizontal and vertical lines
9. Graphical interpretation of the solution of a system of two linear equations: intersection point of the
lines corresponding to the equations
10. Applications and word problems involving linear functions, more specifically: setting up an equation
for one or more linear functions given a description in words and using these equations to solve a
1
problem by calculating a function value, solving a linear equation, solving a linear inequality, solving
a system of linear equations, …
Examples
Example 1. Which of the following functions are linear? Moreover, in case the function is linear, find out
whether the function is increasing or decreasing.
A. y  x 3
B.
y  3x
C.
y  3x  5
D. y  5
E.
y  2(3x  5)
F.
y  x(3x  5)
G. y  x(3x  5)  3x 2
H. y  3x  5
Example 2. Find the equation of the line through the points P(2, 1) and Q(1, 2) .
Example 3. Find the equation of the line through the point P(2, 1) and parallel to the line through Q(5, 2)
and R(0, 4.5) .
Example 4. Find the equation of the line through the origin and perpendicular to the line having equation
y  2x  3 .
Example 5. Are the lines 2 x  3 y  4 and 3x  2 y  4 perpendicular to each other?
Example 6. Find the intersection point of the lines 2 x  3 y  4 and 3x  2 y  4 .
Example 7. The cost of c of producing q units of a certain good consists of two parts. There is a fixed cost
of 280 EUR, plus a variable cost of 3 EUR per unit produced. Write an equation giving the cost c in terms
of the number q of units produced.
Example 8. A new product was launched in 2004. The graph below shows how many units of this product
were sold in the years 2004 to 2007. If the sales continue increasing in the same way, how many units of
the product will be sold in 2010?
sales
10000
8000
6000
4000
2000
0
2004
2005
2006
2007
Example 9. The cost c of a taxi ride of x km is given by the equation c  5  0.8 x . Make a graph showing
the price of taxi rides for a distance between 0 and 10 km.
2
Example 10. A first electricity company charges a fixed rate of 100 EUR per year and 0.2 EUR per
kilowatt-hour of electricity consumed. A second company has no fixed rate, but charges 0.22 EUR per
kilowatt-hour consumed. Find out for which yearly consumption of electricity the second company is
cheaper than the first one.
Solutions to the examples
Example 1. B (increasing), C (increasing), E (decreasing), G (decreasing), H (increasing)
Example 2. y   x  3
Example 3. y  0.5x  2
Example 4. y  0.5x
Example 5. yes
 20 4 
Example 6. intersection point has coordinates  , 
 13 13 
Example 7. c  280  3q
Example 8. 16 000
Example 9. see graph below
cost
14
12
10
8
6
4
2
0
0
2
4
6
Example 10. yearly consumption below 5000 kilowatt-hours
3
8
10 km