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FORMATIVE 6.3-6.5
Problem
1. Sketch graphs to help explain what happens to the graph of
a) the coefficient of x increases by 1 each time until it is 6
b) the constant term decreases by 1 each time until it is
when:
2. For the equation
:
a) Explain how to change the equation so the line has a greater slope, then a lesser slope.
b) Explain how to change the equation so the line has a greater y-intercept, then a lesser y-intercept.
c) Rewrite the equation so the new line has a y-intercept that is one less than the given y-intercept, and a
slope that is one more than the given slope.
3. Write an equation to describe this function. Verify the equation.
y
4
2
–4
–2
0
2
4
x
–2
–4
4. An equation of a line is
. Determine the value of m when the line passes through the point J(–5, 2).
5. Francine runs a T-shirt company. For each order she receives, Francine charges a flat fee of $50, plus $8.95
per T-shirt .
a) Write an equation for the total cost, C dollars, for ordering n T-shirts.
b) Marnell ordered 62 T-shirts. What was the total cost?
c) Jakub paid a total cost of $971.85. How many T-shirts did he order?
6. a) Write an equation in slope-point form for this line.
y
4
2
–4
0
–2
2
4
x
–2
–4
b) Write the equation in part a in slope-intercept form. What is the y-intercept of this line?
7. In Canada, the number of girls playing organized ice hockey from January 1990 to January 2010 increased by
approximately 4162 girls per year. In January 2000, there were approximately 45 400 girls playing
organized ice hockey.
a) Write an equation in slope-point form to represent the number of girls, n, playing organized ice hockey as
a function of the number of years, t, after 1990.
b) Use the equation in part a to estimate the number of girls playing organized ice hockey in January 2009.
8. In Jay’s business, the annual cost of operating a car, c, is a linear function of the number of kilometres the car
is driven, k. The annual cost of operating a car that has been driven 19 375 km is approximately $3875. The
annual cost of operating a car that has been driven 20 000 km is approximately $3900.
a) Write an equation in slope-point form to represent this function.
b) Use the equation in part a to determine how many kilometres a car has been driven when the annual
operating cost is approximately $4350.
9. Write an equation for the line that passes through B(–1, 3) and is:
a) parallel to the line
b) perpendicular to the line
FORMATIVE 6.3-6.5
Answer Section
PROBLEM
1. ANS:
a) When the coefficient of x increases by 1, the y-intercept of the graph stays the same and the slope of the
graph increases by 1.
6
y
4
2
–6
–4
–2
0
2
4
6 x
–2
–4
–6
b) When the constant term decreases by 1, the slope of the graph stays the same and the y-intercept of the
graph decreases by 1.
6
y
4
2
–6
–4
–2
0
2
4
6 x
–2
–4
–6
PTS: 1
DIF: Moderate
REF: 6.3 Investigating Graphs of Linear Functions
LOC: 10.RF7
TOP: Relations and Functions
KEY: Communication | Problem-Solving Skills
2. ANS:
a) For the line to have a greater slope, I change the coefficient of x so that it is a number greater than –7. For
the line to have a lesser slope, I change the coefficient of x so that it is a number less than –7.
b) For the line to have a greater y-intercept, I change the constant term so that it is a number greater than 4.
For the line to have a lesser y-intercept, I change the constant term so that it is a number less than 4.
c)
PTS: 1
DIF: Moderate
REF: 6.3 Investigating Graphs of Linear Functions
LOC: 10.RF7
TOP: Relations and Functions
KEY: Communication | Problem-Solving Skills
3. ANS:
Use the equation:
To write the equation of a linear function, determine the slope of the line, m, and its y-intercept, b.
The line intersects the y-axis at ; so,
.
From the graph, the rise is 4 when the run is 5.
So,
Substitute for m and b in
.
An equation for the function is:
To verify the equation, substitute the coordinates of a point on the line into the equation. Choose the point (0,
).
Substitute
and
into the equation:
Since the left side is equal to the right side, the equation is correct.
PTS: 1
DIF: Moderate
REF: 6.4 Slope-Intercept Form of the Equation for a Linear Function
LOC: 10.RF6
TOP: Relations and Functions
KEY: Communication | Problem-Solving Skills
4. ANS:
Substitute the coordinates of point J(–5, 2) into the equation
, then solve for m.
So, when the line passes through the point J(–5, 2), the value of m is
.
PTS: 1
DIF: Difficult
REF: 6.4 Slope-Intercept Form of the Equation for a Linear Function
LOC: 10.RF6
TOP: Relations and Functions
KEY: Problem-Solving Skills
5. ANS:
a) The flat fee is: $50
When n T-shirts are ordered, the additional cost is: 8.95n dollars
So, an equation is:
b) Use the equation:
The total cost was $604.90.
c) Use the equation:
Jakub ordered 103 T-shirts.
PTS: 1
DIF: Moderate
REF: 6.4 Slope-Intercept Form of the Equation for a Linear Function
LOC: 10.RF6
TOP: Relations and Functions
KEY: Problem-Solving Skills
6. ANS:
a)
Identify the coordinates of one point on
the line and calculate the slope.
y
4
The coordinates of one point are (–2, –3).
To calculate the slope, m, use:
2
–4
–2
0
2
4
x
–2
(–2, –3)
–4
Use the slope-point form of the equation.
Substitute
,
, and
.
In slope-point form, the equation of the line is:
b)
In slope-intercept form, the equation of the line is:
From the equation, the y-intercept is –2.
PTS: 1
DIF: Moderate
REF: 6.5 Slope-Point Form of the Equation for a Linear Function
LOC: 10.RF7
TOP: Relations and Functions
KEY: Problem-Solving Skills
7. ANS:
a) The increase in girls playing organized ice hockey is 4162 girls per year.
This is the slope of a graph of this function.
The year 2000 is
years after 1990.
So, an ordered pair that satisfies this function is: (10, 45 400)
Use the slope-point form of the equation of a linear function:
Replace y with n, and replace x with t.
Substitute
,
, and
into the equation.
b) The year 2009 is 19 years after 1990.
Substitute t = 19 in the equation:
There were approximately 82 858 girls playing organized ice hockey in January 2009.
PTS: 1
DIF: Difficult
REF: 6.5 Slope-Point Form of the Equation for a Linear Function
LOC: 10.RF7
TOP: Relations and Functions
KEY: Problem-Solving Skills
8. ANS:
a)
so two points on the graph have coordinates A(19 375, 3875) and B(20 000, 3900).
Use this form for the equation of a linear function:
Substitute:
,
,
, and
In slope-point form, the equation that represents this function is:
b) Use:
Substitute:
When the annual operating cost is approximately $4350, the car has been driven 31 250 km.
PTS: 1
DIF: Difficult
REF: 6.5 Slope-Point Form of the Equation for a Linear Function
LOC: 10.RF7
TOP: Relations and Functions
KEY: Problem-Solving Skills
9. ANS:
Sketch the line with equation:
y
,
and mark a point at B(–1, 3).
B (–1, 3)
Compare the equation:
4
2
with the
equation:
–4
The slope of the line is
–2
0
2
4
x
–2
.
–4
a) Any line parallel to
has slope
.
The required line passes through B(–1, 3).
Use:
Substitute
,
, and
The line that is parallel to the line
.
and passes through B(–1, 3) has equation:
b) Any line perpendicular to
is
has a slope that is the negative reciprocal of
; that is, its slope
.
The required line passes through B(–1, 3).
Use:
Substitute
,
, and
.
The line that is perpendicular to the line
and passes through B(–1, 3) has equation:
PTS: 1
DIF: Moderate
REF: 6.5 Slope-Point Form of the Equation for a Linear Function
LOC: 10.RF7
TOP: Relations and Functions
KEY: Problem-Solving Skills