Algebra II
Unit 2
1. Given the line 2x+4y=8
a) Change to slope-intercept form:
b) What is the slope?
c) What is the y-intercept?
d) Graph:
2. Given the line 2y–6x=–4
a) Change to slope-intercept form:
b) What is the slope?
c) What is the y-intercept?
d) Graph:
For each of the following problems, state the slope and graph the line.
1. x=–2
2. y=–3
3. x=0
4. y= 4
Slope:_____
Slope:_____
Slope:_____
Slope:_____
1
Algebra II
1.
Unit 2
Sketch the line 2x + 3y = 6
y
2.
Sketch the line 2x – 3y = 6
y
3.
Sketch the line 4x – 5y = –20
y
2
Algebra II
1.
Unit 2
3
Sketch the line 3y + 5x = –15
y
2. Write the equation of a line in point slope form of a line through the point (1, 2) with a slope
of 3.
1
3. Write the equation of a line in point slope form through the point (7, –2) with a slope of − .
2
4. Write the equation of a line in point slope form through the point (–4, 3) with a slope of
5. Write the equation of a line in point slope form through the points (1, –3) and (2, –5).
6. Write the equation of a line in point slope form through the points (2, 3) and (4, –5).
7
.
5
Algebra II
Unit 2
4
1. Write the equation of a line in point slope form through the points (7, –3) and (2, –5).
2. Write the equation of a line through the points (7, –5) and (4, –5).
3. Write the equation of a line through the points (1, –3) and (1, –5).
4. Write the equation of a line in point slope form through the points (2, 3) and (4, –5).
5. Find equivalent function rules.
a) (3x + 7) + (2x + 9)
b) (-4x – 5) + ( 2x + 8)
c) (3x + 2)(x – 7)
d) (-2x + 3) – (6x + 8)
e) (9x – 1) – ( –3x + 7)
f) (x +4)(x – 4)
Which of your answers is the only function rule that represents more than a two step process?
Algebra II
Unit 2
5
Find the pattern in the in/out charts below, identify a possible function symbolically, then fill in
the rest of the chart. Show work in space below. These may not all be linear functions!
Given the following functions:
f(x) = 2x + 3
g(x) = x – 5
h(x) = –3x + 5 j(x) = x + 3
l(x) = 2x – 1
m(x) = x – 3
Please combine and simplify each of the following:
k(x) = x + 5
Algebra II
Unit 2
1. Find the pattern for the following function.
2. f (x) = −6x +12
a) What kind of function?
b) y intercept?
c) zero?
d) point (–3, ____)
e) graph
3. f (x) = x 2 − 4x − 5
a) What kind of function?
b) y intercept?
c) zeros?
d) point (–3, ____)
e) graph
6
Algebra II
Unit 2
Given the following functions:
f(x) = 2x + 3
g(x) = x – 5
l(x) = 2x – 1
m(x) = x – 3
h(x) = –3x + 5
Evaluate:
1. f (2)
2. h(−10)
3. k(7)
4. f (?) = 31
5. h(b) = 20
b=?
6. l(z) = −41
z =?
7. f (x) = −8x + 6
a) What kind of function?
b) y intercept?
c) zero?
d) point (–4, ____)
e) graph
7
j(x) = x + 3
k(x) = x + 5
Algebra II
Unit 2
8
1. f (x) = x 2 − 3x + 2
a) What kind of function?
b) y intercept?
c) zeros?
d) point (–3, ____)
e) graph
2.
Write the equation of a line in point slope form through the point (–6, 5) and parallel to
3
the line y = x − 7 .
2
3.
Write the equation of a line in point slope form through the point (4, 3) and perpendicular
1
to the line y = − x .
2
4.
Write the equation of a line through the point (3, 4) and parallel to the line
y = 8 . (What is the slope?)
5.
Write the equation of a line through the point (2, –5) and perpendicular to y = 8 . (What
is the slope?)
6.
Find the x-intercept and y- intercept of the following lines.
a)
4x − 2y = 20
7.
Find the slope of the line through the following points.
a)
(8, 1) (2, –3)
b)
3x + 4y = 5
c)
b)
y = 4x − 5
(0, 9) (9, 0)
Algebra II
Unit 2
9
1.
Write the equation of the line given the following information.
(you decide the BEST form the equation should be)
a)
slope of 5, through the point (2, –3)
b)
slope of 3, y-intercept of 7
c)
vertical through (1, –9)
d)
horizontal through (2, –7)
e)
undefined slope through (8, 5)
f)
zero slope through (0, 5)
g)
parallel to the line y = 4x − 5 , through the point (2, –3)
h)
perpendicular to the line y = 3x + 2 , through the point (0, 7)
i)
perpendicular to the line y =
j)
through the points (2, 5) and (–3, 7)
2
x − 4 , through the point (5, 6)
3
Algebra II
Unit 2
10
Systems of Equations and Solving
What is a system of equations? A set of two or more equations
What does it mean to solve a system? To find the point(s) of intersection(s) of the graphs of the
equations. This will be the solution to the problem.
To begin, we will be solving linear systems or systems composed of lines. What are the different
types of solutions that are possible?
Case 1: Intersecting
Case 2: Parallel
Case 3: Overlapping
Number of solutions:
Number of solutions:
Number of solutions:
one
none
infinite
There are four ways to solve any system: graphing, using a graphing calculator, substitution, or
linear combination (addition method). We will now explore these methods and determine when
to use each.
METHOD 1: SOLVE BY GRAPHING
Graph both lines and find the point of intersection. This method is least accurate and is helpful
primarily in gaining a visual understanding of what solving a system means.
Case 1: Intersecting Lines
Solve:
y = -x+5
y = x-1
Graph:
The solution is the point of intersection, (3, 2).
Check this solution by plugging (3, 2) into
both equations.
Algebra II
Unit 2
Case 2: Parallel Lines
1
y = − x+2
Solve:
3
3y = −x + 12
Put the second equation into slope-intercept form.
3y
x 12
=− +
3
3 3
1
y = − x+4
3
Graph:
There is no point of intersection, so
there is no solution.
Notice the lines have the same slope
(parallel).
Case 3: Overlapping Lines
Solve:
y = –3x – 4
2y = –6x – 8
Put the second equation into slope-intercept form.
2y −6x 8
=
−
2
2
2 --> You can see they are the same equation!
y = −3x − 4
Graph:
Solution: All points on the
line (an infinite number)
11
Algebra II
Unit 2
12
Method 2: Solve with a Graphing Calculator
This method is helpful when the numbers in the system are messy. The system equations must be
in “y=” form to enter them into the calculator.
Example 1: Using a graphing calculator, solve the following system of equations:
y = 2x + 1
y = −x − 3
Step 1: Type the
key.
Step 2: Type in the 2 equations:
Hit:
Your screen should look like this:
Step 3: Now graph the 2 lines:
Hit:
Your screen should look like this:
Step 4: Now calculate the point of intersection:
Hit:
Your screen should look like this:
So the solution is (–1.33, –1.67).
Algebra II
Unit 2
13
Example 2: Using a graphing calculator, solve the following system of equations:
1
3y − 2x = 9
y + x = 22
2
Step 1: First you must solve each equation for y = .
Step 2: Type the
key and type in the 2 equations.
Your screen should look like this:
Step 3: Now graph the 2 lines:
Your screen should look like this:
Step 4: Notice there is only one line in the window(–10 < x < 10 and –10 < y < 10).
Let's zoom out:
Hit:
Your screen should look like this:
Step 5: Now calculate the point of intersection:
Your screen should look like this:
So the solution is (16.29, 13.86).
Algebra II
Unit 2
14
Use the graphing calculator to answer questions 1 and 2. Round all answers to 3 decimal places:
1. f (x) = .23x − 6.254
a) What is the y intercept?
b) What is the x intercept?
c) What is f (4)?
d) What is f (12)?
2. Sketch 2x + 5y = 9 and 3x − 8y = 10 and find the point of intersection.
3. Graph the equations of the two lines (without your calculator) and find the point of
intersection.
x= 1
4.
x= 5
y = –4
x= –8
Solution: _______
Solution: _______
Algebra II
Unit 2
15
Use the addition method of solving systems of equations to solve for x and y:
1.
–x + y = 4
2.
x
−4
2
2y + x = −8
y=
2x + y = 7
3.
4.
y = 8x - 12
y=−
x
+ 23
3
x+y = 40
x – y = -30
Algebra II
Unit 2
16
5.
6.
y = 2x – 15
y + 2x = – 14
y + 4x = 18
y – 3x = 10
Solve the following systems with the graphing calculator.
7.
2x–y = 10
8.
x+3y = –9
9.
11.
3x–2y=0
x+y = –5
y = 2x
x
y = −3
2
10.
y = 2x-3
y+3x = 7
12.
x+y = 4
x-y = 2
y = 3x-4
y - 3x = 1
Case 1: Intersecting Lines
Solve:
3x+7y = 17
y – 2x = 0
Solve one equation for x or y. Here, rewrite the second equation as y = 2x.
In the other equation, substitute the solved for variable with the expression it equals (replace y
with 2x) and solve for the remaining variable (x).
3x + 7y = 17
3x + 7(2x) = 17
3x + 14x = 17
17x = 17
Algebra II
Unit 2
17
x=1
Plug in this solution to evaluate the other variable.
y = 2x
y = 2(1) = 2
Solution: x = 1 and y = 2 or (1,2)
Case 2: Parallel Lines
Solve:
x=y+5
x - y =6
Solve one equation for x or y. This is already done in the first equation: x = y + 5
In the other equation, substitute the solved for variable with the expression it equals (replace x
with y+5) and solve for the remaining variable (y).
x – y =6
(y+5) – y = 6
5=6
Notice that the variable cancels out and you are left with a FALSE statement (5=6). There is no
solution. The lines are parallel and there is no point of intersection.
Case 3: Overlapping Lines
Solve:
x
+2
3
6y − 2x = 12
y=
Solve one equation for x or y. This is done in the first equation y =
x
+2
3
In the other equation, substitute the solved for variable with the expression it equals.
6y – 2x = 12
x
6( + 2 )– 2x = 12
3
2x + 12 –2x = 12
12= 12
Notice that the variable cancels out again but you are left with an ALWAYS TRUE statement.
The lines are overlapping (the same) and there are an infinite number of solutions.
Algebra II
Unit 2
18
Problems: Solve by Addition method.
1.
3x+2y = 9
y = 3x
3.
x = 10y
x - 4y = 12
2.
y = 4x - 12
3x+4y = 66
4.
y = x+4
3x + y = 4
Algebra II
5.
x = 5 - 3y
-3x+2y = 18
7.
y = 37 – 3x
2x – 3y = 21
Unit 2
19
6.
8.
y = –3x+7
x+y = 1
3x–5y = 32
y = 17 – 2x
Algebra II
Unit 2
20
9. x = y+4
2x+3y = –2
11.
2x+7y = 31
x = 3y – 17
13.
y = 4–2x
3x – 2y = –50
14.
10.
3x –3y = –12
x=y–4
12.
3x+y = 2
y = –3x
y = 3x – 1
9x+2y = 3
Algebra II
Unit 2
21
Solve each of the following system of equations using addition method:
Be sure to check your answers.
1.
x+y = 4
2.
4x+ y = –5
x –y = 2
2x – 3y = –13
3.
2x–7y = 8
3x–4y = –1
4.
7x–5y = –2
-8x–y = 9
Algebra II
Unit 2
22
5.
5x+y = 15
3x+2y = 9
6.
4x+3y = –2
8x–2y = 12
7.
5x–2y = 8
3x–5y = 1
8.
4x+3y = –6
10x+2y = 7
Algebra II
Unit 2
23
Set up variables, then write the systems for each of the following equations and solve.
1. The first time Damon and Jen went to the restaurant, they ordered four hamburgers
and three hot dogs. The bill was $15. the next time they ordered two hamburgers and
five hot dogs and the bill was $11. How much did each hamburger and each hot dog
cost?
2. The ice cream person sold 3 times as many popsicles as ice cream bars. The popsicles
sold for $.20 and the ice cream bars sold for $.30 for a total of $8.10. How many of each
treat were sold?
Algebra II
Unit 2
24
3. In your pocket you have a handful of change worth $2.40. Your pocket contains 24
coins, all either pennies or quarters. How many of each coin do you have?
4. Paula bought six Cub hats and four Red Sox hats for a total of $38. Nancy bought 3
Cub hats and three Red Sox hats for a total of $24. How much did each hat cost?
Algebra II
Unit 2
25
5. A certain toll booth charges vehicles passing through according to vehicle type. Cars
are charged $.50 and trucks are charged $1.25. During one afternoon, a total of $72.00
was collected from 108 vehicles that were either cars or trucks. How many cars and how
many trucks were there?
Review Problems: