Lesson 4B.1 – y = x2 + k
Warm-Up: “The Base Parabola”
y = x2 is called the base parabola. (no transformations have been applied)
Confirm y = x2 is a quadratic relation by completing the following table of values. Then graph.
x
-3
y
First
Differences
Second
Differences
-2
-1
0
1
2
3
y
Vertex =

Step Pattern =
x





1
Investigation: “y = x2 + k”
Q1:
You were randomly assigned the equation of a parabola. Write it here: ________________
What do you think the graph of your parabola will look like?
A1:
 Test your above idea by completing the following table of values for your parabola.
 Neatly graph it on a piece of chart paper with the equation stated
x
clearly.
-3
 Check your graph with Desmos or a similar graphing App.
-2
 What did happen to the graph of y = x2 when it changed to your
parabola’s equation?
y
-1
0
1
 Compare everyone’s parabola graphs. What is the same? What is
different?
2
3
 Compare everyone’s equations. What can you now conclude:
Conclusion:
1) If k > 0, then y = x2 + k will __________________________________________
2) If k < 0, then y = x2 + k will __________________________________________
2
Q2:
How can we graph y = x2 + k without using a table of values?
A2:
Exs:
Graph each parabola, showing at least 5 clear points for each. Label each parabola.
a)
y = x2 – 9
b) y = x2 + 5
y

x





3
4B.1 Practice Questions
1.
Graph each parabola, showing at least 5 clear points for each. Label each parabola.
a)
y = x2 – 3
b) y = x2 + 1
c) y = x2 + 6
d) y = x2 - 10
y

x





2.
A parabola is shifted down 7 units. What would be the parabola’s equation?
3.
A parabola is shifted up 8 units. What would be the parabola’s equation?
4
4.
a) What do you think y = -x2 looks like?
b) Confirm your above idea by completing the following table of values for y = -x2. Then
graph. Check your graph with Desmos or a similar graphing App.
x
-3
y
First
Differences
Second
Differences
-2
-1
0
1
2
3
c)
Now graph y = -x2 + 9.
y

x





5
Lesson 4B.2 – y = (x - h)2
Warm-Up: “The Base Parabola”
y = x2 is called the base parabola. (no transformations have been applied)
Recall:
Graph y = x2.
y
Vertex =
Step Pattern =

x





Recall:
3) If k > 0, then y = x2 + k will __________________________________________
4) If k < 0, then y = x2 + k will __________________________________________
6
Investigation: “y = (x – h)2 ”
Q1:
You were randomly given the equation of a parabola. Write it here: ________________
What do you think the graph of your parabola will look like?
A1:
 Test your above idea by completing the table of values for your parabola that is on your slip
of paper.
 Neatly graph it on a piece of chart paper with the equation stated clearly.
 Check your graph with Desmos or a similar graphing App.
 What did happen to the graph of y = x2 when it changed to your parabola’s equation?
 Compare everyone’s parabola graphs. What is the same? What is different?
 Compare everyone’s equations. What can you now conclude:
Conclusion:
5) If h > 0, then y = (x – h)2 will __________________________________________
6) If h < 0, then y = (x – h)2 will __________________________________________
7
Q2:
How can we graph y = (x – h)2 without using a table of values?
A2:
Exs:
Graph each parabola, showing at least 5 clear points for each. Label each parabola.
b)
y = (x – 4)2
b) y = (x + 5)2
y

x





8
4B.2 Practice Questions
1.
Graph each parabola, showing at least 5 clear points for each. Label each parabola.
d)
y = (x – 3)2
b) y = (x + 1)2
c)
y = (x + 6)2
d) y = (x – 7)2
y

x





2.
A parabola is shifted left 10 units. What would be the parabola’s equation?
3.
A parabola is shifted right 8 units. What would be the parabola’s equation?
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4.
a) What do you think y = -(x – 3)2 looks like?
e) Confirm your above idea by completing the following table of values for y = -(x – 3)2. Then
graph. Check your graph with Desmos or a similar graphing App.
x
0
y
First
Differences
Second
Differences
1
2
3
4
5
6
y

x





10
Lesson 4B.3 – y = (x - h)2 + k
Warm-Up: “The Base Parabola ”
y = x2 is called the base parabola. (no transformations have been applied)
Recall:
Graph y = x2.
y
Vertex =
Step Pattern =

x





Recall:
7) If k > 0, then y = x2 + k will __________________________________________
8) If k < 0, then y = x2 + k will __________________________________________
9) If h > 0, then y = (x – h)2 will __________________________________________
10) If h < 0, then y = (x – h)2 will __________________________________________
11
Investigation: “y = (x – h)2 + k ”
Q1:
What do you think would happen if the base parabola’s equation was changed to
y = (x - 3)2 + 2 ?
A1:
 Test your above idea by completing the following table of values for y = (x - 3)2 + 2and then
graphing.
 Check your graph with Desmos or a similar graphing App.
 What did happen to the graph of y = x2 when it changed to y = (x - 3)2 + 2?
x
6
y
y
5
4

3
2
1
0
x





12
Dice Game “Rolling With My Parabola”
1.
2.
3.
4.
5.
6.
7.
8.
With a partner:
Roll the dice. Write your number here: _______
Even number = up that many units
Odd number = down that many units
Which way and how many units must your parabola move? _______________________
Roll the dice. Write your number here: _______
Even number = right that many units
Odd number = left that many units
Which way and how many units must your parabola move? _______________________
Combine your above two moves and graph your parabola on a piece of chart paper. Think
about where your vertex should be first.
What equation describes your parabola? Write it here and on the chart paper beside your
graph.
Conclusion:
How can we graph y = (x – h)2 + k without using a table of values?
13
Exs:
Graph each parabola, showing at least 5 clear points for each. Label each parabola.
c)
y = (x – 4)2 - 6
b) y = (x + 5)2 + 3
c) y = (x – 2)2 + 4
d) y = (x + 3)2 - 5
y

x





14
4B.3 Practice Questions
1.
Graph each parabola, showing at least 5 clear points for each. Label each parabola.
f)
y = (x – 3)2 - 8
b) y = (x + 1)2 + 5
c)
y = (x + 6)2 - 4
d) y = (x – 7)2 + 2
y

x





2.
A parabola is shifted left 10 units and up 2 units. What would be the parabola’s equation?
3.
A parabola is shifted right 8 units and down 1 unit. What would be the parabola’s equation?
15
4.
a) What do you think y = -(x – 3)2 + 2 looks like?
g) Confirm your above idea by completing the following table of values for y = -(x – 3)2 + 2.
Then graph. Check your graph with Desmos or a similar graphing App.
x
0
y
First
Differences
Second
Differences
1
2
3
4
5
6
y

x





16
Lesson 4B.4 – y = a(x-h)2+k
Recall:
1) If k > 0, then y = x2 + k will __________________________________________
2) If k < 0, then y = x2 + k will __________________________________________
3) If h > 0, then y = (x – h)2 will __________________________________________
4) If h < 0, then y = (x – h)2 will __________________________________________
Investigation: “y = a (x – h)2 + k ”
Q1:
What do you think would happen if the base parabola’s equation was changed to
y = 2 (x - 3)2 - 5 ?
A1:
 Test your above idea by completing the following table of values for y = 2 (x - 3)2 - 5 and then
graphing.
 Check your graph with Desmos or a similar graphing App.
 What did happen to the graph of y = x2 when it changed to y = 2 (x - 3)2 -5 ?
y
x
6
y

5
4
3
2
x



1
0


17
4B.4 Practice Questions
1.
Graph each parabola, showing at least 5 clear points for each. Label each parabola.
a)
y = 3 (x – 3)2 - 8
b) y = -3 (x + 1)2 + 5
c)
y = 2 (x + 6)2 - 4
d) y = - (x – 7)2 + 2
y

x





2.
A parabola is shifted left 10 units and up 2 units and flipped upside down. What would be
the parabola’s equation?
3.
A parabola is shifted right 8 units, down 1 unit and has a step pattern of 2, 6, 10…. What
would be the parabola’s equation?
18
Midterm Exam Review Lesson: Units 1-4
Date: _____________
1.
Graph line A: y = -x + 1
2.
Graph line B: x – 2y + 8 = 0
3.
Graphing lines A and B together has created a linear _________________. The point of
intersection is _______________.
4.
Perform LS/RS checks to prove your P.O.I. is correct.
19
5.
Solve the previous linear system by using the Equating Equations method.
y = -x + 1
x – 2y + 8 = 0
6.
Solve
a) 7(2x + 3) = 35x – 21
b)
3
5
𝑥−6=3
20
7.
State the following features for the parabola shown.
Vertex ____________
Zeros ______________
Axis of Symmetry _________ Y-Intercept _________
Direction of Opening ___________
Optimal Value ____________
8.
Max or Min? __________
a) Draw a parabola with:


zeros at 2 and 8
maximum value of 5
b) State three more features that are true for this
parabola.
9.
a) State an equation for a linear relation.
___________________________
b) State the equations of two parallel lines. ___________________ ___________________
c) State the equation of a horizontal line. __________________
d) State the equation of a vertical line. ________________
e) State an equation for a quadratic relation. _________________________
21
Midterm Test Practice Questions
Date: _____________
1.
Graph each line.
y
slope
y-intercept
y = 2x + 7

3
y = −5𝑥 + 1
x
y = 4x



y = -10


2.
Calculate the slope of the line that passes through points (5, 8) and (2, -10).
22
3.
Solve.
a)
4.
12x – 7 = 9x – 22
b)
4
7
𝑥−3=5
Express 6x + 4y – 12 = 0 in y = mx + b form and then graph the line.
y

x





5.
Rearrange the following area formula to isolate “h”.
𝑨=
𝒃𝒉
𝟐
23
6.
Solve the following linear system using each method stated.
y = 2x – 7
2x + 4y = -8
a) Graphing
y

x




b)
Equating Equations

24
7.
Stephen wants to rent a motor boat. He finds the following two rental companies:
“Speedsters”
“Water Sports”
charges $250 plus $70 per day
charges $170 plus $90 per day
Determine when it is cheaper to rent from “Speedsters”.
8.
State the following for the given parabola.
a) Vertex
__________________
b) Max/min value & which one
c) Zeros
__________________
d) Axis of Symmetry ______________
e) Y-intercept ________________
f) Direction of Opening __________
9.
How can we tell a relation is quadratic when given the:
a) table of values
b) equation
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