Going Around the Curve 1
Experiment A
A particular mould grows in the following way: If there is one “blob” of mould today, then there will be 4 tomorrow, 9 the next day, 16 the next day, and so on.
Model this relationship using linking cubes.
Purpose
Find the relationship between the side length and the number of cubes.
Hypothesis
What type of relationship do you think exists between the side length and the number of cubes?
Procedure
1. Build the following sequence of models, using the cubes.
2. Build the next model in the sequence.
Mathematical Models
Complete the table, including first and second differences.
Make a scatter plot and a line of best fit.
First Differences
0 / 0 / Second Difference
Going Around the Curve 2
Experiment B
Jenny wants to build a square pool for her pet iguana. She plans to buy tiles to place around the edge to make a full play area for her pet.
Model the relationship, comparing total play area (pool combined within the edging) to the side length of the pool, using linking cubes.
Purpose
Find the relationship between the side length of the pool (shaded inside square) and the total play area.
Hypothesis
What type of relationship do you think exists between the side length and the play area?
Procedure
1. Build the following sequence of models using the cubes.
Note: The pool is the shaded square, the tiles are white.
2. Build the next model in the sequence.
Mathematical Models
Complete the table, including first and second differences.
Make a scatter plot and a line of best fit.
First Differences
1 / Second Difference
Going Around the Curve 3
Experiment C
A particular mould grows in the following way: If there is one “blob” of mould today, then there will be 3 tomorrow, and 6 the next day.
Model this relationship using linking cubes.
Purpose
Find the relationship between the number of cubes in the bottom row and the total number of cubes.
Hypothesis
What type of relationship do you think exists between the number of cubes in the bottom row and the total number of cubes?
Procedure
1. Build the following sequence of models using the cubes.
2. Build the next model in the sequence.
Mathematical Models
Complete the table, including first and second differences.
Make a scatter plot and a line of best fit.
First Differences
Second Difference
Going Around the Curve 4
Experiment D
Luisa is designing an apartment building in a pyramid design. Each apartment is a square.
She wants to know how many apartments can be built in this design as the number of apartments on the ground floor increases.
Model this relationship, using linking cubes.
Purpose
Find the relationship between the number of cubes in the bottom row and the total number of cubes.
Hypothesis
What type of relationship do you think exists between the number of cubes in the bottom row and the total number of cubes?
Procedure
1. Build the following sequence of models using the cubes.
2. Build the next model in the sequence.
Mathematical Models
Complete the table, including first and second differences.
Make a scatter plot and a line of best fit.
First Differences
0 / 0 / Second Difference
Going Around the Curve 5
Experiment E
Liz has a beautiful pond in her yard and wants to build a tower beside it using rocks. She is unsure how big she will make it and how many rocks she will need. She is particularly concerned to have the nicest rocks showing.
Model the relationship comparing the length of the base to the number of visible rocks using linking cubes.
Purpose
Find the relationship between the number of cubes on the side of the base and the total number of unhidden cubes.
Hypothesis
What type of relationship do you think exists between the length of the side of the base and the number of visible cubes?
Procedure
1. Build the following sequence of models using the cubes.
2. Build the next model in the sequence.
Mathematical Models
Complete the table, including first and second differences.
Make a scatter plot and a line of best fit.
First Differences
Second Difference
Key Features of Quadratic Relations 6
Terminology / Definition / How Do ILabel It? / Graph A / Graph B
Vertex / The maximum or minimum point on the graph. It is the point where the graph changes direction. / (x,y)
Minimum/ maximum value
Axis of symmetry
y-intercept
x-intercepts
Zeros
Label the graphs using the correct terminology.
Graph A Graph B
Key Features of a Parabola 7
Write the feature of a parabola that you were given in the centre of the graphic. Complete the chart. Include sketches and graphs with your work.
Definition / Facts/CharacteristicsExamples / Non-examples
Quadratics Practice 8
Each parabola has the following: an axis of symmetry
a vertex
a y – intercept
0, 1, or 2 x-intercepts (aka “roots” or “zeros”)
a direction of opening (up or down)
a maximum or minimum value
For the parabolas in #1 – 6, state the required information.
1.
Equation of the axis of symmetry: ______
Vertex: ______
Zero(s): ______
Direction of opening: ______
y-intercept: ______
max/min value (circle): ______
2.
Equation of the axis of symmetry: ______
Vertex: ______
Zero(s): ______
Direction of opening: ______
y-intercept: ______
max/min value (circle): ______
3.
Equation of the axis of symmetry: ______
Vertex: ______
Zero(s): ______
Direction of opening: ______
y-intercept: ______
max/min value (circle): ______
4. 9
Equation of the axis of symmetry: ______
Vertex: ______
Zero(s): ______
Direction of opening: ______
y-intercept: ______
max/min value (circle): ______
5.
Equation of the axis of symmetry: ______
Vertex: ______
Zero(s): ______
Direction of opening: ______
y-intercept: ______
max/min value (circle): ______
6.
Equation of the axis of symmetry: ______
Vertex: ______
Zero(s): ______
Direction of opening: ______
y-intercept: ______
max/min value (circle): ______
For #7 – 12, sketch the parabola that has the given description. 10
7.
Equation of the axis of symmetry: x = 2
Zero(s): -1 and 5
Minimum Value: -4
8.
Vertex: (-2, 4)
y-intercept: -1
9.
Zero(s): -2 and 2
Minimum value: -3
11
10.
Zero: -2
y-intercept: 5
11.
Equation of the axis of symmetry: x = 2
y-intercept: -2
Maximum: 3
12.
Vertex: (-3, -1)
y-intercept: -3
Is it a Line or a Parabola? 12
Fill in the following table, then state whether each set of data best models a line or a parabola.
1.
x / y / First Difference / Second Difference0 / 12
1 / 3
2 / 0
3 / 3
4 / 12
5 / 27
Therefore, the data best models a ______
2.
x / y / First Difference / Second Difference5 / 21
7 / 28
9 / 35
11 / 42
13 / 49
15 / 56
Therefore, the data best models a ______
3. 13
x / y / First Difference / Second Difference11 / 1.4
12 / 1.1
13 / 0.8
14 / 0.5
15 / 0.2
16 / -0.1
Therefore, ______
4.
x / y / First Difference / Second Difference1 / -4.9
2 / -19.6
3 / -44.1
4 / -78.4
5 / -122.5
6 / -176.4
Therefore, ______
5.
x / y / First Difference / Second Difference11 / 1.4
12 / 1.1
13 / 0.8
14 / 0.5
15 / 0.2
16 / -0.1
Therefore, the data best models a ______
6. 14
x / y / First Difference / Second Difference4
-3
8
-3
12
-3
16 / -2
-3
20
-3
24
Therefore, ______
7.
x / y / First Difference / Second Difference-5
5 / 2
15 / 4 / 2
1
25 / 2
35 / 2
45
Therefore, ______
Summary of Quadratics…..so far 15
1. Label the diagram using the terms given below.
Note: not all terms need to be used and some terms can be used more than once.
Terms: axis of symmetry vertex maximum
zero (x-intercept) y-intercept minimum
2. Describe the function below, remembering to include all key parts.
16
3. Indicate whether the tables show a relation that is linear, quadratic, or neither. Explain how you know.
x / Y0 / 1
1 / 1
2 / 2
3 / 3
4 / 5
5 / 8
6 / 13
x / y
0 / 16
1 / 9
2 / 4
3 / 1
4 / 0
5 / 1
6 / 4
x / Y
0 / 1
1 / 3
2 / 5
3 / 7
4 / 9
5 / 11
6 / 13
4. True or False:
_____ The axis of symmetry always goes through the y-intercept.
_____ The vertex is always located halfway between the zeroes.
_____ The y-coordinate of the vertex is always the same as the maximum or minimum value.
_____ The x-coordinate of the vertex is always the same as the axis of symmetry.
_____ A parabola must open up.
_____ The y-intercept is always positive.
17
5. Complete the table below using the diagrams of the parabolas.
Parabola GraphVertex
Maximum or Minimum Value
Axis of Symmetry
Zeroes
(x-intercept)
Direction of opening
y-intercept
6. Shelby has walked to a water tower beside a nearby gorge in order to launch her newly designed paper airplane. The graph shows the flight of the paper airplane. A negative height means the airplane is below the level of the ground. (Height in feet and time in seconds)
1. Estimate the height of the water tower. ______
2. How long does it take for the paper airplane to reach its minimum height? ______
3. How high is the minimum height? ______
4. When has the paper airplane reached ground level?
______
5. Write the vertex of this parabola. ______
6. Will the airplane continue in a parabolic path? Explain why or why not.
Algebra Tile Template 18
1. y = ( )( ) = ______/ 2. y = ( )( ) = ______3. y = ( )( ) = ______/ 4. y = ( )( ) = ______
5. y = ( )( ) = ______/ 6. y = ( )( ) = ______
Multiply a Binomial by a Binomial 19
Name:
Part A
Use algebra tiles to multiply binomials and simplify the following:
1. y = (x + 1)(x + 3) = ______/ 2. y = (x + 2)(x + 3) = ______Part B
Use the chart method to multiply and simplify the following:
1. y = (x + 1)(x + 3) = ______/ 2. y = (x + 2)(x +3) = ______3. y = (x + 2)(x – 1) = ______/ 4. y = (x – 2)(x + 3) = ______
5. y = (x – 1)(x – 1) = ______/ 6. y = (x – 1) (x – 2) = ______
Multiply a Binomial by a Binomial (continued) 20
Part C
Multiply and simplify the two binomials, using the chart method and the distributive property.
x / +4x
–3
1. (x + 4)(x – 3) / x / –3
x
–3
2. (x – 3)(x – 3)
x / +2
x
+2
3. (x + 2)2 / x / +2
x
–1
4. (x + 2)(x – 1)
x / –2
x
+1
5. (x – 2) (x + 1) / x / –1
x
–1
6. (x – 1)2
x / –1
x
–2
7. (x – 1)(x – 2) / x / –3
x
–4
8. (x – 3)(x – 4)
Chart Template for Distributive Property 21
1. y = ( )( ) = ______/ 2. y = ( )( ) = ______3. y = ( )( ) = ______/ 4. y = ( )( ) = ______
5. y = ( )( ) = ______/ 6. y = ( )( ) = ______
7. y = ( )( ) = ______/ 8. y = ( )( ) = ______
Multiplying Binomials Practice Sheet 22
Expand and simplify. Check by graphing, using a graphing calculator.
1. (x + 3)(x + 2) 2. (7 + a)(4 + a) 3. (b + 5)(10 + b)
4. (2v + 5)(v + 4) 5. (4x + 5)(x – 5) 6. (m – 2)(m + 3)
7. (4 – n)(7 + n) 8. (2k – 1)(3k + 2) 9. (4x – 6)(3 + 4x)
10. (2m + 1)(2m – 1) 11. (5a + 3)(5a – 3) 12. (6 – y)(6 + y)
13. (x – 4)2 14. (3v + 5)2 15. (7 – a)2
Determine an expression for the area of each rectangle below:
16. 17.
x - 11 2x - 26
x + 1 x + 2
Finding the y-Intercept of a Quadratic Equation 23
1. Use the graphing calculator to find the y-intercept for each of the equations:
Note any patterns you see.
Equation / y-intercept2. How can you determine the y-intercept by looking at a quadratic equation?
3. Which form of the quadratic equation is easiest to use to determine the y-intercept?
Explain your choice.
4. Using your conclusion from question 2, state the y-intercept of each and check using a graphing calculator.
Equation / y-intercept / Does it check?Yes / No
5. Explain the connection between the y-intercept and the value of y when x = 0.
Quadratic Equations 24
1. Find the y-intercept for each of the following quadratic equations given in factored form. Write the equations in standard form. Show your work.
a) / standard form:y-intercept:
b) / standard form:
y-intercept:
c) / standard form:
y-intercept:
standard form:
y-intercept:
2. Find the y-intercept for each of the following quadratic equations:
a) / b)y-intercept: / y-intercept:
Finding the x-Intercepts of a Quadratic Equation 25