Pumpkin Regression & Models
At a pumpkin patch, four fields were fertilized a different number of times. The table shows the yield of pumpkins per acre.
| Field | # times fertilized | Yield per acre |
|---|---|---|
| A | 1 | 3 |
| B | 2 | 5 |
| C | 3 | 7 |
| D | 4 | 9 |
Which linear equation best represents the given data?
A. y = 3x + 2
B. y = 2x + 1
C. y = −3x + 2
D. y = −2x + 1
B. y = 2x + 1
The yield increases by 2 each time. Using (1, 3), 3 = 2(1) + b, so b = 1.
Witchy Vertex Challenge
Write the quadratic function that has a vertex at (2, −1) and passes through (0, 1).
Choose a number for each box. Not all answers will be used.
Number bank: 2 | −1 | −2 | 1 | 0 | 4 | 0.5 | −0.5
f(x) = [ ](x − [ ])2 + [ ]
Blanks: 0.5, 2, −1.
f(x) = 0.5(x − 2)2 − 1
1 = a(0 − 2)2 − 1, so 2 = 4a and a = 0.5.
Ghostly Conjugates
Determine the product of (1 + i)(1 − i).
Answer: [ ]
2
(1 + i)(1 − i) = 1 − i2 = 1 + 1 = 2.
Spooky Quadratic Challenge
Solve the following equation.
x2 + 2x + 3 = 0
A. 1 ± 2i√2
B. 1 ± i√2
C. −1 ± 2i√2
D. −1 ± i√2
D. x = −1 ± i√2
Haunted Roots
For which of the following equations is x = 2 the only solution?
A. (x − 2)2 = 0
B. (x − 2)(x + 2) = 0
C. (x + 2)2 = 0
D. (2x)2 = 0
A. (x − 2)2 = 0
The repeated factor gives only x = 2.
A parabola in a pumpkin-patch design passes through (−1, −2), (0, −2), and (2, 4). Given y = ax2 + bx + c, what are the values of a, b, and c?
A. a = 1, b = 2, c = 1
B. a = 2, b = 1, c = −2
C. a = −2, b = 1, c = 1
D. a = 1, b = 1, c = −2
D. a = 1, b = 1, c = −2
x = 0 gives c = −2. The other points give a − b = 0 and 4a + 2b = 6, so a = b = 1.
Witchy Vertex Challenge
Which function represents a parabola that opens downward, has a vertex of (1, −1), and goes through (3, −3)?
A. f(x) = −0.5(x − 1)2 + 1
B. f(x) = −(x − 1)2 − 1
C. f(x) = (x − 1)2 − 1
D. f(x) = −0.5(x − 1)2 − 1
D. f(x) = −0.5(x − 1)2 − 1
−3 = a(3 − 1)2 − 1, so −2 = 4a and a = −0.5.
Ghostly Complex Numbers
Which number is equivalent to (−2 − i) + 2i − (3 + 3i)?
A. −5 + 6i
B. −5 − 2i
C. 1 − 2i
D. 1 + 6i
B. −5 − 2i
Real: −2 − 3 = −5. Imaginary: −1 + 2 − 3 = −2.
Spooky Quadratic Challenge
Function r is defined as r(x) = 2x2 + x − 1. What are the solutions to r(x) = 0?
Select TWO correct answers.
A. x = −2
B. x = −1
C. x = 1
D. x = 2
E. x = 1/2
B and E. x = −1 and x = 1/2
Haunted Vertex Challenge
Transform f(x) = x2 + 2x + 2 to vertex form and identify the vertex.
A. f(x) = (x + 1)2 − 1; Vertex: (1, −1)
B. f(x) = (x + 1)2 + 1; Vertex: (1, −1)
C. f(x) = (x + 1)2 + 1; Vertex: (−1, 1)
D. f(x) = (x + 1)2 − 1; Vertex: (−1, −1)
C. f(x) = (x + 1)2 + 1; Vertex: (−1, 1)
x2 + 2x + 2 = (x + 1)2 + 1. The vertex is (−1, 1).
At a pumpkin patch, four fields were fertilized a different number of times. The table shows the yield of pumpkins per acre.
| Field | # times fertilized | Yield per acre |
|---|---|---|
| A | 1 | 4 |
| B | 2 | 5 |
| C | 3 | 7 |
| D | 4 | 8 |
Which linear equation best represents the given data?
A. y = 1.5x + 3
B. y = 1.4x + 2.5
C. y = −1.5x + 3
D. y = −1.4x + 2.5
B. y = 1.4x + 2.5
Linear regression gives slope 1.4 and intercept 2.5.
Witchy Vertex Challenge
Write the quadratic function that has a vertex at (3, −2) and passes through (1, 0).
Choose a number for each box. Not all answers will be used.
Number bank: 3 | −2 | −3 | 2 | 1 | 0 | 0.5 | −0.5
f(x) = [ ](x − [ ])2 + [ ]
Blanks: 0.5, 3, −2.
f(x) = 0.5(x − 3)2 − 2
0 = a(1 − 3)2 − 2, so 2 = 4a and a = 0.5.
Ghostly Conjugates
Determine the product of (2 + i)(2 − i).
Answer: [ ]
5
(2 + i)(2 − i) = 4 − i2 = 4 + 1 = 5.
Spooky Quadratic Challenge
Solve the following equation.
x2 + 4x + 7 = 0
A. 2 ± 2i√3
B. 2 ± i√3
C. −2 ± 2i√3
D. −2 ± i√3
D. x = −2 ± i√3
Haunted Roots
For which of the following equations is x = 3 the only solution?
A. (x − 3)2 = 0
B. (x − 3)(x + 3) = 0
C. (x + 3)2 = 0
D. (3x)2 = 0
A. (x − 3)2 = 0
The repeated factor gives only x = 3.
A parabola passes through the points (−1, -6), (0, -8), and (2, 18). Given the standard form of a parabola, y = ax2 + bx + c, what are the values of a, b, and c?
A. a = 5, b = 6, c = 3
B. a = 3, b = 5, c = -8
C. a = -8, b = 5, c = 3
D. a = 5, b = 3, c = -8
D. a = 5, b = 3, c = -8
x = 0 gives c = -8. Substituting the other two points gives a = 5, b = 3.
Which function represents a parabola that opens downward, has a vertex of (8, -5), and goes through the point (14, -11)?
A. f(x) = −(1/6)(x − 8)2 + 5
B. f(x) = −(x − 8)2 − 5
C. f(x) = (x − 8)2 − 5
D. f(x) = −(1/6)(x − 8)2 − 5
D. f(x) = −(1/6)(x − 8)2 − 5
-11 = a(6)2 + (-5), so a = −1/6.
Which number is equivalent to (−10 − 5i) + 5i − (9 + 15i)?
A. -19 + 25i
B. -19 − 15i
C. -1 − 15i
D. -1 + 25i
B. -19 − 15i
Real: −10 − 9 = -19. Imaginary: −5 + 5 − 15 = -15.
Function r is defined as r(x) = 18x2 + 53x − 35. What are the solutions to r(x) = 0?
Select TWO correct answers.
A. x = −9/5
B. x = −7/2
C. x = 7/2
D. x = 9/5
E. x = 5/9
B and E. x = −7/2 and x = 5/9
Transform f(x) = 2x2 + 20x + 42 to vertex form and identify the vertex.
A. f(x) = (x + 5)2 − 10; Vertex: (5, -10)
B. f(x) = 2(x + 5)2 − 8; Vertex: (5, 8)
C. f(x) = 2(x + 5)2 − 8; Vertex: (−5, -8)
D. f(x) = (x + 5)2 − 10; Vertex: (−5, -10)
C. f(x) = 2(x + 5)2 − 8; Vertex: (−5, -8)
Complete the square: 2(x2 + 10x) + 42 = 2(x + 5)2 + (-8).
The following data represents the yield of a crop per acre for four fields fertilized a different number of times.
| Field | # times fertilized | Yield per acre |
|---|---|---|
| A | 1 | 55 |
| B | 2 | 62 |
| C | 3 | 66 |
| D | 4 | 69 |
Which linear equation best represents the given data?
A. y = 5.1x + 60
B. y = 4.6x + 51.5
C. y = −5.1x + 60
D. y = −4.6x + 51.5
B. y = 4.6x + 51.5
Linear regression gives slope 4.6 and intercept 51.5.
Write the quadratic function that has a vertex at (9, -6) and passes through the point (3, 12).
Choose the correct answer for each box. Not all answers will be used.
Number bank: 9 | -6 | -9 | 6 | 3 | 12 | 0.5 | -0.5
f(x) = [ ](x − [ ])2 + [ ]
Blanks: 0.5, 9, -6.
f(x) = 0.5(x − 9)2 − 6
12 = a(3 − 9)2 + (-6), so a = 18/36 = 0.5.
Determine the product of (9 + 6i)(9 − 6i).
Answer: [ ]
117
92 − (6i)2 = 81 + 36 = 117.
Solve the following equation.
2x2 + 28x + 124 = 0
A. 7 ± 8i√2
B. 7 ± i√13
C. −7 ± 8i√2
D. −7 ± i√13
D. x = −7 ± i√13
For which of the following equations is x = 11 the only solution?
A. (x − 11)2 = 0
B. (x − 11)(x + 11) = 0
C. (x + 11)2 = 0
D. (11x)2 = 0
A. (x − 11)2 = 0
The repeated factor gives only x = 11.