#### Solved Examples and Worksheet for Writing Quadratic Equations for Line of Fit

Q1Which of the following equations best represents the graph? A. y = x2 - 2x + 1
B. y = - x2 - 2x + 1
C. y = - x2 - 2x3 + 1
D. y = x3 + 2x + 1

Step: 1
The equations y = x3 + 2x + 1 and y = - x2 - 2x3 + 1 do not represent curves as they are not quadratic.
Step: 2
In the equation, y = x2 - 2x + 1, the values of a, b and c are 1, - 2 and 1 respectively.
Step: 3
The x-coordinate of the vertex = -b2a = -(-2)2(1)= 1
[Replace b with - 2 and a with 1.]
Step: 4
The y-coordinate of the vertex = (1)2 - 2(1) + 1 = 0
[Replace x with 1 in the equation.]
Step: 5
Vertex (1, 0) is not a point on the graph, so the equation is not a solution for the graph.
Step: 6
In the equation, y = - x2 - 2x + 1, the values of a, b and c are - 1, - 2 and 1 respectively.
Step: 7
The x-coordinate of the vertex = -b2a = -(-2)2(-1)= - 1
[Replace b with - 2 and a with - 1.]
Step: 8
The y-coordinate of the vertex = - (- 1)2 - 2(- 1) + 1 = 2
[Replace x with - 1 in the equation.]
Step: 9
From the graph the x-coordinate of the vertex = - 1 and the y-coordinate of the vertex = 2.
Step: 10
So, the equation y = - x2 - 2x + 1 represents the graph.
Correct Answer is :   y = - x2 - 2x + 1
Q2Which of the following equations represents the graph shown? A. y = 2x2 + 4x + 1
B. y = - 2x2 - 4x + 1
C. y = - 2x2 + 4x + 1
D. y = 2x2 - 4x + 1

Step: 1
y = 2x2 - 4x + 1, the x-coordinate of the vertex = - b2a = - (- 4)2(2) = 1 and the y-coordinate of the vertex = - 1.
Step: 2
y = - 2x2 - 4x + 1, the x-coordinate of the vertex = - b2a = - (- 4)2(- 2) = - 1 and the y-coordinate of the vertex = 3.
Step: 3
y = 2x2 + 4x + 1, the x-coordinate of the vertex = - b2a = - 42(2) = - 1 and the y-coordinate of the vertex = - 1.
Step: 4
y = - 2x2 + 4x + 1, the x-coordinate of the vertex = - b2a = - 42(- 2) = 1 and the y-coordinate of the vertex = 3.
Step: 5
From the graph the x-coordinate of the vertex = 1 and the y-coordinate of the vertex = - 1.
Step: 6
y = 2x2 - 4x + 1 represents the graph.
Correct Answer is :   y = 2x2 - 4x + 1
Q3Which equation is represented by the graph shown? A. y = - x2 - 4
B. y = x2 + 2
C. y = x2 - 2
D. y = - x2 + 4

Step: 1
y = x2 + 2, the x-coordinate of the vertex = - b2a= -(0)2(1) = 0 and the y-coordinate of the vertex = (0)2 + 2 = 2.
Step: 2
y = - x2 + 4, the x-coordinate of the vertex = - b2a = -(0)2(-1) = 0 and the y-coordinate of the vertex = - (0)2 + 4 = 4.
Step: 3
y = x2 - 2, the x-coordinate of the vertex = - b2a= -(0)2(1) = 0 and the y-coordinate of the vertex = (0)2 - 2 = - 2.
Step: 4
y = - x2 - 4, the x-coordinate of the vertex = - b2a = -(0)2(-1) = 0 and the y-coordinate of the vertex = - (0)2 - 4 = - 4.
Step: 5
From the graph the x-coordinate of the vertex is 0 and the y-coordinate of the vertex is - 2.
Step: 6
So, y = x2 - 2 represents the graph.
Correct Answer is :   y = x2 - 2
Q4The table shows the data of the volume of air present inside a balloon as time elapses. Find a quadratic model for the data.
 Elapsed time Volume 0s 25 cm3 5s 23 cm3 10s 20 cm3 15s 15 cm3 20s 13 cm3 25s 9 cm3

A. y = - 0.02x2 - 0.3x + 25
B. y = 25x2
C. y = - 0.34x2 - 1.3x + 25
D. y = 38x2 - 13x + 25

Step: 1
The standard form of equation of a parabola is y = ax2 + bx + c - - - - (1)
Step: 2
Let′s pick the points (0, 25), (5, 23) and (10, 20) from the given table.
Step: 3
Substitute each point in equation (1)
Step: 4
25 = a(0)2 + b(0) + c
[Substititue the values.]
Step: 5
c = 25
[Simplify.]
Step: 6
25a + 5b + c = 23 and 100a + 10b + c = 20
[Substitute the values.]
Step: 7
c = 25 reduces the two new equations to:
25a + 5b = - 2 - - - - (2) and
100a + 10b = - 5 - - - - (3)
Step: 8
Solve (2) and (3) to get the values of a and b.
Step: 9
a = - 0.02 and b = - 0.3
Step: 10
So, y = - 0.02x2 - 0.3x + 25
[Substitute the values.]
Correct Answer is :    y = - 0.02x2 - 0.3x + 25
Q5The total number of dots on different patterns is shown. Calculate the number of dots in the 10th pattern.
(Hint: Find a quadratic model for the data.)
 Pattern number: x 1 2 3 4 Number of dots: y 1 6 15 28 A. 210
B. 10
C. 190
D. 30

Step: 1
First find the quadratic model for the given pattern.
Step: 2
The standard form of equation of a parabola is y = ax2 + bx + c - - - - (1)
Step: 3
Equation (1) involves 3 unknowns a, b and c. So we need 3 pairs of data.
Step: 4
The number of dots in the 3rd pattern is 15.
[Use the figure given.]
Step: 5
So, we have (1, 1), (2, 6) and (3, 15).
Step: 6
Substitute each point in equation (1)
Step: 7
a + b + c = 1 - - - - (2)
4a + 2b + c = 6 - - - - (3)
9a + 3b + c = 15 - - - - (4)
Step: 8
Solving the equations (2), (3) and (4), we get the values of a, b and c as: a = 2, b = -1 and c = 0
Step: 9
So, y = 2x2 - x
[Replace a = 2, b = - 1 and c = 0 in (1).]
Step: 10
Number of dots in the 10th pattern = 2(10)2 - (10)
[Replace x = 10.]
Step: 11
= 2(100) - 10 = 190
Step: 12
So, there are 190 dots in the 10th pattern.
Q6Find an equation for the set of data shown.
 x -2 -1 0 1 2 f(x) 14 -1 0 17 50

A. y = 9x2 + 8x
B. y = 8x2 + 9x
C. y = 8x + 9
D. y = 17x2 + 50x

Step: 1
The equation of the given data could be a linear one or a quadratic one. Let′s consider the quadratic equation only. Because, when the coefficient of x2 (that is ′a′) is zero, it will reduce to a linear equation.
Step: 2
Recall that the equation of a quadratic function is of the form: y = ax2 + bx + c- - - - (1)
Step: 3
Pick out any 3 data pairs from the table. Let′s take: (0, 0), (1, 17) and (2, 50)
Step: 4
Substitute each pair in (1).
Step: 5
On substituting (0, 0), we get, c = 0
Step: 6
On substituting (1, 17) and (2, 50), we get, a + b + c = 17 and 4a + 2b + c = 50.
Step: 7
c = 0 reduces the two new equations to:
a + b = 17 - - - - (2)
4a + 2b = 50 - - - - (3)
Step: 8
Solve (2) and (3) to get the values of a and b. We get a = 8 and b = 9.
Step: 9
So, y = 8x2 + 9x
[Replace a = 8, b = 9 and c = 0 in (1).]
Correct Answer is :   y = 8x2 + 9x
Q7Which equation is represented by the graph shown? A. y = x2 + 7x - 6
B. y = - x2 + 7x + 6
C. y = x2 + 7x + 6
D. y = x2 - 7x + 6

Step: 1
y = x2 - 7x + 6, the x-coordinate of the vertex = - b2a = - (- 7)2(1) = 72 and the y-coordinate of the vertex = (72 )2 - 7(72 ) + 6 = - 254
Step: 2
y = x2 + 7x + 6, the x-coordinate of the vertex = - b2a = - 72(1) = - 72 and the y-coordinate of the vertex = (- 72 )2 + 7 (- 72 ) + 6 = - 434
Step: 3
y = - x2 + 7x + 6, the x-coordinate of the vertex = - b2a = - 72(- 1) = 72 and the y-coordinate of the vertex = - (72 )2 + 7(72 ) + 6 = - 434
Step: 4
y = x2 + 7x - 6, the x-coordinate of the vertex = - b2a = - 72 and the y-coordinate of the vertex = (- 72 )2 + 7 (72 ) - 6 = 254
Step: 5
From the graph the x-coordinate of the vertex is 72 and the y-coordinate of the vertex is - 254.
Step: 6
So, y = x2 - 7x + 6 represents the graph.
Correct Answer is :   y = x2 - 7x + 6
Q8Which of the following equations best represents the graph? A. y = - x2 + 2x - 6
B. y = x2 - 2x - 6
C. y = - x2 + 2x3 + 6
D. y = x3 - 2x + 6

Step: 1
The equations y = x3 - 2x + 6 and y = - x2 + 2x3 + 6 do not represent curves as they are not quadratic.
Step: 2
In the equation, y = - x2 + 2x - 6, the values of a, b, and c are - 1, 2, and - 6 respectively.
Step: 3
x-coordinate of the vertex = - b2a = -(2)2(-1) = 1
[Replace b with 2 and a with - 1.]
Step: 4
y-coordinate of the vertex = - (1)2 + 2(1) + 1 = 2
[Replace x with 1 in the equation.]
Step: 5
Vertex (1, 2) is not a point on the graph, so the equation is not a solution for the graph.
Step: 6
In the equation, y = x2 - 2x - 6, the values of a, b, and c are 1, - 2, and - 6 respectively.
Step: 7
x-coordinate of the vertex = - b2a = -(-2)2(1) = 1
[Replace b with - 2 and a with 1.]
Step: 8
y-coordinate of the vertex = (1)2 - 2(1) - 6 = - 7
[Replace x with 1 in the equation.]
Step: 9
From the graph, x-coordinate of the vertex = 1 and y-coordinate of the vertex = - 7.
Step: 10
So, the equation y = x2 - 2x - 6 represents the graph.
Correct Answer is :   y = x2 - 2x - 6
Q9Which of the following equations best represents the graph shown? A. y = x2 + 2x - 3
B. y = x2 - 2x - 3
C. y = - x2 + 2x - 3
D. y = x2 - 2x + 3

Step: 1
y = x2 + 2x - 3, the x-coordinate of the vertex = - b2a = -(2)2(1) = - 1 and the y-coordinate of the vertex = (- 1)2 + 2(- 1) - 3 = - 4.
Step: 2
y = - x2 + 2x - 3, the x-coordinate of the vertex = - b2a = -(2)2(-1) = 1 and the y-coordinate of the vertex = - (1)2 + 2(1) - 3 = - 2.
Step: 3
y = x2 - 2x - 3, the x-coordinate of the vertex = - b2a = -(-2)2(1) = 1 and the y-coordinate of the vertex = (1)2 - 2(1) - 3 = - 4.
Step: 4
y = x2 - 2x + 3, the x-coordinate of the vertex = - b2a = -(-2)2(1) = 1 and the y-coordinate of the vertex = (1)2 - 2(1) + 3 = 2
Step: 5
From the graph the x-coordinate of the vertex = - 1 and
the y-coordinate of the vertex = - 4
Step: 6
So, the equation y = x2 + 2x - 3 represents the graph.
Correct Answer is :   y = x2 + 2x - 3
Q10John throws a ball from the top of a building 60 feet high. The table below shows the height of the ball from the ground after a time t seconds. Find the quadratic model.
 Time (t) Height (h) 1 72 2 52 3 0

A. h = - (16t2 + 28t + 60)
B. h = 16t2 + 28t + 60
C. h = - 16t2 + 28t - 60
D. h = - 16t2 + 28t + 60

Step: 1
The standard form of equation of a parabola is y = ax2 + bx + c ............ (1)
Step: 2
Let's pick the points (0, 60), (1, 72), and (3, 0) from the given table.
[Height of the building gives the point (0, 60).]
Step: 3
Substitute each point in equation (1)
Step: 4
60 = a(0)2 + b(0) + c
[Substitute the point (0, 60).]
Step: 5
c = 60
Step: 6
a + b + c = 72 and 9a + 3b + c = 0
[Substitute the points (1, 72) and (3, 0).]
Step: 7
c = 60 reduces the two new equations to a + b = 12 ........(2) and
9a + 3b = - 60 .......(3)
Step: 8
Solve (2) and (3) to get the values of a and b
Step: 9
a = - 16 and b = 28
Step: 10
So, h = - 16t2 + 28t + 60
[Replace a = - 16, b = 28, and c = 60 in (1).]
Correct Answer is :   h = - 16t2 + 28t + 60