Step: 1

The equations y = x ^{3} + 2x + 1 and y = - x ^{2} - 2x ^{3} + 1 do not represent curves as they are not quadratic.

Step: 2

In the equation, y = x ^{2} - 2x + 1, the values of a ,
b and c are 1, - 2 and 1 respectively.

Step: 3

The x -coordinate of the vertex = - b 2 a = -(-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 = - x ^{2} - 2x + 1, the values of a , b and c are - 1, - 2 and 1 respectively.

Step: 7

The x -coordinate of the vertex = - b 2 a = -(-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 = - x ^{2} - 2x + 1 represents the graph.

Correct Answer is : y = - x ^{2} - 2x + 1

Step: 1

Step: 2

Step: 3

Step: 4

Step: 5

From the graph the x -coordinate of the vertex = 1 and the y -coordinate of the vertex = - 1.

Step: 6

Correct Answer is : y = 2x ^{2} - 4x + 1

Step: 1

Step: 2

Step: 3

Step: 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 = x ^{2} - 2 represents the graph.

Correct Answer is : y = x ^{2} - 2

Elapsed time | Volume |

0s | 25 cm^{3} |

5s | 23 cm^{3} |

10s | 20 cm^{3} |

15s | 15 cm^{3} |

20s | 13 cm^{3} |

25s | 9 cm^{3} |

Step: 1

The standard form of equation of a parabola is y = ax ^{2} + b x + 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

[Simplify.]

Step: 6

25a + 5b + c = 23 and 100a + 10b + c = 20

[Substitute the values.]

Step: 7

25

100

Step: 8

Solve (2) and (3) to get the values of a and b .

Step: 9

Step: 10

So, y = - 0.02x ^{2} - 0.3x + 25

[Substitute the values.]

Correct Answer is : y = - 0.02x ^{2} - 0.3x + 25

(Hint: Find a quadratic model for the data.)

Pattern number: | 1 | 2 | 3 | 4 |

Number of dots: | 1 | 6 | 15 | 28 |

Step: 1

First find the quadratic model for the given pattern.

Step: 2

The standard form of equation of a parabola is y = ax ^{2} + b x + 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 3^{rd} 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

4

9

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 = 2x ^{2} - x

[Replace a = 2, b = - 1 and c = 0 in (1).]

Step: 10

Number of dots in the 10^{th} pattern = 2(10)^{2} - (10)

[Replace x = 10.]

Step: 11

= 2(100) - 10 = 190

Step: 12

So, there are 190 dots in the 10^{th} pattern.

Correct Answer is : 190

-2 | -1 | 0 | 1 | 2 | |

14 | -1 | 0 | 17 | 50 |

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 x ^{2} (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 = a x ^{2} + b x + 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

4

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 = 8x ^{2} + 9x

[Replace a = 8, b = 9 and c = 0 in (1).]

Correct Answer is : y = 8x ^{2} + 9x

Step: 1

Step: 2

Step: 3

Step: 4

Step: 5

From the graph the x -coordinate of the vertex is 7 2 and the y -coordinate of the vertex is - 25 4 .

Step: 6

So, y = x ^{2} - 7x + 6 represents the graph.

Correct Answer is : y = x ^{2} - 7x + 6

Step: 1

The equations y = x ^{3} - 2x + 6 and y = - x ^{2} + 2x ^{3} + 6 do not represent curves as they are not quadratic.

Step: 2

In the equation, y = - x ^{2} + 2x - 6, the values of a ,
b , and c are - 1, 2, and - 6 respectively.

Step: 3

[Replace b with 2 and a with - 1.]

Step: 4

[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 = x ^{2} - 2x - 6, the values of a , b , and c are 1, - 2, and - 6 respectively.

Step: 7

[Replace b with - 2 and a with 1.]

Step: 8

[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 = x ^{2} - 2x - 6 represents the graph.

Correct Answer is : y = x ^{2} - 2x - 6

Step: 1

Step: 2

Step: 3

Step: 4

Step: 5

From the graph the x -coordinate of the vertex = - 1 and

they -coordinate of the vertex = - 4

the

Step: 6

So, the equation y = x ^{2} + 2x - 3 represents the graph.

Correct Answer is : y = x ^{2} + 2x - 3

Time (t) | Height (h) |

1 | 72 |

2 | 52 |

3 | 0 |

Step: 1

The standard form of equation of a parabola is y = a x ^{2} + b x + 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

Step: 6

[Substitute the points (1, 72) and (3, 0).]

Step: 7

9

Step: 8

Solve (2) and (3) to get the values of a and b

Step: 9

Step: 10

So, h = - 16t ^{2} + 28t + 60

[Replace a = - 16, b = 28, and c = 60 in (1).]

Correct Answer is : h = - 16t ^{2} + 28t + 60

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