Modeling Quadratic Functions Worksheet

**Page 1**

1.

Determine which of the graphs shown represent a quadratic function.

a. | Graph 2 and Graph 3 | ||

b. | Graph 2 and Graph 1 | ||

c. | Graph 2 only | ||

d. | Graph 3 only |

Graph-3 is a parabola.

Graph-1 is a straight line.

Graph-4 shows the graph of an absolute function.

LetÃ¢â‚¬â„¢s check Graph-2.

The points that lie on graph-2 are: (- 2, 11), (- 1, 5) and (0, 3).

Substitute each point in

3 =

[Substitute the values.]

[Solve for

4

[Substitute the values.]

4

Solve (1) and (2) to get the values of

So,

[Substitute the values.]

So the Graph 2 and Graph 3 represent quadratic functions.

Correct answer : (1)

2.

Determine which of the graphs represent a quadratic function.

a. | Graph 1 and Graph 4 | ||

b. | Graph 1 only | ||

c. | Graph 1 and Graph 3 | ||

d. | Graph 4 only |

Graph-1 is a parabola.

Graph-2 is a straight line.

Graph-3 shows the graph of an absolute function.

LetÃ¢â‚¬â„¢s check Graph-4.

The points that lie on graph 4 are: (- 2, 2), (- 1, 3.5) and (0, 4)

Substitute each point in

4 =

[Substitute the values.]

4

[Substitute the values.]

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

So,

[Substitute the values.]

So the graph 1 and graph 4 represent quadratic functions.

Correct answer : (1)

3.

Find the minimum number of data pairs required to find a linear model for a data set.

a. | 2 | ||

b. | 1 | ||

c. | 3 | ||

d. | None of the above |

The number of data pairs needed = The number of unknowns involved in the standard equation of the model.

The linear model is of the form

So, the minimum number of data pairs needed to find a linear model for a data set is 2.

Correct answer : (1)

4.

Determine if the function $y$ = (5$x$ + 24)($x$ - 5) is linear or quadratic.

a. | linear | ||

b. | quadratic |

= 5

[Multiply using FOIL.]

A function of the form

So, the given function is quadratic.

Correct answer : (2)

5.

Obtain an equation for the set of data shown.

$x$ | -1 | 0 | 1 | 2 | 3 |

$f$($x$) | 12 | 11 | 10 | 9 | 8 |

a. | $y$ = $x$ - 11 | ||

b. | $y$ = 11$x$ - 1 | ||

c. | $y$ = - $x$ ^{2} + 11 | ||

d. | $y$ = - $x$ + 11 |

Recall that the equation of a quadratic function is of the form:

Pick out any 3 data pairs from the table. Let′s take: (0, 11), (1, 10) and (2, 9)

Substitute each pair in (1).

On substituting (0, 11), we get,

On substituting (1, 10) and (2, 9), we get,

4

By solving (2) and (3) we get

So,

[Replace

Correct answer : (4)

6.

Find an equation for the set of data shown.

$x$ | -2 | -1 | 0 | 1 | 2 |

$f$($x$) | 22 | -1 | 0 | 25 | 74 |

a. | $y$ = 25$x$ ^{2} + 74$x$ | ||

b. | $y$ = 12$x$ + 13 | ||

c. | $y$ = 13$x$ ^{2} + 12$x$ | ||

d. | $y$ = 12$x$ ^{2} + 13$x$ |

Recall that the equation of a quadratic function is of the form:

Pick out any 3 data pairs from the table. Let′s take: (0, 0), (1, 25) and (2, 74)

Substitute each pair in (1).

On substituting (0, 0), we get,

On substituting (1, 25) and (2, 74), we get,

4

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

So,

[Replace

Correct answer : (4)

7.

The volume of air present inside a balloon as time elapses is as shown in the table. Find a linear model for the data.

Elapsed Time | Volume |

0s | 17 cm^{3} |

5s | 14 cm^{3} |

10s | 11 cm^{3} |

15s | 6 cm^{3} |

20s | 4 cm^{3} |

25s | 2 cm^{3} |

a. | $y$ = 17$x$ - 0.6 | ||

b. | $x$ = - 0.6$y$ + 17 | ||

c. | $y$ = - 0.64$x$ | ||

d. | $y$ = - 0.6$x$ + 17 |

LetÃ¢â‚¬â„¢s pick the points (0, 17) and (10, 11) from the given table.

17 =

[Replace

10

[Replace

[Simplify.]

So,

[Replace

Correct answer : (4)

8.

Find a quadratic model for the data shown.

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

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

a. | $y$ = 2$x$ ^{2} - $x$ + 1 | ||

b. | $y$ = 3$x$ ^{2} - $x$ | ||

c. | $y$ = 2$x$ ^{2} - $x$ | ||

d. | $y$ = 2$x$ ^{2} - 2$x$ |

Equation (1) involves 3 unknowns

The number of dots in the 3

[Use the figure given.]

So, we have (1, 1), (2, 6) and (3, 15).

Substitute each point in equation (1)

4

9

Solving the equations (2), (3) and (4), we get the values of

So,

[Substitute the values.]

Correct answer : (3)

9.

The 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 cm^{3} |

5s | 23 cm^{3} |

10s | 20 cm^{3} |

15s | 15 cm^{3} |

20s | 13 cm^{3} |

25s | 9 cm^{3} |

a. | $y$ = - 0.34$x$ ^{2} - 1.3$x$ + 25 | ||

b. | $y$ = 38$x$ ^{2} - 13$x$ + 25 | ||

c. | $y$ = - 0.02$x$ ^{2} - 0.3$x$ + 25 | ||

d. | $y$ = 25$x$ ^{2} |

LetÃ¢â‚¬â„¢s pick the points (0, 25), (5, 23) and (10, 20) from the given table.

Substitute each point in equation (1)

25 =

[Substititue the values.]

[Simplify.]

25

[Substitute the values.]

25

100

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

So,

[Substitute the values.]

Correct answer : (3)

10.

The total number of dots on different patterns is shown. Calculate the number of dots in the 10^{th} pattern.

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

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

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

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

a. | 190 | ||

b. | 30 | ||

c. | 10 | ||

d. | 210 |

The standard form of equation of a parabola is

Equation (1) involves 3 unknowns

The number of dots in the 3

[Use the figure given.]

So, we have (1, 1), (2, 6) and (3, 15).

Substitute each point in equation (1)

4

9

Solving the equations (2), (3) and (4), we get the values of

So,

[Replace

Number of dots in the 10

[Replace

= 2(100) - 10 = 190

So, there are 190 dots in the 10

Correct answer : (1)