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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

#### Solution:

The graph of a quadratic function is a parabola.

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 y = ax2 + bx + c, the standard form of equation of a parabola.

3 = a(0)2 + b(0) + c
[Substitute the values.]

c = 3
[Solve for c.]

4a - 2b + c = 11 and a - b + c = 5
[Substitute the values.]

c = 3 reduces the two new equations to:

4a - 2b = 8 - - - - (1) and
a - b = 2 - - - - (2)

Solve (1) and (2) to get the values of a and b.

a = 2 and b = 0

So, y = 2x2 + 0x + 3
[Substitute the values.]

y = 2x2 + 3, which is an equation of a parabola.

So the Graph 2 and Graph 3 represent quadratic functions.

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

#### Solution:

The graph of a quadratic function is a parabola.

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 y = ax2 + bx + c, - - - - (1) the standard form of equation of a parabola.

4 = a(0)2 + b(0) + c

c = 4
[Substitute the values.]

4a - 2b + c = 2 and a - b + c = 3.5
[Substitute the values.]

c = 4 reduces the two new equations to: 4a - 2b = - 2 - - - - (2) and
a - b = - 0.5 - - - - (3)

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

a = - 12and b = 0

So, y = - (12) x2 + 0x + 4
[Substitute the values.]

y = - (12) x2 + 4, which is an equation of a parabola.

So the graph 1 and graph 4 represent quadratic functions.

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

#### Solution:

The minimum number of data pairs needed to find a model depends on the number of unknowns involved in the standard equation of the model.

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 y = mx + c, where m and c are the unknowns.

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

4.
Determine if the function $y$ = (5$x$ + 24)($x$ - 5) is linear or quadratic.

#### Solution:

y = (5x + 24)(x - 5)

= 5x2 - x - 120
[Multiply using FOIL.]

A function of the form y = ax2 + bx + c, where a ≠ 0 is called a quadratic function.

So, the given function is quadratic.

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

#### Solution:

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.

Recall that the equation of a quadratic function is of the form: y = ax2 + bx + c - - - - (1)

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, c = 11

On substituting (1, 10) and (2, 9), we get, a + b + c = 10 and 4a + 2b + c = 9.

c = 11 reduces the two new equations to:
a + b = - 1 - - - - (2)
4a + 2b = - 2 - - - - (3)

By solving (2) and (3) we get a = 0 and b = - 1.

So, y = - x + 11
[Replace a = 0, b = - 1 and c = 11 in (1).]

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$

#### Solution:

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.

Recall that the equation of a quadratic function is of the form: y = ax2 + bx + c- - - - (1)

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, c = 0

On substituting (1, 25) and (2, 74), we get, a + b + c = 25 and 4a + 2b + c = 74.

c = 0 reduces the two new equations to:
a + b = 25 - - - - (2)
4a + 2b = 74 - - - - (3)

Solve (2) and (3) to get the values of a and b. We get a = 12 and b = 13.

So, y = 12x2 + 13x
[Replace a = 12, b = 13 and c = 0 in (1).]

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 cm3 5s 14 cm3 10s 11 cm3 15s 6 cm3 20s 4 cm3 25s 2 cm3

 a. $y$ = 17$x$ - 0.6 b. $x$ = - 0.6$y$ + 17 c. $y$ = - 0.64$x$ d. $y$ = - 0.6$x$ + 17

#### Solution:

The slope-intercept form of an equation of a linear model is given by: y = mx + c - - - - (1)

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

17 = m(0) + c

c = 17
[Replace x = 0 and y = 17.]

10m + c = 11
[Replace x = 10 and y = 11.]

m = - 35
[Simplify.]

So, y = - 0.6x + 17
[Replace m = - 0.6 and c = 17.]

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$

#### Solution:

The standard form of equation of a parabola is y = ax2 + bx + c - - - - (1)

Equation (1) involves 3 unknowns a, b and c. So we need 3 pairs of data.

The number of dots in the 3rd pattern is 15.
[Use the figure given.]

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

Substitute each point in equation (1)

a + b + c = 1 - - - - (2)
4a + 2b + c = 6 - - - - (3) and
9a + 3b + c = 15 - - - - (4)

Solving the equations (2), (3) and (4), we get the values of a, b and c as: a = 2, b = - 1 and c = 0

So, y = 2x2 - x is the required quadratic model.
[Substitute the values.]

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 cm3 5s 23 cm3 10s 20 cm3 15s 15 cm3 20s 13 cm3 25s 9 cm3
 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

#### Solution:

The standard form of equation of a parabola is y = ax2 + bx + c - - - - (1)

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 = a(0)2 + b(0) + c
[Substititue the values.]

c = 25
[Simplify.]

25a + 5b + c = 23 and 100a + 10b + c = 20
[Substitute the values.]

c = 25 reduces the two new equations to:
25a + 5b = - 2 - - - - (2) and
100a + 10b = - 5 - - - - (3)

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

a = - 0.02 and b = - 0.3

So, y = - 0.02x2 - 0.3x + 25
[Substitute the values.]

10.
The 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. 190 b. 30 c. 10 d. 210

#### Solution:

First find the quadratic model for the given pattern.

The standard form of equation of a parabola is y = ax2 + bx + c - - - - (1)

Equation (1) involves 3 unknowns a, b and c. So we need 3 pairs of data.

The number of dots in the 3rd pattern is 15.
[Use the figure given.]

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

Substitute each point in equation (1)

a + b + c = 1 - - - - (2)
4a + 2b + c = 6 - - - - (3)
9a + 3b + c = 15 - - - - (4)

Solving the equations (2), (3) and (4), we get the values of a, b and c as: a = 2, b = -1 and c = 0

So, y = 2x2 - x
[Replace a = 2, b = - 1 and c = 0 in (1).]

Number of dots in the 10th pattern = 2(10)2 - (10)
[Replace x = 10.]

= 2(100) - 10 = 190

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