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natka813 [3]
3 years ago
5

ABCD is a trapezoid with median EF. a. AB = 15, DC = 11, then EF = ? b. EF = 14, DC = 10, then AB = ? c. EF = 18, AB = (5n - 9),

and DC = (2n + 3). Find n, AB, and DC. d. EF = (2y + 4), AB = (5y + 2), and DC = (-3y + 8). Find y, EF, AB, and DC.
Mathematics
1 answer:
Reptile [31]3 years ago
5 0
The median of the trapezoid is the average of the bases. If we draw the trapezoid that is being described in this item, we will deduce that AB and DC are the bases and EF is given to be the median.

 For this item,
                        EF = (AB + CD) / 2

Part A:
                      EF = (15 + 11) / 2 = 13

Part B:
                      AB = 2EF - CD
                      AB = (2)(14) - 10 = 18

Part C:
                       18 = ((5n - 9) + (2n + 3))/2
                        18 = (7n - 6) / 2
                                 n = 6

Part D:
                  2y + 4 = ((5y + 2) + (-3y + 8))/2

                     y = 1

EF = 2(1) + 4 = 6
AB = (5(1) + 2 = 7
AB = -3(1) + 8 = 5

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Even function:

A function is said to be even if its graph is symmetric with respect to the , that is:

Odd function:

A function is said to be odd if its graph is symmetric with respect to the origin, that is:

So let's analyze each question for each type of functions using examples of polynomial functions. Thus:

FOR EVEN FUNCTIONS:

1. When  becomes  

1.1 Effects on the y-intercept

We need to find out the effects on the y-intercept when shifting the function  into:

We know that the graph  intersects the y-axis when , therefore:

So:

So the y-intercept of  is one unit less than the y-intercept of

1.2. Effects on the regions where the graph is increasing and decreasing

Given that you are shifting the graph downward on the y-axis, there is no any effect on the intervals of the domain. The function  increases and decreases in the same intervals of

1.3 The end behavior when the following changes are made.

The function is shifted one unit downward, so each point of  has the same x-coordinate but the output is one unit less than the output of . Thus, each point will be sketched as:

FOR ODD FUNCTIONS:

2. When  becomes  

2.1 Effects on the y-intercept

In this case happens the same as in the previous case. The new y-intercept is one unit less. So the graph is shifted one unit downward again.

An example is shown in Figure 1. The graph in blue is the function:

and the function in red is:

So you can see that:

2.2. Effects on the regions where the graph is increasing and decreasing

The effects are the same just as in the previous case. So the new function increases and decreases in the same intervals of

In Figure 1 you can see that both functions increase at:

and decrease at:

2.3 The end behavior when the following changes are made.

It happens the same, the output is one unit less than the output of . So, you can write the points just as they were written before.

So you can realize this concept by taking a point with the same x-coordinate of both graphs in Figure 1.

FOR EVEN FUNCTIONS:

3. When  becomes  

3.1 Effects on the y-intercept

We need to find out the effects on the y-intercept when shifting the function  into:

As we know, the graph  intersects the y-axis when , therefore:

And:

So the new y-intercept is the negative of the previous intercept shifted one unit upward.

3.2. Effects on the regions where the graph is increasing and decreasing

In the intervals when the function  increases, the function  decreases. On the other hand, in the intervals when the function  decreases, the function  increases.

3.3 The end behavior when the following changes are made.

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FOR ODD FUNCTIONS:

4. When  becomes  

4.1 Effects on the y-intercept

In this case happens the same as in the previous case. The new y-intercept is the negative of the previous intercept shifted one unit upward.

4.2. Effects on the regions where the graph is increasing and decreasing

In this case it happens the same. So in the intervals when the function  increases, the function  decreases. On the other hand, in the intervals when the function  decreases, the function  increases.

4.3 The end behavior when the following changes are made.

Similarly, each point of the function  has the same x-coordinate just as the function  and the y-coordinate is the negative of the previous coordinate shifted one unit upward.

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