### 1.

On of the very best examples of a plot with many, many line segments is the Stock Market prices.

This line has two segments and is not a straight line.

In math, we write multiple equations and then specify when they should be used. For example:

For phone calls:

10 cents/min (midnight to 8 a.m.)

15 cents/min (8 a.m. to midnight)

For this problem:

Stock price is:

$10 + ((12-10)/31)*day (for January days)

$12 - ((-9-12)/28)*day (for February days)

### 2.

**1)** This equation will be in the form y=mx+b , where m is slope and b is y-intercept.

First, we need to find the slope between the two points.

m = (y2 - y1) / (x2 - x1)

m = (-1 - 2) / (3 - 1)

m = -3 / 2

The equation of the line so far is

y = (-3/2)x + b

Now we plug in the values of the coordinate point (1, 2) into the equation to find b.

2 = (-3 / 2)(1) + b

2 = (-3 / 2) + b

7 / 2 = b

The equation you want is

y = (-3/2)x + (7/2)

**2)** To find the vertex, we put f(x) into vertex form. Any quadratic function in vertex form is

f(x) = a(x - h)2 + k

where:

a is the coefficient of the x2 term

The coordinate of the vertex is (h, k)

f(x) = (x2 - 6x + 9) - 6

f(x) = (x - 3)(x - 3) - 6

f(x) = (x - 3)2 - 6

h = 3

k = -6

Therefore, the vertex is (3, -6).

**3)** When drawing a smiling face, we know the feature it has is a mouth and two eyes. The mouth represents a parabola that opens upward or the negative half of a circle. The eyes represents a single point, or a small line. Knowing this fact, we can create a piecewise function for each type of curve to represent different facial features of a smiling face.

We can use the origin as our reference point to start off.

For the mouth, lets use the parabola f(x) = (1/5)x2 - 3. One point of the parabola will be (0, -3). Now we want to pick the endpoints of this parabola. To do this, we can find the x-intercepts. Set f(x) equal to zero and solve for x.

(1 / 5)x2 - 3 = 0

(1 / 5)x2 = 3

x2 = 15

x = -√15 and x = √15

So the domain of this parabolic function is -√15 ≤ x ≤ √15.

Now to draw the eyes, we will draw two short horizontal lines that are symmetrical to each other with reference to the y-axis.

f(x) = 2

The domain of this function on the left of the y-axis is -2 ≤ x ≤ -1. The domain on the right side is 1 ≤ x ≤ 2.

.

Here is your piecewise function:

f(x) = (1/5)x2 - 3 for -√15 ≤ x ≤ √15

```
2 for -2 ≤ x ≤ -1 and 1 ≤ x ≤ 2
```

*****From wyzant.com