What is the vertex point of y = |x - 2| + 3? Enter your answer as an ordered pair, e.g. (1,2).

Vertex: (2, 3).

Step-by-step explanation:

Definition of terms:

The general form of an absolute function is given by: y = a|x - h| + k, where:

| a | ⇒ Determines the vertical stretch or compression factor of the function.

(h, k) ⇒ Coordinates of the vertex, which is either the minimum or maximum point on the graph.

x = h ⇒ Axis of symmetry, which is the imaginary vertical line that splits the graph of the function into two symmetrical parts.

Explanation"

Given the absolute value function, y = |x - 2| + 3:

Based on the general form of absolute value functions described in the previous section of this post, y = a|x - h| + k:

We can assume that the value of "a" in the given absolute value function is 1, because if we distribute 1 into the terms inside the bars, "|    |," the constant value of a = 1 will not change the value of those terms.  

However, there are other instances where there is a given value for "a," which could either "stretch" or "compress" the graph of the absolute value function.  If the value of | a | > 1, then it represents the vertical stretch (the graph appears narrower than the parent graph of the absolute value function). In contrast, if the the value of "a" is 0 < | a | < 1, then it represents a vertical compression (the graph appears wider than the parent graph of the absolute value function).

In terms of the vertex, it occurs at point, (2, 3), where h = 2, and k = 3.  

Therefore, the vertex of the given absolute value function is (2, 3).

0.02%

Step-by-step explanation:

In 20 tosses getting at least 18 tails is the same that getting 18, 19 or 20 tails. We need to find every of these probabilities and sum them. We have:

P(at least 18 t) = P (18 t) + P(19 t) + P (20 t)

We will suppose the probability of getting a tail is 1/2, it is, that the coin is fair. So, the prob of 20 tails is simply the product of 1/2 20 times:

P(20t) = 1/2 * 1/2 * 1/2 * ... * 1/2 = (1/2)^20 = 0.00000095

For the 19 tails we need to get 1 head and then 19 tails. What we need to manage is the order, it is, in which toss the heads comes. We calculate the probability of 19 tails if the heads comes on the first toss, then the prob if the head comes on the second toss and so on until the head comes on the 20th toss, then we sum all of them. However, as the probability of head is equal to the probability of tail, the probability of each of the 19 tails will be the same as the 20 t. Lets see:

P(19t if head 1st) = P(head) * P(tail) * P(tail) * *P(tail) = P(head) * P(tail)^19

And as P(tail) = P(head) = 0.5

P(19t if head 1st)=P(head)*P(tail)^19=P(tail)*P(tail)^19=P(tail)^20=0.00000095

This is the prob if head comes on the 1st toss. The prob if head comes on the 2nd is equal and so on to the 20th toss. So we can say:

P(19 tails) = 20*(0.00000095) = 0.000019

For the 18 tails is the same, we need to manage where the heads come. They can come on 1st and 2nd, 5th and  9th, 18th and 19th, or any pair of tosses. Notice that there are 20 tosses for 2 heads, so we can use combinations of 20 taken by 2 as we dont care about which head comes first. So, the different forms of getting to heads is:

C(20,2) = 20C2=190

So, there are 190 different forms of getting 2 heads in 20 tosses. Noy we need the probability that 2 heads come out with 18 tails. But, thinking as we did with 19tails and 1 head, as the P(head) = P(tails), the result will be the same:

P(18t if head 1st and 2nd) = P(head)*P(head)* P(tail) * P(tail) * *P(tail) = P(head)^2 * P(tail)^18 = P(tail)^20 = 0.00000095

So,

P(18 tail) = 190 * 0.00000095 = 0.00018

So, summing all:

P (at least 18 t) = P (18 t) + P(19 t) + P (20 t)

P (at least 18 t) = 0.00018 + 0.000019 + 0.00000095 = 0.0002 = 0.2%