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2 years ago

Express y = -2x² - 4x + 7 in the form of perfect square and find the maximum points.

Mathematics

1 answer:

Marat540 [252]2 years ago

3 0

Answer:

<em>Maximum: (-1,9)</em>

Step-by-step explanation:

<u>Vertex form of the quadratic function</u>

If the graph of the quadratic function has a vertex at the point (h,k), then the function can be written as:

$y=a(x-h)^2+k$

Where a is the leading coefficient.

We are given the following function:

$y =-2x^2-4x+7$

To find the vertex, we need to complete squares. First, factor -2 on the first two terms:

$y =-2(x^2+2x)+7$

The expression in parentheses must be completed to represent the square of a binomial. Adding 1 and subtracting 1:

$y =-2(x^2+2x+1- 1)+7$

Taking out the -1:

$y =-2(x^2+2x+1)+2+7$

Factoring the trinomial and operating:

$y =-2(x+1)^2+9$

Comparing with the vertex form we have

Vertex (-1,9)

Leading coefficient: -2

Since the leading coefficient is negative, the function has a maximum value at its vertex, i.e.

Maximum: (-1,9)

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The lifespan (in days) of the common housefly is best modeled using a normal curve having mean 22 days and standard deviation 5.

Natasha_Volkova [10]

Answer:

Yes, it would be unusual.

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

If $Z \leq -2$ or $Z \geq 2$ , the outcome X is considered unusual.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$ , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$ .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this question, we have that:

$\mu = 22, \sigma = 5, n = 25, s = \frac{5}{\sqrt{25}} = 1$

Would it be unusual for this sample mean to be less than 19 days?

We have to find Z when X = 19. So

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{19 - 22}{1}$

$Z = -3$

$Z = -3 \leq -2$ , so yes, the sample mean being less than 19 days would be considered an unusual outcome.

7 0

3 years ago

How many 5 digit numbers are even if repetition is not allowed?

shtirl [24]

23308 is the answer of how many is even with no repitition

7 0

3 years ago

If f(x) = -x + 8 and g(x) = x^4, what is (gºf)(2)?

kotykmax [81]

Answer:

$\large\boxed{(g\circ f)(2)=1296}$

Step-by-step explanation:

$(g\circ f)(x)=g\bigg(f(x)\bigg)\\\\f(x)=-x+8,\ g(x)=x^4\\\\(g\circ f)(x)=\g\bigg(f(x)\bigg)=(-x+8)^4\\\\(g\circ f)(2)\to\text{put x = 2 to the equation}\ (g\circ f)(x):\\\\(g\circ f)(2)=(-2+8)^4=(6)^4=1296$