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posledela
2 years ago
15

The quadratic function f(x)=ax^2+4x+c has a maximum at the point (1, 9) . What are the values of a and c?

Mathematics
2 answers:
valina [46]2 years ago
6 0

Answer: a = -2  and  c = 7

Step-by-step explanation:

(this answer uses some basic calculus)

We know that f'(x) = 2ax + 4, and since (1,9) is a maximum of f, then f'(1) = 0. That means 2a + 4 = 0, so \boxed{a = -2}. Then since f(1) = 9, we have -2(1)^2 + 4(1) + c = 9, so \boxed{c = 7}.

valina [46]2 years ago
5 0

Answer:

a = -2

c = 7

Step-by-step explanation:

Given the quadratic function, f(x) = ax² + 4x + c, for which its vertex is the maximum point occurring at (1, 9), and b = 4.

<h3>Solve for the value of <em>a</em>:</h3>

Since the x-coordinate of the vertex, x = 1, can be calculated using the formula, x = \frac{-b}{2a}:

We can substitute the value of the x-coordinate and b = 4 into the formula, and solve for the value of <em>a</em>:

x = \frac{-b}{2a}

1 = \frac{-4}{2a}

Multiply both sides by 2a:

(2a) 1 =  \frac{-4}{2a} (2a)

2a = -4

Divide both sides by 2 to solve for a:

\frac{2a}{2} = \frac{-4}{2}

a = -2

Therefore, the value of a = -2.

<h3>Solve for the value of c:</h3>

Next, to solve for c, substitute the coordinate values of the vertex, (1, 9) into the given quadratic function:

f(x) = ax² + 4x + c

9 = -2(1)² + 4(1) + c

9 = -2 + 4 + c

9 = 2 + c

Subtract 2 from both sides to isolate c:

9 - 2 = 2 - 2  + c

7 = c

<h3>Double-check:</h3>

In order to double-check the validity of our values for a and c, substitute a = -2, and c = 7 into the function, along with the coordinates of the vertex, (1, 9):

f(x) = -2x² + 4x + 7

9 = -2(1)² + 4(1) + 7

9 = -2 + 4 + 7

9 = 9 (True statement).

Therefore, the correct answers are: a = -2, and c = 7.

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Step-by-step explanation:

y=-3x+5

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The revenue from selling xshirts is r(x) = 12x.
algol [13]

Answer:

D. p(x)=7x-20

Step-by-step explanation:

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4 0
3 years ago
Suppose that an airline overbooks seats on their flights. In particular, it sells 300 tickets for a flight when there are only 2
vladimir1956 [14]

Using the <u>normal approximation to the binomial</u>, it is found that there is a 0.994 = 99.4% probability that we will have enough seats for everyone who shows up.

In a normal distribution with mean \mu and standard deviation \sigma, the z-score of a measure X is given by:

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

  • It 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, which is the percentile of X.
  • The binomial distribution is the probability of <u>x successes on n trials</u>, with <u>p probability</u> of a success on each trial. It can be approximated to the normal distribution with \mu = np, \sigma = \sqrt{np(1-p)}.

In this problem:

  • 15% do not show up, so 100 - 15 = 85% show up, which means that p = 0.85.
  • 300 tickets are sold, hence n = 300.

The mean and the standard deviation are given by:

\mu = np = 300(0.85) = 255

\sigma = \sqrt{np(1-p)} = \sqrt{300(0.85)(0.15)} = 6.185

The probability that we will have enough seats for everyone who shows up is the probability of at most <u>270 people showing up</u>, which, using continuity correction, is P(X \leq 270 + 0.5) = P(X \leq 270.5), which is the <u>p-value of Z when X = 270.5</u>.

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

Z = \frac{270.5 - 255}{6.185}

Z = 2.51

Z = 2.51 has a p-value of 0.994.

0.994 = 99.4% probability that we will have enough seats for everyone who shows up.

A similar problem is given at brainly.com/question/24261244

8 0
3 years ago
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