Using the completing-the-square method, find the vertex of the function f(x) =- 3x2 + 6x − 2 and indicate whether it is a minimum or a maximum and at what point.

Answer:

Part A) The vertex is the point [tex](0.5,-0.5)[/tex]

Part B) The axis of symmetry is [tex]x=0.5[/tex]

Part C) The graph in the attached figure

Step-by-step explanation:

we know that

The equation of a vertical parabola into vertex form is equal to

[tex]y=a(x-h)^{2}+k[/tex]

where

(h,k) is the vertex of the parabola

and the axis of symmetry is equal to the x-coordinate of the vertex

so

[tex]x=h[/tex] -----> equation of the axis of symmetry

In this problem we have  

[tex]f(x)=-2x^{2}+2x-1[/tex]

Convert to vertex form

Group terms that contain the same variable, and move the constant to the opposite side of the equation

[tex]f(x)+1=-2x^{2}+2x[/tex]

Factor the leading coefficient

[tex]f(x)+1=-2(x^{2}-x)[/tex]

Complete the square. Remember to balance the equation by adding the same constants to each side

[tex]f(x)+1-(1/2)=-2(x^{2}-x+(1/4))[/tex]

[tex]f(x)+(1/2)=-2(x^{2}-x+(1/4))[/tex]

Rewrite as perfect squares

[tex]f(x)+(1/2)=-2(x-(1/2))^{2}[/tex]

[tex]f(x)=-2(x-(1/2))^{2}-(1/2)[/tex] -----> equation in vertex form

The vertex of the parabola is the point [tex](0.5,-0.5)[/tex]

Is a vertical parabola open downward

The axis of symmetry is equal to

[tex]x=0.5[/tex]

see the attached figure to better understand the problem

The number of houses left to be sold is 9, calculated from the initial total and the proportions sold each month.

The number of houses left to be sold is 9.

Initially, 1/3 of the 84 new houses were sold, which is 84 × (1/3) = 28 houses sold.

There were 84 - 28 = 56 houses remaining.

From these remaining houses, 5/7 were sold this month, which equates to 56 × (5/7) = 40 houses sold. Thus, 56 - 40 = 16 houses are left to be sold.

The answer is:

vertex: (4,-5); focus: (4,2); directrix: y=-12

The vertex of the parabola is (4, -5), the focus is (4, 2), and the directrix is the line y = -12.

To determine the vertex, focus, and directrix of a parabola given by the equation y = {1{28}(x - 4)² - 5, we must first identify the standard form of a parabolic equation, which is y = a(x - h)² + k where (h, k) is the vertex of the parabola. In this case, we can see that the vertex (h, k) is (4, -5).

The focus of a parabola is a fixed point from which the distance to any point on the parabola is equal to the distance from that point to the directrix.

The directrix is a fixed straight line. The distance between the vertex and the focus, denoted as 4p, can be determined from the coefficient a in the standard form, where a = {1}/{4p} or p = {1}/{4a}. Thus, p = {1}/{4({1}/{28})} = 7. The focus is then at (h, k + p) = (4, -5 + 7) = (4, 2).

The directrix has the equation y = k - p, so it will be y = -5 - 7 = -12. The parabola opens upwards because the value of a is positive.

In summary, the vertex of the parabola is (4, -5), the focus is (4, 2), and the directrix is the line y = -12.

Final answer:

To form a polynomial with real coefficients, given non-real zeros 4i and 5i, include their conjugates -4i and -5i. Multiply the corresponding factors to get f(x) = (x^2 + 16)(x^2 + 25), which simplifies to f(x) = x^4 + 41x^2 + 400.

Explanation:

To form a polynomial, f(x), with real coefficients having a specified degree and given zeros, you must use the fact that non-real roots of polynomials with real coefficients come in conjugate pairs. Since 4i and 5i are given as zeros of the polynomial, their conjugates, -4i and -5i, must also be zeros of the polynomial.

The factors of the polynomial that correspond to these zeros are (x - 4i), (x + 4i), (x - 5i), and (x + 5i). To find the polynomial, these factors are multiplied together.

Multiplying the factors that come in conjugate pairs first:

Next, multiply these two results to obtain the polynomial:

(x^2 + 16)(x^2 + 25) = x^4 + 25x^2 + 16x^2 + 400 = x^4 + 41x^2 + 400

Therefore, the polynomial with degree 4 and zeros 4i and 5i is f(x) = x^4 + 41x^2 + 400.

To form the polynomial f(x) with real coefficients and given the zeros 4i and 5i, we use the conjugate pairs and multiply the resulting factors to get x⁴ + 41x² + 400.

To form a polynomial f(x) with real coefficients given the degree and zeros, we start by noting that complex roots always come in conjugate pairs when dealing with polynomials that have real coefficients. Hence, for the zeros 4i and 5i, the polynomial must also include the conjugate roots -4i and -5i.

Thus, the polynomial f(x) is x⁴ + 41x² + 400.

Answer:

0.36

Step-by-step explanation:

One - hundredth =[tex]\frac{1}{100}[/tex]

one hundredth means we have to divide 1 by hundred.

When we  divide one by hundred we get 0.01

To find one-hundredth of 36, than we have to multiply 36 by 0.01

 After multiplying we get

36×0.01=0.36

OR

36×[tex]\frac{1}{100}[/tex]=0.36

Hence, the correct answer is 0.36

One-hundredth of 36 is 0.36.

To find one-hundredth of 36, we simply need to divide 36 by 100. Dividing by 100 means shifting the decimal point two places to the left. Therefore, one-hundredth of 36 is 0.36.

For example, if we have the number 36, we can represent this as 36/1 to show that we have 36 whole parts. When we want one-hundredth of these 36 parts, we are seeking 1 part out of every 100 parts. So, we would write this as (36/1)/ (100/1) which simplifies to 36/100. After simplification, we get 0.36 as the final answer.

This concept is similar to understanding that 1 cm is 1/100th of a meter. By using division or understanding fractions and decimal places, we can easily find one-hundredth of any given number.

Answer:  The correct option is (D) (x - 3).

Step-by-step explanation:  Given that a polynomial function has a zero at x = 3.

We are to select the correct expression that must be a factor of the polynomial.

FACTOR THEOREM :   The theorem states that if x = a is a zero of a polynomial p(x), then (x - a) will be a factor of the polynomial p(x).

In case of our given polynomial, since x = 3 is a zero of the polynomial, so by factor theorem, we can say that

(x - 3) is a factor of the given polynomial.

Thus, the required factor is (x - 3).

Option (D) is CORRECT.

Final answer:

701.5 is equal to 701.50 because in decimals, trailing zeroes after the decimal point do not change the value of the number; they only indicate precision.

Explanation:

The question is about whether 701.5 is equal to 701.50. In mathematics, when comparing numbers with different numbers of decimal places, it's important to understand the significance of trailing zeroes in the decimal part.

A zero after the decimal point but at the end of a number (trailing zero) does not change the value of the number.

Therefore, 701.5 is exactly equal to 701.50

This is because the trailing zero in 701.50 does not add any value but merely serves as a placeholder, indicating precision to the hundredth place without changing the number's value.

The function f(x) = xe⁻ˣ is concave down for intervals where the second derivative f''(x) is negative, which occurs when x > 0.

To determine the intervals where the function f(x) = xe⁻ˣ is concave down, we need to find the second derivative of the function and identify where it is negative. The first derivative of the function f'(x) = e⁻ˣ - xe⁻ˣ is obtained using the product rule. Then, we take the second derivative: f''(x) = -e⁻ˣ - e⁻ˣ + xe⁻ˣ = -2e⁻ˣ + xe⁻ˣ.

To find where the function is concave down, we need to find the intervals where f''(x) < 0. By analyzing the second derivative, we can see that the term -2e⁻ˣ is always negative, while the term xe⁻ˣ can be either positive or negative depending on the value of x.

For x > 0, the term xe⁻ˣ is positive, making f''(x) more negative, therefore, the function is concave down for x > 0.

To convert 0.427 kg to pounds, multiply 0.427 by 2.2, resulting in approximately 0.939 pounds. Thus, the correct answer is Option B.

To convert 0.427 kg to pounds, we use the given conversion factor: 1 kilogram (kg) = 2.2 pounds (lb). Here is the step-by-step explanation:

Write down the value to be converted: 0.427 kg.

Multiply by the conversion factor: 0.427 kg × 2.2 lb/kg.

Perform the multiplication: 0.427 × 2.2 = 0.9394 lb.

So, 0.427 kg is approximately 0.939 lb. Therefore, the correct answer is Option B.

Final answer:

The quadratic equation 5x² - 5x - 100 is factored completely by first taking out the greatest common factor of 5, and then finding two binomials that multiply to give the remaining quadratic. The correct factorization is 5(x - 5)(x + 4), which is option A.

Explanation:

To factor the quadratic equation 5x² − 5x − 100 completely, we need to find two binomials that, when multiplied together, will give us the original equation. First, we will factor out the greatest common factor (GCF), which is 5. This leaves us with:

5(x² − x − 20)

Now, we need to factor the quadratic expression inside the parentheses. The factors of −20 (the constant term) that add up to −1 (the coefficient of the middle term) are −5 and 4. Thus:

5(x − 5)(x + 4)

We can confirm that none of the other choices provide the correct factors:

Therefore, the correct answer is 5(x − 5)(x + 4), which corresponds to option A.

Answer:

The probability that a ticket that is randomly chosen will award a larger prize is:

                        1/5=0.2

Step-by-step explanation:

Let A denote the event that the ticket is a winning ticket.

B denote the event that there is a larger prize.

A∩B denote the event that there is a larger prize on the winning ticket.

Let P denote the probability of an event.

Now according to the given information we have:

[tex]P(A)=\dfrac{6}{10}[/tex]

Also, [tex]P(B|A)=\dfrac{1}{3}[/tex]

Hence, we are asked to find: P(A∩B)

We know that:

[tex]P(B|A)=\dfrac{P(A\bigcap B)}{P(A)}\\\\\\\dfrac{1}{3}=\dfrac{P(A\bigcap B)}{\dfrac{6}{10}}\\\\\\P(A\bigcap B)=\dfrac{1}{3}\times \dfrac{6}{10}\\\\P(A\bigcap B)=\dfrac{2}{10}=\dfrac{1}{5}=0.2[/tex]

            The probability is:

              1/5=0.2

The overall probability that a randomly chosen ticket will award a larger prize is 0.2, or 20%, obtained by multiplying the probability of drawing a winning ticket (0.6) with the probability of winning a larger prize if the ticket is a winner (0.333).

For the probability that a ticket randomly chosen will award a larger prize, given that 6 out of every 10 tickets are winners and of those, 1 out of every 3 is a larger prize.

First, we calculate the probability of drawing a winning ticket, which is 6/10 or 0.6.

Then we compute the probability of the winning ticket awarding a larger prize, which is 1/3 or approximately 0.333.

To find the overall probability of a randomly chosen ticket awarding a larger prize, we multiply these two probabilities together:

Overall probability = Probability of winning ticket×Probability of larger prize on winning ticket

Overall probability = 0.6 ×0.333

Overall probability ≈ 0.2

So, the probability that a ticket awards a larger prize is 0.2, or 20%.

Answer:

C)18

Step-by-step explanation:

Let Sara present age = x years.

According to question Ralph present  age =3x

Sara age after 6 years =x+6.

Ralph age after 6 years =3x+6.

After 6 years Ralph will be only twice as old as Sara will be then.

3x+6=2(x+6)

Solving for x:

3x+6=2x+12

3x-2x=12-6

x=6.

Ralph present age = 3x=3(6)= 18 years.

Option C.