Given the function f(x)= x^4+6x^3-x^2-30x+4
Use the intermediate value Theorem to decide which of the following intervals contains at least one zero (there are 4 answers)

A) [-5,-4]
B) [-4,-3]
C) [-3,-2]
D) [-1,0]
E) [0,1]
F) [1,2]

An algebraic description of the transformation that translates a point (x, y) nine units to the right and five units down is written as the new point (x + 9, y - 5).

The algebraic description of a transformation that translates a point (x, y) nine units to the right and five units down can be written as the following ordered pair: (x + 9, y - 5).

In this transformation, the x-coordinate of each point is increased by 9 units, reflecting a horizontal translation to the right, and the y-coordinate is decreased by 5 units, representing a vertical translation downwards. Thus, if you have a point (x, y), after the transformation, it becomes (x + 9, y - 5).

For example, if your initial point is (3, 4), after the transformation it would be at (3 + 9, 4 - 5) = (12, -1).

The standard form of the equation of the parabola with focus at (1, 1) and directrix 'y = 7' is (x - 1)² = 12(y - 4). The vertex is found by determining the midpoint between the focus and directrix, and the 'a' value is found by calculating half the distance between the focus and directrix.

To find the standard form of a parabola given the focus and directrix, we'll use the definition of a parabola itself. A parabola is defined as the set of points that are equidistant from a point, known as the focus, and a line, known as the directrix.

The equation representing a parabola is: (x - h)² = 4a(y - k) where (h, k) is the vertex of the parabola and 'a' denotes the distance between the vertex and the focus or the vertex and the directrix.

The mid-point of the focus (1, 1) and the directrix y=7 gives the coordinates of the vertex. So, h = 1 and k = (1+7)/2 = 4.

Now, 'a' is half the distance between the focus and directrix, a = (7-1)/2 = 3.

So, the standard form of the equation is (x - 1)² = 12(y - 4)

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The interval of x-values over which the function is increasing is (-3 , -1), the correct option is B.

A function is a law that relates two variables namely, a dependent and an independent variable.

A function always has a defined range and domain.

The x values and the corresponding function values are shown in the figure.

The x and y values are:

-6      -34

-5       3

-4      -10

-3      -11

-2      -6

-1        -1

0        -2

1        -15

The  interval of x-values over which the function is increasing has to be determined.

The first option is (-6, -3) , the function is not increasing in this interval.

The second option is (-3 , -1) , the function is increasing in this interval.

The third option is(–3, 0), the function is not increasing in this interval.

The fourth option is (–6, –5), the function is not increasing in this interval.

Your question seems incomplete, the complete question is attached with the answer.

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We can actually solve this problem using the cosine law. We know that Gustavo ran 23° above the horizontal, while Aiden ran 33° below, hence the total angle between them is 56°. Therefore the distance between them is:

c^2 = a^2 + b^2 – 2 a b cos θ

c^2 = (300)^2 + (250)^2 – 2 (300) (250) cos 56

c^2 = 68,621.06

c = 262 m

They are 262 meters apart.

Gustavo ran 23° above the east horizontal, while Aiden ran 33° below the east horizontal. So, they are 262 meters apart.

Cosine law is a formula relating the length of the sides of a triangle to the cosine of one angle of the triangle.

Gustavo ran 23° above the east horizontal, while Aiden ran 33° below the east horizontal.

The total angle between them = 23 + 33 = 56°.

Let the distance between them be c. by using cosine law we have,

c² = a² + b² – 2ab cos θ

Where a and b are distances traveled by Gustavo and Aiden. θ is the angle between them.

Therefore,

[tex]c^2 = a^2 + b^2 - 2 a b cos \theta[/tex]

[tex]c^2 = (300)^2 + (250)^2 -2 (300) (250) cos 56[/tex]

[tex]c^2 = 68,621.06[/tex]

c = 262 m

Thus, the distance between them is 262 meters.

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we know that

if the lines bd and xy are parallel. then

[tex]y+70\°=180\°[/tex] --------> by consecutive interior angles

so

Solve for y

Subtract [tex]70\°[/tex] both sides

[tex]y+70\°-70\°=180\°-70\°[/tex]

[tex]y=110\°[/tex]

therefore

the answer is

[tex]y=110\°[/tex]

Answer:

Option C. y = 110°

Step-by-step explanation:

In this question BD and XY are two parallel line and a transverse is intersecting both the lines at A and E.

Now ∠BAE = ∠AEY = 70° [ alternate interior angles ]

and ∠EAD + ∠BAE = 180° [ supplementary angles ]

or y + 70° = 180°

⇒ y + 70 - 70 = 180 - 70

⇒ y = 110°

Therefore Option C. measurement of y = 110° is the answer.

Dublin is 3 hours ahead of Moscow.

The difference between how time is measured in different ways; the length of a period of time. the time difference between two or more distinct time zones in terms of clock time.

Given Dublin time = 11:30 am

Moscow  time = 2:30

we can military time and subtract.

11:30 a.m. = 11:30

2:30 p.m. = 2:30 + 12:00 = 14:30

14:30 - 11:30 = 3:00

Hence the time difference is 3 hours.

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Answer:

Step-by-step explanation:

Given that the sophomore class sold a total of 700 adult and student tickets to the school play. adult tickets sold for $4.50 each, and student tickets cost $2.00 each.

Total = 1200

x, the number of student tickets sold, and y, the number of adult tickets sold

Then we have

[tex]x+y =700\\4.5x+2y = 1200[/tex]

These are the system to solve for x

Answer:

Step-by-step explanation:

y=3.14x

Answer:

4mph

5mph

slower

Step-by-step explanation:

i did it

To calculate the average rate of change, divide the total change in distance by the total change in time. Without the specific distance covered in the given durations, it's impossible to provide absolute answers, but an example calculation has been given using an average speed of 9.5 mph.

In your question, the average rate of change refers to the change in distance with respect to time, also known as velocity or speed. Since there are no concrete values mentioned in your question for the distances covered by a runner between time 0.5 and 2 hours, and 1.5 to 2.5 hours, it would be impossible to provide an accurate answer without such data.

However, allow me to explain how you could calculate these values. If you assume that a runner has the marathon runner's average speed of 9.5 miles per hour, then the average rate of change, or average speed, over a certain time period is just the total change in distance divided by the total change in time. For instance, if the runner covered 9.5 miles between 0.5 and 2 hours, then the average rate of change would be 9.5 miles per 1.5 hours, or approximately 6.33 mph. This same method would apply to calculations between 1.5 and 2.5 hours.

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Answer:

[tex]10^5 * 10^6 = 10^{5+6} = 10^{11}[/tex]

Step-by-step explanation:

find the product of 100,000 and 1,000,000 using scientific notation

All whole numbers have a decimal point at the end

convert each number into scientific notation. To get scientific notation move the decimal point after 1

100000.  -----> 1.00000 ----> 1* 10^5 ----> 10^5

1000000.  -----> 1.000000 ----> 1* 10^6 ----> 10^6

a^m * a^n = a^m+n

[tex]10^5 * 10^6 = 10^{5+6} = 10^{11}[/tex]

The step that would not be necessary in solving 5x + 15 + 2x = 24 + 4x is D. Use the distributive property.

An equation is a pair of algebraic equations with the equal sign (=) in the middle and the same value.

To solve the equation 5x + 15 + 2x = 24 + 4x, we need to simplify and manipulate the equation to isolate the variable on one side.

First, we can combine like terms:

5x + 2x - 4x = 24 - 15

3x = 9

Then, we can isolate x by dividing both sides by 3:

x = 3

Therefore, the correct steps to solve this equation are:

B. Collect variable terms on one side.

C. Collect like terms.

A. Isolate the variable.

We do not need to use the distributive property in this problem.

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Answer:

C - Opposite sides are congruent  

D - Corresponding parts of congruent triangles are congruent

Step-by-step explanation:

A parallelogram is a quadrilateral whose both pairs of opposite sides are parallel. In a parallelogram, the 2 sets of opposite sides are congruent. Also it has 2 sets of opposite angles congruent.

Therefore, the following are necessary when proving the opposite sides of a parallelogram are congruent.

C - Opposite sides are congruent  

D - Corresponding parts of congruent triangles are congruent

Answer:

Option A    

Step-by-step explanation:

Given : Equation [tex]y-1 = 2(x -2)[/tex]

To find : Which graph represents the given equation

Solution :

First thing we plot the graph of the equation.

We see that graph cuts at point (1.5,0) and (0,-3) to the coordinates.

From the given graph Option A graph cuts to these points.

Therefore, Option A is correct.

Refer the attached graph A.

ANSWER

We need to rewrite [tex]x+2y=6[/tex], in slope intercept form.

[tex]\Rightarrow 2y=-x+6[/tex],

[tex]\Rightarrow y=-\frac{1}{2}x+3[/tex]

The given equation has a slope of [tex]-\frac{1}{2}[/tex].

ANSWER TO QUESTION 1

We now the write given options also in slope intercept form.

For the first one, we have

[tex]y=-\frac{1}{2}x+5[/tex]

This first equation also has a slope  of [tex]-\frac{1}{2}[/tex].

Hence [tex]y=-\frac{1}{2}x+5[/tex] is parallel to [tex]x+2y=6[/tex].

ANSWER TO QUESTION 2:

The second equation is;

[tex]-2x+y=-4[/tex].

We need to write this one too in slope intercept form.

[tex]y=2x-4[/tex]

This second equation has a slope of 2

Since [tex]2\times -\frac{1}{2}=-1[/tex], the line [tex]-2x+y=-4[/tex]

is perpendicular to [tex]x+2y=6[/tex].

ANSWER TO QUESTION 3

We rewrite the equation [tex]-x+2y=2[/tex] in slope intercept form.

[tex]\Rightarrow 2y=x+2[/tex]

[tex]\Rightarrow y=\frac{1}{2}x+1[/tex]

This equation also has a slope of [tex]\frac{1}{2}[\tex]

since the slope of [tex]-x+2y=2[/tex] is not equal to the slope of [tex]x+2y=6[/tex], and the product of these two slopes [tex]\frac{1}{2} \times - \frac{1}{2} \ne -1[/tex], the two lines are neither parallel nor perpendicular.

QUESTION 4

The given line has equation [tex]x+2y=-2[/tex]. We rewrite this equation in the slope intercept form.

[tex]\Rightarrow 2y=-x-2[/tex]

[tex]\Rightarrow y=-\frac{1}{2}x-1[/tex]

This line has a slope of [tex]-\frac{1}{2}[/tex].

Since this slope is the same as the slope of the given line,

[tex]y=-\frac{1}{2}x-1[/tex] is parallel to  [tex]x+2y=6[/tex].

See graph in attachment.

Expression after the simplification is 62.5.

In mathematics, expression is the combining of variables with the application of operations and prescribed rules. It could take the shape of an equation, some numbers, etc.

Given, expression

55 - 4.5 +12

Simplifying,

first, add the numbers.

55 - 4.5 +12

= 67 - 4.5

= 62.5

Hence, the value of the expression is 62.5.

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Answer: C. The amount earned is a function of the number of hours worked.

Explanation:

Since, a function is always the type of y= f(x), where x is independent variable and y is dependent variable which is dependent on the value of x.

Here,  a graph of Olivia's earning per hour is given.

Since, it is the obvious thing that the earning will depend on hours she works.

Thus, we can say that earning is dependent variable while hours is independent variable.

Therefore, earning makes the y-axis and number of hours makes the x-axis.

So, option C is correct.

The dimensions of the swimming pool are 51 feet (length) and 12 feet (width), determined by setting up equations based on the given information (perimeter and difference between the sides) and solving these equations using substitution.

The problem at hand is a classic algebraic word problem. It involves determining the length (x) and width (y) of a swimming pool given the perimeter and the difference between the length and width. The formulas we'll use are for the perimeter of a rectangle (2x + 2y = perimeter) and a given equation based on the description (x - y = difference).

So the dimensions of the swimming pool are 51 feet (length) and 12 feet (width).

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Final answer:

To evaluate the given integral, split the function at x = 3 and integrate separately. The integral evaluates to 18, which makes the correct answer 18.

Explanation:

The function we're integrating is 1, which makes this the same integral as the one in example 3 on p. 178. To evaluate this integral, we need to split it at x = 3 and integrate the corresponding functions separately.

For x ≤ 3, the function is f(x) = 3. So, the integral from 0 to 3 of 3 dx equals 9.

For x > 3, the function is f(x) = 6 - x. Integrate this from 3 to 6: ∫(6 - x) dx = 6x - 0.5x^2. Evaluating this from 3 to 6 gives 9.