Quadratic Functions
Instruction Active
TRY IT Finding the Zeroes of a Quadratic Function
-2
-1
0
1
2
a = -2
b=0
c=2
f(x)=x²+x+6
X
0
X
6
X 4
a
4
b
6
с
The axis of symmetry is
x x=-1
x x=0
✓
x = ½
COMPLETE
The smaller zero is
The larger zero is
DONE

1. To determine if the pair of planes 3x + 2y + z = 8 and x + 3y – z = 5 are parallel or intersecting, we can check if their normal vectors are parallel or not.

The normal vector of the first plane is <3, 2, 1>, and the normal vector of the second plane is <1, 3, -1>. These normal vectors are not parallel, so the planes intersect at a line.

To find a parametrization for the line of intersection, we can set one of the variables to be a parameter and solve for the other two. Let's choose z as the parameter.

From the second plane equation, we have z = x + 3y - 5. Substituting this into the first plane equation, we get: 3x + 2y + (x + 3y - 5) = 8

Simplifying, we get: 4x + 5y = 13

Solving for y in terms of x, we get: y = (13 - 4x) / 5

So a parametrization for the line of intersection is: x = t

y = (13 - 4t) / 5, z = t + 3((13 - 4t) / 5) - 5 = (2t + 2) / 5

2. To determine if the pair of planes 2c – y + z = 4 and 4c – 2y + 2z = 3 are parallel or intersecting, we can again check if their normal vectors are parallel or not.

The normal vector of the first plane is <2, -1, 1>, and the normal vector of the second plane is <4, -2, 2>. These normal vectors are parallel (in fact, they are scalar multiples of each other), so the planes are parallel.

To find the distance between the two parallel planes, we can take any point on one plane and find its distance to the other plane. Let's choose the point (0, 0, 4) on the first plane. The distance between this point and the second plane is: |4c - 2(0) + 2(4) - 3| / sqrt(4^2 + (-2)^2 + 2^2) = |4c - 5| / 2sqrt(6)

So the distance between the two parallel planes is |4c - 5| / 2sqrt(6).

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As a result, the answer is (e) 2.4 square inches, which is the difference between Richard's measured and real circle area.

In mathematics, area is a measure of the amount of space occupied by a two-dimensional object, such as a rectangle, triangle, circle, or any other shape. It's a scalar quantity that describes the size of a region in two-dimensional space. The units of area are typically square units, such as square inches, square centimeters, square meters, etc. In general, the area of a shape is a measure of how much space it occupies, and it is an important concept in geometry, engineering, and many other fields.

Here,

The formula for the area of a circle is given by:

A = πr²

Where r is the radius of the circle.

Using the incorrect radius of 3.6 inches, the calculated area would be:

A = π * (3.6 inches)² = 40.44 square inches

Using the correct radius of 3.5 inches, the actual area would be:

A = π * (3.5 inches)² = 38.5 square inches

So, the difference between Richard's measured area of the circle and the actual area of the circle would be:

40.44 square inches - 38.5 square inches = 1.94 square inches

Therefore, the answer is (e) 2.4 square inches that is the difference between Richard’s measured area of the circle and the actual area of the circle.

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The correct initial value problem is: y'' + (9.81/0.75) y = 0, y(0) = -0.36, y'(0) = -0.45.

Here's how you can set up the initial value problem

Let y(t) be the displacement of the mass from its spring-mass equilibrium position at time t. Then, the force acting on the mass is given by Hooke's Law:

F = -ky

where k is the spring constant.

Since the mass is at rest at its equilibrium position, the force acting on it is balanced by the force of gravity.

mg = ky_eq

where y_eq is the equilibrium position of the spring-mass system.

Substituting F = -ky into the equation and solving for k, we get:

k = mg/y_eq

Substituting k and the values given, we get:

y'' + (9.81/0.75) y = 0

where y'' is the second derivative of y with respect to time.

When the mass is displaced 36 cm upward from its spring-mass equilibrium and then given a sharp downward tap, the initial conditions are:

y(0) = -0.36 m

y'(0) = -0.45 m/s (note that the velocity is downward, so it's negative)

Therefore, the initial value problem that models this experiment is:

y'' + (9.81/0.75) y = 0

y(0) = -0.36

y'(0) = -0.45

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The exact value of all six trigonometric functions:

sin(θ) = 0.55

cos(θ) = 0.83

tan(θ) = 0.66

cot(θ) = 1.51

sec(θ) = 1.2

cosec(θ) = 1.81

Let us assume that x represents the horizontal distance(x-axis), y represents the vertical distance and r be the radius.

here, r = 6 units and x = 5 units

Using Pythagoras theorem,

y = [tex]\sqrt{r^2-x^2}[/tex]

y =  [tex]\sqrt{6^2-5^2}[/tex]

y = 3.32 units

Now we find the exact value of all six trigonometric functions.

Let us assume that θ be an angle made by radius with positive x-axis.

sin(θ) = y/r

sin(θ) = 3.32 / 6

sin(θ) = 0.55

cos(θ) = x/r

cos(θ) = 5/6

cos(θ) = 0.83

tan(θ) = y/x

tan(θ) = 3.32 / 5

tan(θ) = 0.66

cot(θ) = x/y

cot(θ) = 5/3.32

cot(θ) = 1.51

sec(θ) = r/x

sec(θ) = 6/5

sec(θ) = 1.2

cosec(θ) = r/y

cosec(θ) = 6/3.32

cosec(θ) = 1.81

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Andre's number is 43.

Generally, Let's call the tens digit of Andre's number "t" and the ones digit "o".

We know that Andre can make the number in 8 different ways. This means that there are 8 different combinations of t and o that can make up the number.

One of these ways has 1 more ten than ones, which means that the number is 10 more than the sum of the digits. In other words:

t + o + 10 = 10t + o + 1

Simplifying this equation, we get:

9t = 9o + 9

Dividing both sides by 9, we get:

t = o + 1

This means that the tens digit is 1 more than the ones digit, which narrows down our possibilities.

Since the number is a 2-digit number, the tens digit must be between 1 and 9 (inclusive), and the ones digit must be between 0 and 9 (inclusive). We also know that the tens digit is 1 more than the ones digit.

Using these constraints, we can create a table of all the possible combinations of t and o that satisfy these conditions:( table attached below)

Out of these 8 numbers, only one of them can be made in a way where there is 1 more ten than ones. Looking at the table, we can see that the number is 43, since it is the only number where the tens digit is 1 more than the ones digit, and it can be made by combining the digits in a way where there is 1 more ten than ones (i.e., 3 tens and 4 ones).

Therefore, Andre's number is 43.

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a. The average American spend on Food & Beverage is $152.2. b. The average American spend on Entertainment/Recreation is $98.93. c. They spent $53.27 more on food than on entertainment.

In order to show mathematical issues, the "pie chart," often referred to as a "circle chart," divides the circular statistical visual into sectors or portions. A proportionate amount of the entire is indicated by each sector. The Pie-chart is now the most effective method for determining the composition of something. Pie charts typically take the role of other graphs, such as bar graphs, line plots, histograms, etc.

Given that, the average American spent $761.00.

a. Money spent on Food and Beverages is:

F = 761(20/100) = $152.2

b. Money spent on Entertainment is:

E = 761(13/100) = $98.93

c. The money spent on food is more than entertainment by:

A = 152.2 - 98.93 = $53.27

Hence, a. The average American spend on Food & Beverage is $152.2. b. The average American spend on Entertainment/Recreation is $98.93. c. They spent $53.27 more on food than on entertainment.

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The 45th Percentile from the data is 7.65

Percentiles are a method for dividing data into 100 equal parts. So, there are 99 percentile values.

single data percentile formula is

Px = [tex]\frac{x (y+1)}{100}[/tex]

Px = Percentile x

y = lots of data

How to find the 45th percentile weight of newborn babies?

first, we must be sorted from the smallest to the largest.

5.5   6.5   6.5   7.1   7.3   7.4   7.6   7.7   7.8   8.1   8.3   8.4   8.5   8.6   9.4

After sorting the data, we can plug it into a single data percentile formula

Px = P45

x = 45

y = 15

P45 = [tex]\frac{45 (15+1)}{100}[/tex]

P45 = [tex]\frac{36}{5}[/tex]

P45 = 7.2

The 45th percentile is in the order of the 7th & 8th digits

the order of the 7th & 8th numbers is 7.6 & 7.7

after that we average the 7th & 8th numbers to find out the 45th percentile value

[tex]\frac{7.6 + 7.7}{2}[/tex] = [tex]\frac{15.3}{2}[/tex] = 7.65

the 45th percentile of newborn weight data is 7.65

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The areas of the triangles are as follows:

1. 52.5  ft²

2. 369 inches²

The area of a triangle can be represented as follows:

area of a triangle = 1 / 2 bh

where

Therefore, let's find the areas of the triangle using the formula as follows:

1.

area of the triangle = 1 / 2 × 15 × 7

area of the triangle = 1 / 2 × 105

area of the triangle =  52.5  ft²

2.

area of the triangle = 1 / 2 × 41 × 18

area of the triangle = 1 / 2 × 738

area of the triangle = 369 inches²

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The matching parts of vectors a and c to find a + c. For instance, (1, -2, 4) + (2, 3, -1) = (3, 1, 3).

The equations and calculations to determine the vectors a and c's sum are as follows:

Assume that vector an is shown as (a1, a2, a3) and that vector c is shown as (c1, c2, c3).

Then, by adding each of their respective components, it is possible to determine the vectors a and c's sum, denoted as a + c:

A plus C equals (A+C1, A+C2, A+C3, etc.)

For instance, if a = (2, 3, -1) and c = (1, -2, 4), we can use the following formula to get their sum:

a + c = (2 + 1, 3 - 2, -1 + 4) = (3, 1, 3) (3, 1, 3)

As a result, the vectors a and c's sum is (3, 1, 3).

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The missing values in the table are

3/2, 4/3 and 4/9

The table shows a proportion of

oats =  flours * 4/9

the proportion was gotten from

1/3 / 3/4 = 4/9

The missing parts are

let the missing value be x

x * 4/9 = 2/3

x =  2/3 / 4/9

x = 3/2

3 * 4/9 = x

x = 4/3

1 * 4/9 = x

x = 4/9

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Answer:

We can use the formula:

distance = rate x time

The total distance traveled by the plane is 3,000 + 600 = 3,600 miles. The time it took to cover this distance is 2 hours and 30 minutes, or 2.5 hours. So we have:

3,600 = rate x 2.5

Solving for the rate, we get:

rate = 3,600 / 2.5 = 1440 mph

Therefore, the answer is (A) 1440mph

The coordinates of both positions of point B is (2√5,0) and B2: (0,2√5).

The intersection of the graph of the function with the y-axis gives y-intercept of that function. The y-intercept is the value of y on the y-axis at which the considered function intersects it.

Assume that we've got: y = f(x)

At y-axis, we've got x = 0, so putting it will give us the y-intercept.

Thus, y-intercept of y = f(x) is y = f(0)

We are given that;

Point b=  2 root 5 units

Now,

Since point B is 2√5 units away from point A, we can draw two circles with radius 2√5 centered at point A. The two intersections of these circles with the gridlines will give us the two possible positions of point B.

For example, if point A has coordinates (0,0) and the gridlines are the x-axis and y-axis, then the two possible positions of point B are:

B1: (2√5,0) - the intersection of the circle with the positive x-axis

B2: (0,2√5) - the intersection of the circle with the positive y-axis

If the gridlines are different, then the positions of point B will be different.

Therefore, by the intercept the answer will be (2√5,0) and B2: (0,2√5).

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There was 19% percent error of the baker's prediction.

Percent error is the difference between the estimated value and the actual value in comparison to the actual value and is expressed as a percentage.

Given that, a bakery sold 105 cupcakes in one day, the head baker predicted he would sell 85 cupcakes that day.

We are asked to find the percent error of the baker's prediction,

Percent error = |expected value - exact value| / exact value × 100 %

The expected value is the prediction of the Chef = 85

The exact value = 105

Percent error = |85-105| / 105 × 100 %

= 20/105 × 100 %

= 19%

Hence, there was 19% percent error of the baker's prediction.

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The measure of arc AD is 25°.

AC and BD are chords of the circle. The two chords intersect at the point E which makes an angle 93°. The measure of arc BC is 161°. To find the measure of arc AD:

The measure of arc AD by using the property that "if two chords intersect in the interior of the circle, then the measure of each angle is half the sum of the arcs intercepted by the angles and its vertical angle".

Thus, applying the above theorem:

93° = 1 /2 (arcBC + arcAD)

Here, measure of angle E is 93°.

93° = 1 / 2( 161 + AD)

93 = 161 / 2 + AD / 2

93 = 161/ 2 +  AD/2

solving further ( multiplying equation by 2 on both sides)

186 = AD + 161

AD = 186 - 161

arc AD = 25°

So, the measure of arc AD is  25°.

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____The given question is incomplete, the complete question is given below:

Find each value or measure. Assume that segments that appear to be tangent are tangent. Find arc AD.

The solution of the inequality is -3.

Mathematical expressions with inequalities are those in which the two sides are not equal. Contrary to equations, we compare two values in inequality. Less than (or less than or equal to), greater than (or greater than or equal to), or not equal to signs are used in place of the equal sign.

Given:

22 ≤-7y+1

Solving the inequality

22 ≤-7y+1

22 - 1 ≤-7y

21 ≤-7y

21/7 ≤-y

3 ≤-y

y≤ -3.

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a) The pet shelter has 40% cats (20% black, 5% adopted), and 60% dogs (10% black, 10% adopted). 50% are adopted in non-black cats and dogs, b) 0.45, c) 0.19, d) 0.15, e) 0.375, f) 0.6.

a) The probability tree would seem like this:

P(Cat) = 40%

P(Black Cat) = 20%

P(Adopted Black Cat) = 5%

P(Not Adopted Black Cat) = 95%

P(Not Black Cat) = 80%

P(Adopted Not Black Cat) = 50%

P(Not Adopted Not Black Cat) = 50%

P(Dog) = 60%

P(Black Dog) = 10%

P(Adopted Black Dog) = 10%

P(Not Adopted Black Dog) = 90%

P(Not Black Dog) = 90%

P(Adopted Not Black Dog) = 50%

P(Not Adopted Not Black Dog) = 50%

b) P ( Adopted & Not Black Dog ) = P(Not Black Dog) x P(Adopted | Not Black Dog) = 0.90 * 0.50 = 0.45

c) P(Not Adopted & Black Cat) = P(Black Cat) x P(Not Adopted | Black Cat) = 0.20 * 0.95 = 0.19

d) P(Adopted) = P(Adopted Black Cat) + P(Adopted Not Black Cat) + P(Adopted Black Dog) + P(Adopted Not Black Dog) = 0.05 + 0.50 + 0.10 + 0.50 = 0.15

e) P(Adopted | Cat) = P(Adopted & Cat) / P(Cat) = (0.05 + 0.50) / 0.40 = 0.375

f) P(Cat | Adopted) = P(Cat & Adopted) / P(Adopted) = (0.05 + 0.50) / 0.15 = 0.6.

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The image point will be (0, -4).

When rotating a point 180 degrees counterclockwise about the origin our point A(x,y) becomes A'(-x,-y). So all we do is make both x and y negative.

Given:

Coordinate of a point → (0, 1)

Rule for the transformation has been given as,

[tex]R_{180[/tex] o  D₄

First use [tex]R_{180[/tex] → Rotation of the point by 180° counterclockwise

Rule :

(x, y) → (-x, -y)

(0, 1) → (0, -1)

Now, D₄ → Dilation of the point by a scale factor of 4 about the origin

If a point (x, y) is dilated by a scale factor 'k' about the origin rule for the dilation is,

(x, y) → (kx, ky)

By applying this rule,

(0, -1) → (0, -4)

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Answer:

$60  per TV sold

Step-by-step explanation:

Multiply $25 by 17 to find out how much she earned from the extended warranties

so 25 x 17 = 425

1445-425= 1020

now divide 1020 by 17 to find out how much neda earned from each TV

1020/17 = 60

Answer:

  September, October, November

Step-by-step explanation:

When monthly rainfall in centimeters is represented by A(t) = 2.3sin(πt/6)+9.25 and A(0) represents July's rainfall, you want to know the months in which average rainfall is at least 10.5 cm.

We want to find the values of t that make A(t) ≥ 10.5:

  2.3sin(πt/6) +9.25 ≥ 10.5

  2.3sin(πt/6) ≥ 1.25 . . . . . . subtract 9.25

  sin(πt/6) ≥ 0.543478 . . . . divide by 2.3

  πt/6 ≥ 0.574575 . . . . . . . take inverse sine

  t ≥ 1.097

The sine function is symmetrical about π/2, so this also means solutions will be of the form ...

  πt/6 ≤ π -0.574575

  t ≤ 6 -1.097 ≈ 4.903

The month numbers that will have rainfall at least 10.5 cm will fall in the range ...

  1.097 ≤ t ≤ 4.903

  t ∈ {2, 3, 4}

If July is month 0, then these months are September, October, November.

Jacy's hometown will get at least 10.5 cm of rain in September, October, and November.

__

Additional comment

The attachment confirms this result. We have shifted the rainfall function so it can use conventional month numbers. It shows months 9, 10, 11 have rainfall above 10.5 cm.

<95141404393>

The requried properties are true for all the parallelograms are,
1. Opposite sides are parallel     2. Opposite sides are congruent

A parallelogram is a quadrilateral consisting of pairs of parallel sides.

The following properties are true for all parallelograms,
The opposite sides are parallel.
The Opposite sides are congruent.

All parallelograms have four congruent sides (property 1), four right angles (property 2), or exactly one pair of parallel sides (property 3). These properties only apply to specific types of parallelograms, such as rectangles or squares.

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The surface area of the composite figure is 415. 25 in²

From the image shown, we can see that the figure is made up of a cuboid and a semicircle.

The formula for calculating the surface area of a cuboid is given as;

Surface area =2lw+2lh+2hw

Given that, l is the length, h is the height and w, the width.

Surface area = 2(6×10) + 2(6×8) + 2(8 × 10)

expand the brakect

Surface area = 120 + 96+ 160

Surface area = 376 in²

Surface area of a semicircle = 1/2(πr2 ),

Substitute the values

Surface area = 1/2 ×22/7 ×5² = 39.25 in²

Hence, the value is the sum of the surface areas

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The mass of the water is 7880.1 grams.

Specific heat capacity is the amount of heat energy required to raise the temperature of a substance per unit of mass.

Mathematically -

Q = mcΔT

where -

{Q} = heat energy

{m} = mass

{c} = specific heat capacity

{ΔT} =change in temperature

Given is the water in a container changes its temperature from 12°C to 38°C, when 205 calories are transferred to it.

Specific heat capacity of water = 4.186 J/g°C

205 calories = 857720 Joules

We can write -

Q = mcΔT

857720 = m x 4.186 x (38 - 12)

857720 = 108.836 x m

m = 857720/108.836

m = 7880.1 grams

Therefore, the mass of the water is 7880.1 grams.

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{QUESTION IN ENGLISH -

The water in a container changes its temperature from 12°C to 38°C, when 205 calories are transferred to it. What is the mass of water in the container?}

The solution to the system of equations is x = -8, y = 9.

To solve the system of equations by elimination, we must manipulate the two equations so that when the equations are added or subtracted, one of the variables is eliminated.

One method is to multiply one or both equations by a constant, resulting in opposite coefficients for one of the variables. In this case, we can multiply the first equation by 4 and the second equation by -5, yielding the following y coefficients:

code :

(4)(5x + 4y = -4) => 20x + 16y = -16 (-5)

(4x + 4y = 4) => -20x - 20y = -20

We can now combine the two equations to eliminate y:

code :

20x + 16y = -16 \s-20x - 20y = -20 \s———————

0x - 4y = -36

When we simplify the result, we get:

code :

-4y = -36 y = 9

Now we can plug y = 9 back into one of the original equations to find x. Let's look at the first equation:

code :

5x+4y = -4 5x+4(9) = -4 5x+36 = -4 5x = -40 x = -8

As a result, the system of equations solution is x = -8, y = 9.

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a. To find f(-4), we simply substitute -4 for x in the function f(x):

f(-4) = (-4)² - 3(-4) = 16 + 12 = 28

Therefore, f(-4) = 28.

b. To find g(1)-f(-3), we first need to find g(1) and f(-3) separately.

g(1) = 2(1) - 5 = -3

f(-3) = (-3)² - 3(-3) = 9 + 9 = 18

Then, we can substitute these values into the expression:

g(1)-f(-3) = -3 - 18 = -21

Therefore, g(1)-f(-3) = -21.

c. To find f(g(-4)), we first need to find g(-4) and then substitute this value into f(x).

g(-4) = 2(-4) - 5 = -8 - 5 = -13

Then, we can substitute -13 for x in the function f(x):

f(g(-4)) = (-13)² - 3(-13) = 169 + 39 = 208

Therefore, f(g(-4)) = 208.

Answer: 17x = 54

x represents chau's savings before it increased

Since the optimal production level is within the company's maximum capacity of 150 items, the company should produce and sell 75 items to maximize profit.

A relation between a collection of inputs and outputs is known as a function. A function is, to put it simply, a relationship between inputs in which each input is connected to precisely one output.

To find the optimal production level a that maximizes profit, we need to determine the point at which marginal revenue equals marginal cost.

The marginal cost function is the derivative of the cost function, which is C'(a) = 50.

The marginal revenue function is the derivative of the revenue function, which is R'(a) = -0.5(3/8 - 110).

Setting these two equal and solving for a gives:

50 = -0.5(3/8 - 110)

a = 75

Since the optimal production level is within the company's maximum capacity of 150 items, the company should produce and sell 75 items to maximize profit.

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The edge of the cube is equal to 2 m³.

There are different quadrilaterals, for example: square, rectangle, rhombus, trapezoid and parallelogram.  Each type is defined accordingly to its length of sides and angles. For example,  in a square, all angles are 90° and all sides present the same value.

A square is a 2D figure, when the square is associated with a 3D figure, the figure is called a cube.

The cube presents 6 faces with length (l), height (h) and width (w) equal. Thus, the cube presents congruent edges. The volume is given from the formula V=a³, where a is a congruent edge.

The exercise gives the volume of the cube 8m³. As presented previously, the edges of the cube are congruent and the formula by the volume is V=a³. Thus,

V=a³

8=a³

[tex]\sqrt[3]{a^3}[/tex]=[tex]\sqrt[3]{8}[/tex]

a=[tex]\sqrt[3]{2^3}[/tex]

a=2 m

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Answer:

(2x3) / (3x7) = 6/21

We simplify to get 2/7

Answer: 5/21 ?

im not 100% sure this is correct but I think so. sorry if its wrong

Strictly using the limit definition of the derivative to find the derivative of the function:

The limit definition of the derivative looks like this:

[tex]\lim_{h \to \zero} \frac{f(x+h) - f(x) }{h}[/tex]

If we replace f(x) with −3x, we get:

[tex]\lim_{h \to \zero} \frac{-3(h+x) - 3x }{h}\\\\\lim_{h \to \zero} \frac{-3h-3x) - (-3x) }{h}\\[/tex]

Simplifying, we get:

[tex]\lim_{h \to \zero} \frac{-3h }{h}\\[/tex]

So, the derivative of −3x must be −3.

We know that the secondary means the rate of change of the function. Graphically, this means that the outgrowth is the pitch of the graph of that function.

Since −3x is a first degree polynomial, we know that it'll always have the same pitch, and therefor the same outgrowth. We can also corroborate this by looking at the graph, noticing that it's a straight line:

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