Write a formula for quadratic function if its graph has the vertex at point ( 1/3 ,−3) and passes through the point (1,1).

Note that [tex]+\vee-[/tex] stands for plus or minus.

For the quadratic equation of the form [tex]ax^2+bx+c=0[/tex] the solutions are [tex]x_{1,2}=\dfrac{-b+\vee-\sqrt{b^2-4ac}}{2a}[/tex] that means that for [tex]a=4, b=-10, c=5\Longrightarrow x_{1,2}=\dfrac{-(-10)+\vee-\sqrt{(-10)^2-4\cdot4\cdot5}}{2\cdot4}[/tex] this simplifies to [tex]\boxed{x_1=\dfrac{5+\sqrt{5}}{4}}, \boxed{x_2=\dfrac{5-\sqrt{5}}{4}}[/tex]

Hope this helps.

Answer:

[tex]\large\boxed{x=\dfrac{5-\sqrt5}{4},\ x=\dfrac{5+\sqrt5}{4}}[/tex]

Step-by-step explanation:

[tex]\text{The quadratic formula for}\ ax^2+bx+c=0\\\\\text{if}\ b^2-4ac<0,\ \text{then the equation has no real solution}\\\\\text{if}\ b^2-4ac=0,\ \text{then the equation has one solution:}\ x=\dfrac{-b}{2a}\\\\\text{if}\ b^2-4ac,\ ,\ \text{then the equation has two solutions:}\ x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\==========================================[/tex]

[tex]\text{We have the equation:}\ 4x^2-10x+5=0\\\\a=4,\ b=-10,\ c=5\\\\b^2-4ac=(-10)^2-4(4)(5)=100-80=20>0\\\\x=\dfrac{-(-10)\pm\sqrt{20}}{2(4)}=\dfrac{10\pm\sqrt{4\cdot5}}{8}=\dfrac{10\pm\sqrt4\cdot\sqrt5}{8}=\dfrac{10\pm2\sqrt5}{8}\\\\=\dfrac{2(5\pm\sqrt5)}{8}=\dfrac{5\pm\sqrt5}{4}[/tex]

2,8,8,8

8,2,8,8

8,8,2,8

8,8,8,2

4 combinations.

Hope this helps!

Answer:

[tex]P(B)=\frac{1}{3}[/tex]

Step-by-step explanation:

The number of red marbles in the bag is; [tex]n(R)=3[/tex].

The number of blue marbles in the bag is; [tex]n(B)=4[/tex].

The number of white marbles in the bag is; [tex]n(W)=5[/tex].

The total number of marbles in the bag is: [tex]n(S)=3+4+5=12[/tex].

The probability of selecting a blue marble is [tex]P(B)=\frac{n(B)}{n(S)}[/tex]

We substitute the given information to obtain:

[tex]P(B)=\frac{4}{12}[/tex]

We simplify to obtain:

[tex]P(B)=\frac{1}{3}[/tex]

she went down the slope 10.2 meters, -10.2

then she jumped down to the lake 1.5 meters, -1.5

-10.2 - 1.5 = -11.7 meters.

so she pretty much went 11.7 meters down from her original location.

Answer:

nonagon: 140°

100-gon: 176.4°

Step-by-step explanation:

A regular polygon is one that has all sides the same length and all internal angles the same measure. A "nonagon" has nine (9) sides. An image of one is attached. There are some interesting relationships among the angles shown.

The figure can be divided into 9 congruent triangles. Each has a vertex at the center of the figure, and the other two vertices are each end of one side. The central angle (40°) is the supplement of the interior angle at each vertex of the nonagon (140°). Obviously, the central angles sum to 360° (one full circle), so each one has a measure that is 360° divided by the number of sides:

360°/9 = 40°

So, we can figure the interior angle (at the vertex) by subtracting from 180° the value of 360° divided by the number of sides.

For the 100-gon, with 100 sides, the interior angle will be ...

180° -360°/100 = 180° -3.6° = 176.4°

Final answer:

The interior angle of a regular nonagon is 140°, and for a regular 100-gon, it's 176.4°.

Explanation:

To find the interior angles of a regular polygon, we use the formula: (n - 2) × 180°, where n is the number of sides of the polygon. This formula gives us the sum of all interior angles of the polygon.

To find the measure of one interior angle in a regular polygon (where all angles are equal), we divide this sum by the number of sides.

Interior Angle of a Regular Nonagon:

A nonagon has 9 sides. Using our formula:

Interior Angle of a Regular 100-gon:

A 100-gon has 100 sides. Following the same process:

Therefore, each interior angle of a regular nonagon is 140°, and each interior angle of a regular 100-gon is 176.4°.

For this case we have the following system of equations:

[tex]y = x ^ 2-4x + 3\\y = -x + 3[/tex]

Matching we have:

[tex]x ^ 2-4x + 3 = -x + 3\\x ^ 2-4x + x + 3-3 = 0\\x ^ 2-3x = 0[/tex]

We solve by means of

[tex]x = \frac {-b \pm \sqrt {b ^ 2-4 (a) (c)}} {2a}[/tex]

Where:

[tex]a = 1\\b = -3\\c = 0[/tex]

Substituting:

[tex]x = \frac {- (- 3) \pm \sqrt {(- 3) ^ 2-4 (1) (0)}} {2 (1)}\\x = \frac {3 \pm \sqrt {9}} {2}\\x = \frac {3 \pm3} {2}[/tex]

Finally, the roots are:

[tex]x_ {1} = \frac {3-3} {2} = 0\\x_ {2} = \frac {3 + 3} {2} = \frac {6} {2} = 3[/tex]

Answer:

[tex]x_ {1} = 0\\x_ {2} = 3[/tex]

Answer:

Step-by-step explanation:

Dilation does not change the slope, so it remains ...

  m = ∆y/∆x = (8-2)/(3-2) = 6/1

  m = 6

The length is multiplied by the dilation factor. The original length (d) is given by ...

  d = √((∆x)^2 +(∆y)^2) = √(1^2 +6^2) = √37

Then the dilated length is ...

  3.5d = 3.5√37 ≈ 21.290

Answer:

[tex]10.4\ years[/tex]

Step-by-step explanation:

In this problem we have a exponential function of the form

[tex]y=a(b)^{x}[/tex]

where

y is the population of a town

x is the number of years since 2010

a is the initial value

b is the base

[tex]a=72,000\ people[/tex]

[tex]b=1+0.08=1.08[/tex]

substitute

[tex]y=72,000(1.08)^{x}[/tex]

For [tex]y=160,000\ people[/tex]

substitute in the equation and solve for x

[tex]160,000=72,000(1.08)^{x}[/tex]

[tex](160/72)=(1.08)^{x}[/tex]

Apply log both sides

[tex]log(160/72)=(x)log(1.08)[/tex]

[tex]x=log(160/72)/log(1.08)[/tex]

[tex]x=10.4\ years[/tex]

Answer:

  3.5 in

Step-by-step explanation:

Put the given numbers in the formula for the volume of a cylinder, then solve for the unknown.

  V = πr²h

  73.5π = πr²·6 . . . . . fill in V and h

  12.25 = r² . . . . . . . . divide by the coefficient of r²

  3.5 = r . . . . . . . . . . . take the square root

The radius of the can is 3.5 inches.

Answer:

A

Step-by-step explanation:

The bottles are cylindrical.  The volume of a cylinder is:

V = πr²h

where r is the radius (half the diameter) and h is the height.

The bottles are similar, so we can write a proportion to find the height of the smaller bottle:

3/10 = 1.5/h

h = 5

The volume of the big bottle is:

V = π(3/2)²(10)

V ≈ 70.7

The volume of the small bottle is:

V = π(1.5/2)²(5)

V ≈ 8.8

So the difference in volume is:

V = 70.7 - 8.8

V = 61.9

Answer:

A

Step-by-step explanation:

got it right

ANSWER

The curve is a parabola with a vertex at left parenthesis 3 comma negative 4 right parenthesis and is traced from left to right for increasing values of t.

EXPLANATION

The given curve is defined parametrically as;

[tex]x = 3+ t[/tex]

[tex]y = {t}^{2} - 4[/tex]

We need to eliminate the parameter by making t the subject in the first equation and substitute into the second equation.

[tex]t = x - 3[/tex]

[tex]y = {(x - 3)}^{2} - 4[/tex]

This is a parabola that has its vertex at (3,-4).

This parabola opens upwards.

The correct description is a parabola with a vertex at (-3, -4) and, because x increases linearly with t, it is traced from left to right as t increases. This corresponds to option 3.

The student's question involves finding the description of a curve represented by parametric equations x = -3 + t and y = t^2 - 4. Considering the equation for y which is a second-order polynomial in t, it indicates that the graph is a parabola. The presence of t^2 indicates the parabola opens upwards as the coefficient is positive. If we were to eliminate t from the parametric equations, the resultant equation would still describe this parabola.

To analyze the vertex, the standard form for a parabola's equation is y = ax^2 + bx + c. Given that y = t^2 - 4 has a constant term of -4, this suggests the vertex's y-coordinate is -4. For the x-coordinate of the vertex, we must consider the constant -3 in the x equation, which adjusts the x-coordinate of the vertex. Since the parametric equation for x alone does not yield an obvious vertex, we realize that the parabola is shifted from the origin, and the -3 signifies a leftward shift from the y-axis.

Therefore, The correct description is a parabola with a vertex at (-3, -4) and, because x increases linearly with t, it is traced from left to right as t increases. This corresponds to option 3.

b=-7

Please look at the attached picture to see what I did

The value of b=-7.

y=3x+b

point(2,-1)

x=2, y=-1

subtract into equation

-1 = 3(2) +b

-1 = 6+b

-1-6 = b

b= -7

Coordinates are a couple of numbers that describe the precise function of a factor on a cartesian aircraft through the use of the horizontal and vertical lines known as the coordinates. commonly represented by (x, y) the x cost and y price of the point on a graph. Each factor or an ordered pair consists of two coordinates.

A factor-to-factor graph also referred to as a line graph, is a pictorial rendition of records wherein specific values of a feature are plotted as dots on a coordinate aircraft.

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Answer:

D

Step-by-step explanation:

(cos t) (sec t − cos t)

1 − cos² t

sin² t

The expression cos t(sec t - cos t) simplifies to sin² t, which matches option D.

The student has asked to simplify the expression cos t(sec t − cos t).

To simplify this expression, we start by distributing cos t across the parentheses:

cos t × sec t − cos t ×cos t

1 − cos² t (since cos t × sec t = 1)

1 − (1 − sin² t) (using the Pythagorean identity cos² t + sin² t = 1)

sin² t

Thus, the expression simplifies to sin² t, which corresponds to option D.

Given equation is [tex]f\left(x\right)=x^2-b^2[/tex].

Now we need to find about what are the key aspects of the graph of [tex]f\left(x\right)=x^2-b^2[/tex], where b is a real number.

We know that square of any number is always positive.

then [tex]b^2[/tex] must be a positive number.

So that means for any real number b, as the value of b increases then graph of f(x) shifts downward by [tex]b^2[/tex] units as compared to the graph of parent function [tex]f\left(x\right)=x^2[/tex]

Answer with explanation:

The graph of the function is:

   f(x)=x² -b²

Here, b is any Real Number.

f(x)=x² - k, where, k=b².

→y+k=x²

The given curve represents a Parabola having vertex at ,(0, -k) which can be Obtained by , putting, x=0 and, y+k=0→y= -k.

→The curve will open vertically Upwards having y axis as Line of Axis.

→It will cut, x axis at two points, if , k<0 and does not cuts the x axis , if k>0.

→Line, x=0, divides the Parabola into two equal Parts.

Answer:

x = 29.2

Step-by-step explanation:

Given

Perpendicuar=10

Angle=20 degrees

Hypotenuse=x

We will use the trigonometric ratios of right angled triangle to solve this question. Since, we have to find x which is the hypotenuse of the given triangle. So we will use the ratio which involves perpendicular and hypotenuse.

sin⁡ 20=Perpendicular/Hypotenuse

sin⁡ 20=10/x

x=10/sin ⁡20  

x=10/0.3420

x=29.239

Rounding off to nearest tenth.

x=29.2

[tex]\displaystyle\\\text{If }~~180^o<\theta<270^o~~\text{then }~~\theta\in~\text{quadrant 3}\\\\\text{In the 3rd cotangent dial is positive.}\\\\\text{We use the formula: } ~~~\boxed{1+\cot^2\theta=\csc^2\theta}[/tex]

[tex]\displaystyle\\1+\cot^2\theta=\csc^2\theta\\\\\cot^2\theta=\csc^2\theta-1\\\\\cot^2\theta=\left(-\frac{3}{2}\right)^2-1\\\\\\\cot^2\theta=\left(\frac{3}{2}\right)^2-1\\\\\\\cot^2\theta=\frac{9}{4}-\frac{4}{4}\\\\\\\cot^2\theta=\frac{5}{4}\\\\\\\cot\theta=\pm\sqrt{\frac{5}{4}}\\\\\\\text{We will eliminate the negative solution.}\\\\\\\cot\theta=+\sqrt{\frac{5}{4}}\\\\\\\boxed{\bf\cot\theta=\frac{\sqrt{5}}{2}}}[/tex]

Answer:

Step-by-step explanation:

D is correct.  For any x in the domain of your function, there can be ONLY ONE corresponding y value.  If you find more than one y value associated, then this relationship is NOT a function.

Answer: D. A function is a set of ordered pairs in which each first component in the ordered pairs corresponds to exactly one second component.

Step-by-step explanation: In mathematics, Function is a relationship between two groups, in which, each element of the first group corresponds to the elements of the second group only once. They shows how a quantity depends on another. For example, the distance a person travels over time spent.

A function is a technique in which each element (x) of a set X is associated with an element of set Y or f(x). The set X is the domain of a function and set Y or f(x) is the codomain.

Each x is the input of a function or variable and each correspondent y is the value of the function or image of x by f.

Answer:

B. csc²(x)

Step-by-step explanation:

You can use the relations ...

sec(x) = 1/cos(x)

csc(x) = 1/sin(x)

cot(x) = cos(x)/sin(x)

to replace the functions in your expression. Then you have ...

sec²(x)·cot²(x) = (1/cos(x)·cos(x)/sin(x))² = (1/sin(x))² = csc²(x)

___

Alternate solution

You can also use the relation

cot(x) = csc(x)/sec(x)

Then ...

(sec(x)·cot(x))² = (sec(x)·csc(x)/sec(x))² = csc²(x)

Answer:

Yes! The correct answer is option B

Step-by-step explanation:

B.  csc^2x

Answer: 28x + 8y + 2

Answer:

28x +8y +2

Step-by-step explanation:

It can work well to simplify the contents of the parentheses, then apply the overall multiplier.

= -2(x(3-17) +y(12-16) +(-5+4)) . . . . collect terms

= -2(-14x -4y -1)

= 28x +8y +2 . . . . use the distributive property

The answer is:

The x-intercept or roots of the parabola are:

[tex]x_{1}=-5\\x_{2}=-1[/tex]

To solve the problem, we need to find the roots or zeroes of the parabola.

We can find the zeroes of the quadratic equation (parabola) by factoring its equation.

So,we are given the function:

[tex]f(x)=x^{2} +6x+5[/tex]

To factorize the equation, we need to find two numbers which product gives as result the number 5, and its addition gives as result the number "6", these numbers are 5 and 1.

So, rewriting the equation, we have:

[tex]f(x)=x^{2} +6x+5=(x+5)(x+1)=0[/tex]

Therefore, we have that the x-intercept or roots of the parabola are:

[tex]x_{1}=-5\\x_{2}=-1[/tex]

Have a nice day!

Note: I have attached a picture for better understanding.

Answer:

[tex]x = -5\\x = -1[/tex]

Step-by-step explanation:

To find the intercept with the x axis, you must do [tex]f(x) = 0[/tex]

So:

[tex]f(x) = x^2 + 6x + 5 = 0[/tex]

Now you must factor the expression.

To factor the expression you must find two numbers such that when you add them, you obtain 6 and multiplying it will result in 5.

You can verify that these numbers are 5 and 1.

So

[tex]f(x) = (x + 5)(x + 1) = 0[/tex]

Therefore the solutions are

[tex]x = -5\\x = -1[/tex]

Answer:

Strain Hardening as name implies, physical straining of metal is induced to increase strength and thus load carrying capacity of the specimen under consideration. The level of straining is dependent on the increased strength required. Strains are classified into two as 'Lateral Strain' which is decrease of cross sections and 'Linear Strains' which is increase in physical extensions (usually 'length') of the specimen.

Step-by-step explanation:

Answer:

supplementary angle = 60 , complementary angle : none

Final answer:

A supplementary angle of 120 degrees is 60 degrees, and there is no complementary angle for 120 degrees because complementary angles must be less than or equal to 90 degrees.

Explanation:

Finding Supplementary and Complementary Angles

The task is to find the measure of a supplementary angle and a complementary angle for a given angle measure of 120 degrees. To find a supplementary angle, you subtract the given angle from 180 degrees. Thus, the supplementary angle of 120 degrees is 180 - 120 = 60 degrees. On the other hand, a complementary angle adds up to 90 degrees. Since 120 degrees is greater than 90, it is not possible to have a complementary angle to 120 degrees, therefore, we write 'none'. It's important to remember that complementary angles are always less than or equal to 90 degrees and hence, cannot be obtained for angles larger than 90 degrees.

In summary, the supplementary angle to 120 degrees is 60 degrees, and there is no complementary angle for 120 degrees.

Explanation:

The sum of the measures of the interior angles of a triangle is 180°. Then the sum of the given angles is 180°:

m∠P +m∠Q +m∠R = 180°

(5x)° +(5x)° +(8x)° = 180°

18x = 180 . . . . . . . . . . . . . . . collect terms, divide by °

x = 10 . . . . . . . . . . . . . . . . . . divide by 18. This is your desired result.

Explanation:

The sum of the measures of the interior angles of a triangle is 180°. Then the sum of the given angles is 180°:

m∠P +m∠Q +m∠R = 180°

(5x)° +(5x)° +(8x)° = 180°

18x = 180 . . . . . . . . . . . . . . . collect terms, divide by °

x = 10 . . . . . . . . . . . . . . . . . . divide by 18. This is your desired result.

Is there a formula that I could use to solve it?

The area of the shaded region is,

⇒ A = 41.87 cm²

The circle is a closed two dimensional figure , in which the set of all points is equidistance from the center.

We have to given that;

Radius of small circle = 3 cm

And, Radius of large circle = 3 + 4 = 7 cm

Hence, Area of sector for large circle,

A = 120/360 × 3.14 × 7²

A = 51.29 cm²

And, Area of sector for small circle is,

A = 120/360 × 3.14 × 3²

A = 9.42 cm²

Hence, The area of the shaded region is,

⇒ A = 51.29 - 9.42

⇒ A = 41.87 cm²

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ANSWER

The discriminant is 384. Because the discriminant is greater than 0 and is not a perfect square, the two roots are real and irrational.

EXPLANATION

The given quadratic equation is:

[tex] - 3 {x}^{2} - 18x + 5 = 0[/tex]

We compare this equation to:

[tex]a {x}^{2} + bx + c = 0[/tex]

We have a=-3,b=-18, and c=5.

The discriminant of a quadratic equation is calculated using the formula:

[tex]D=b^{2} - 4ac[/tex]

We plug in the values to obtain:

[tex]D= {( - 18)}^{2} - 4( - 3)(5)[/tex]

[tex]D= 324 + 60[/tex]

Simplify:

[tex]D= 384[/tex]

The discriminant is greater than zero, hence there are two distinct real roots.

Since 384 is not a perfect square, the roots are irrational.

Answer:

  564 ft²

Step-by-step explanation:

To account for the extra space between units, we can add 2" to every unit dimension and every box dimension to figure the number of units per box.

Doing that, we find the storage box dimensions (for calculating contents) to be ...

  3 ft 2 in × 4 ft 2 in × 2 ft 2 in = 38 in × 50 in × 26 in

and the unit dimensions to be ...

  (4+2)" = 6" × (6+2)" = 8" × (2+2)" = 4"

A spreadsheet can help with the arithmetic to figure how many units will fit in the box in the different ways they can be arranged. (See attached)

When we say the "packing" is "462", we mean the 4" (first) dimension of the unit is aligned with the 3' (first) dimension of the storage box; the 6" (second) dimension of the unit is aligned with the 4' (second) dimension of the storage box; and the 2" (third) dimension of the unit is aligned with the 2' (third) dimension of the storage box. The "packing" numbers identify the unit dimensions, and their order identifies the corresponding dimension of the storage box.

We can see that three of the four allowed packings result in 216 units being stored in a storage box.

If storage boxes are stacked 4 deep in a 9' space, the 2' dimension must be the vertical dimension, and the floor area of each stack of 4 boxes is 3' × 4' = 12 ft². There are 216×4 = 864 units stored in each 12 ft² area.

If we assume that 2 weeks of production are 80 hours of production, then we need to store 80×500 = 40,000 units. At 864 units per 12 ft² of floor space, we need ceiling(40,000/864) = 47 spaces on the floor for storage boxes. That is ...

  47 × 12 ft² = 564 ft²

of warehouse floor space required for storage.

_____

The second attachment shows the top view and side view of units packed in a storage box.

Check the picture below.

Both [tex]f[/tex] and [tex]g[/tex] satisfy the conditions for Rolle's theorem, which then says there exists [tex]c\in[-1,3][/tex] (for [tex]f[/tex]) such that [tex]f'(c)=0[/tex], and [tex]c\in[-2,1[/tex] (for [tex]g[/tex]) such that [tex]g'(c)=0[/tex].

1.

[tex]f(x)=x^2-2x-8\implies f'(x)=2x-2[/tex]

[tex]f'(c)=2c-2=0\implies c=1[/tex]

2.

[tex]g(t)=2t-t^2-t^3=0\implies g'(t)=2-2t-3t^2[/tex]

[tex]g'(c)=2-2c-3c^2=0\implies c=\dfrac{-1\pm\sqrt7}3[/tex]

Answer:

Step-by-step explanation:

The equation ...

  g(x) = a·f(x -h) +k

indicates a vertical stretch by a factor of "a", a horizontal translation to the right by "h" units, and a translation up by "k" units.

Matching the shapes of the curves, we see that the point of inflection of f(x) is (-2, -1). The corresponding point on g(x) is (2, -1). This is called the "turning point" in your question. It is where the graph turns from being concave downward to being concave upward.

The difference in x-values between g(x) and f(x) for the turning point is ...

  2-(-2) = 4

This is the amount by which the graph of f(x) is translated to the right: 4 units.

The vertical difference between the marked points on f(x) and the turning point is 1 unit. On g(x), those same marked points are 3 units away from the turning point vertically. Hence the vertical stretch factor is 3.

_____

Comment on the transformation of f(x)

Please note that the graph of g(x) is actually related to the graph of f(x) as ...

  g(x) = 3·f(x -4) +2

That is, for x=1 on g(x), the y-coordinate is ...

  g(1) = 3·f(1 -4) +2 = 3·(-2) +2 = -4 . . . . . . . point (1, -4) on g(x)

For x=3 on g(x), the y-coordinate is ...

  g(3) = 3·f(3 -4) +2 = 0 +2 = 2 . . . . . . . . . . point (3, 2) on g(x)

It may seem a little strange that there is a vertical translation of 2 units upward, when the point of inflection has the same vertical location. Actually, that is the clue that there is an upward translation.

The stretch factor operates about the origin, so stretching f(x) by a factor of 3 will make the turning point move from y=-1 to y=3·(-1) = -3. Since it shows on the graph of g(x) at location y=-1, it must have been translated 2 units upward from its stretched location.

Answer:

reflective and slide the y is -1

Step-by-step explanation:

refect over the -1 x axis and the translate x-1 and y-1