What is the range for y=x^2-7x-8 ?

f(x) = x^4 - 5x^3 + x^2 + 21x - 18

=======================================================

Explanation:

The root x = -2 means x+2 is a factor. This is because we add 2 to both sides of x = -2 to get x+2 = 0.

Similarly, x = 3 leads to x-3 being another factor. It's a double root so we really have two copies of (x-3)

The last root is x = 1 which gives the factor x-1.

The four factors are: (x+2)(x-3)(x-3)(x-1)

Let's use the FOIL rule to expand out the first two factors

(x+2)(x-3) = x^2-3x+2x-6 = x^2-x-6

Do the same for the last two factors

(x-3)(x-1) = x^2-1x-3x+3 = x^2-4x+3

--------------------

So far we have:

(x+2)(x-3) = x^2-x-6

(x-3)(x-1) = x^2-4x+3

which leads to  

(x+2)(x-3)(x-3)(x-1) = (x^2-x-6)(x^2-4x+3)

From here I'll use the box method to multiply these trinomials. Check out the diagram below. Each inner cell is the result of multiplying the headers. Example: x^2 times x^2 = x^4 in the upper left corner.

I've color-coded the inner cells to show the like terms. Add up those like terms:

Therefore we end up with

(x^2-x-6)(x^2-4x+3) = x^4 - 5x^3 + x^2 + 21x - 18 which is choice B

--------------------

To verify this answer, plug in x = 0 and you should get y = -18 to show that we have the correct y intercept. If we tried this with choice A, then we'd get y = 18 which helps eliminate choice A.

Furthermore, plug in x = -2 into choice B and you should get y = 0 as an output. The same applies to x = 3 and x = 1. This confirms that they are roots or x intercepts of the polynomial.

Another way to verify the answer is to use something like WolframAlpha to type in (x+2)(x-3)(x-3)(x-1). Under the "expanded form" subsection, it says x^4 - 5x^3 + x^2 + 21x - 18

Answer:

same slope, different intercepts (parallel lines essentially)

Step-by-step explanation:

Answer:

1) All real numbers n such that [tex]n < 5.5[/tex]

2) All real numbers n such that [tex]n > -\frac{19}{7}[/tex]

Step-by-step explanation: How did I come up with such answers? Let me show you!

For the first problem, we just solve the inequality easily.

Subtracting 3 from both sides, we get:

4n < 22

Dividing by 4 from both sides, we get:

n < 5.5

Thus, all real numbers n that is less than 5.5 when plugged in for x will be less than 25.

For the next problem, it will be slightly trickier.

Subtracting 5 from both sides, we get:

-7n < 19

Dividing -7 from both sides, we get:

n > -19/7

Did you catch that? We just flipped the inequality sign and turned it from less than to a greater than. Whenever we divide by a negative number, the signs always change.

Thus, for all real numbers n greater than -19/7, when plugging in for x, you will get less than 24.

Hope this helped!

Answer: 1/12

Step-by-step explanation:

The total area of the rectangle needs to be 1.

Since 20-8=12, the height needs to be 1/12.

Answer:

max=50

min=41

Step-by-step explanation:

graph

Answer:

  -4sin(2x)

Step-by-step explanation:

Angle sum identities can be used to simplify this expression.

The cosine function is even, so cos(-x) = cos(x). The cosine of the sum is ...

  cos(90°+x) = cos(90°)cos(x) -sin(90°)sin(x) = -sin(x)

Then the numerator simplifies to ...

  4cos(-x)cos(90°+x) = -4cos(x)sin(x) = -2sin(2x)

Matching the denominator expression to the sine of a sum relation, we see

  sin(30° -x)cos(x) +sin(x)cos(30° -x) = sin((30°-x) +x) = sin(30°) = 1/2

The simplified expression is the ratio of the simplified numerator to the simplified denominator:

  = -sin(2x)/(1/2)

  = -4sin(2x)

The price of the washer and the dryer set is $720.

The commission earned by Alice is $36.

Percentage can be described as a fraction an amount expressed as a number out of hundred. Percentage is a measure of frequency.

Price after the mark up = ( + percentage mark up) x cost price

(1  + 0.20) x $600  = $720

Commission earned by Alice = 5% x 720 = $36

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Answer:

Anne has n

Fiona has 3n

3n=6

so n=2

so Anne have 2

The angle-bisector construction can be used to bisect any angle and this is proven by:

The angle-bisector construction means that a line is made that can divide an angle into two congruent angles.

This is shown by the fact that all the radii of a circle are congruent and that the corresponding parts of congruent triangles are also congruent (CPCTC) when an angle bisector is made on a circle or triangle respectively.

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The number sentence that applies the commutative property to the example is 4 × 3 = 12

Commutative property can be additive or multiplicative. It is a way of changing the position of a function or value in an expression without affecting the result

Examples are

A +B = B +A

AB = BA

Although the value were interchanged, the result were not affected. Hence the number sentence that applies the commutative property to the example is 4 × 3 = 12 since changing 3 and 4 does not affect the result

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Solution: Check the picture I uploaded.

The surface area of trapezoidal prism is: D. 385.6 yd².

Surface area = (b1 + b2)h + PH.

The parameters are:

b1 = 10 yd

b2 = 6 yd

h = 4.6 yd

Perimeter of base (P) = 6 + 10 + 5 + 5 = 26 yd

H = 12 yd

Plug in the values

Surface area = (10 + 6)4.6 + (26)(12)

Surface area = 385.6 yd²

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Given that, one digit of a two digit number is three times the other digit.

Let assume that digit at ones place be x.

digit at ones place be x.So, digit at tens place be 3x.

digit at ones place be x.So, digit at tens place be 3x.So, two digit original number = 10 × 3x + 1 × x = 30x + x = 31x

Now, number formed on interchanging its digits, i.e.

Reverse number = 10 × x + 1 × 3x = 10x + 3x = 13x

According to statement, if we interchange the digits of this two digit number and add the resulting number to the original number, we get 88.

[tex]\begin{gathered}\sf \: 31x + 13x = 88 \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered}\sf \: 44x = 88 \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered}\sf \: x = \dfrac{88}{44} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered}\sf\implies \: x = 2 \\ \\ \end{gathered}[/tex]

So, two digit number is 31 × 2 = 62

Hence,

[tex]\begin{gathered}\sf\implies \: \boxed{ \bf{ \:Original \: number \: = \: 62 \: \: }} \\ \\ \end{gathered}[/tex]

[tex]\rule{190pt}{2pt}[/tex]

Answer:

26 or 62

Step-by-step explanation:

If one of the two digits of a two-digit number is three times the other digit, then the options for the two-digit number are:

If you interchange the digits of this two-digit number and add the resulting number to the original number, you get 88.

From inspection of the listed pairs of numbers, the only pair that sums to 88 is 26 and 62:

⇒ 26 + 62 = 88

Therefore, the original number is either 26 or 62.

Answer:

6x^5-5x^4+6x^3-x^2+12x-31

Step-by-step explanation:

(−5x^4+6x^3−43)+(6x^5−x^2+12x+12)

To add, combine like terms

6x^5-5x^4+6x^3-x^2+12x-43+12

6x^5-5x^4+6x^3-x^2+12x-31

Standard form means from the highest power of x to the lowest power of x

1. The Year-0 net cash flow is -$1,061,500.

2. The net operating cash flows in Years 1, 2, and 3 are as follows:

Year 1: $288,750 ($386,000 x 1 - 0.25)

Year 2: $288,750 ($386,000 x 1 - 0.25)

Year 3: $288,750 ($386,000 x 1 - 0.25)

3. The additional Year-3 cash flow (i.e, the after-tax salvage and the return of working capital) is $468,750 {($599,000 x 1 - 0.25) + $19,500}.

4. The NPV of the project is ($13,139).

5. The machine should not be purchased because it does not yield a positive NPV.

The net present value (NPV) shows the difference between the present value of cash inflows and the present value of cash outflows over a period of time.

It is determined by calculating the present values of cash flows using their present value factors as below:

Initial cash outlay = $1,061,500 ($1,020,000 + $22,000 + $19,500)

Salvage value = $599,000

Increase in net working capital = $19,500

Annual savings before tax = $386,000

Tax rate = 25%

Annuial savings after tax = $288,750 ($386,000 x 1 - 0.25)

Year      Cashflows        PV Factor     Present Value

0          -$1,061,500               1             -$1,061,500

1              $288,750          0.901           $260,164

2             $288,750          0.812          $234,465

3             $288,750          0.731            $211,076

3             $468,750          0.731          $342,656

Net present value                                -$13,139

Thus, the project should not be undertaken by The Campbell Company due to the negative NPV that it yields.

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the sequence is:

Translation, then reflection.

The correct option is the second one.

We start with figure 1.

In the image, we can see that the image is shifted 4 units up and 4 units left to make figure 2.

Then you can see that the image is reflected across a horizontal line to make figure 3, you can see that because now the "L" is facing upwards.

Then the sequence is:

Translation, then reflection.

The correct option is the second one.

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Answer:

Step-by-step explanation:

The vertex and direction of opening can be read from the vertex-form equation of a quadratic.

Consider the "parent" quadratic function ...

  y = x²

For this function, the value of y cannot be negative. Larger positive values of x will give larger positive values of y. And, negative values of x that have greater magnitude (are farther from the y-axis) will also give larger positive values of y.

This means that the farther away from the y-axis an x-value is, the farther away from the x-axis is the corresponding y-value. The graph of this is said to "open upward." This will be the case for any positive coefficient of x². (Red graph in the first attachment.)

If the coefficient of x² is negative, then larger-magnitude x-values result in more negative y-values. This makes the graph "open downward." (Blue graph in the first attachment.)

The vertex form of a quadratic equation is ...

  y = a(x -h)² +k . . . . . . vertex (h, k); vertical scale factor 'a'

The value of 'a' is the coefficient of the x² term when this is simplified to standard form: y = ax² +bx +c. That is, the sign of 'a' tells you whether the graph opens upward (a > 0) or downward (a < 0).

The other constants in the vertex form equation tell you how the function has been translated. The value of k is a vertical translation quantity. Since it is added to each function value, it tells the number of units the function is translated upward.

The value h is a horizontal translation quantity. It seems slightly counter-intuitive that the function graph is translated to the right h units when h is subtracted from the x-value. That is the case.

As you may have noticed from the graphs in the first attachment, the vertex (turning point) of the parent function graph is at (x, y) = (0, 0). The vertex of the function ...

  y = a(x -h)² +k

is located at (x, y) = (h, k).

You want the opening direction of y = -2(x +3)² -1. You need look no further than the leading minus sign. It tells you the graph opens downward. (Red graph in the second attachment.)

Of course, the -1 at the end of the equation is the vertical translation of the vertex. That vertex is (-3, -1), as shown by the graph.

You want the vertex of y = -1/2(x +5)² +7. Writing this so the binomial term has a minus sign, we have ...

  y = -1/2(x -(-5))² +7

Comparing this to the vertex form ...

  y = a(x -h)² +k

we identify the parameters to be ...

  a = -1/2 . . . . opens downward

  h = -5

  k = 7 . . . . the vertex is (h, k) = (-5, 7)

The graph is the blue graph in the second attachment. It opens downward and has its vertex at (-5, 7).

__

Additional comment

A lot of math is about matching patterns. Here, you're asked to match the given equations to the "vertex form" pattern, and identify corresponding parts of the pattern: the leading coefficient (a), the horizontal translation (h), and the vertical translation (k). Once you know what the parts of the pattern mean, you can answer the questions easily.

Answer:

Option D

Step-by-step explanation:

This function has vertex at origin

Let's verify

Put (0,0)

Hence verified

Answer:

[tex]f(x)=-x^2[/tex]

Step-by-step explanation:

Vertex form of a quadratic function:

  [tex]y=a(x-h)^2+k[/tex]  

where:

If the vertex is at the origin:

Substituting the vertex into the equation:

[tex]\implies y=a(x-0)^2+0[/tex]

[tex]\implies y= ax^2[/tex]

Comparing with the available answer options:

[tex]f(x)=-x^2[/tex]  has its vertex at the origin.

Additional Information:

[tex]\textsf{The vertex of }\:f(x)=(x+4)^2 \: \textsf{ is }\:(-4,0)[/tex]

[tex]\textsf{The vertex of }\:f(x)=x(x-4) \: \textsf{ is }\:(2,-4)[/tex]

[tex]\textsf{The vertex of }\:f(x)=(x-4)(x+4) \: \textsf{ is }\:(0,-16)[/tex]

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Answer:

Russia is so vast that it has eleven time zones.

Explanation:

"Russia has 11 time zones across its vast territory - and its leaders believe that's just too many hours in the day." - NBC News

[tex]y = 2x {}^{2} + e {}^{ - x {}^{4} } [/tex]

[tex] \frac{dy}{dx} = 4x +( e {}^{ - x {}^{4} } )( - 4x {}^{3} )[/tex]

[tex] \frac{dy}{dx} = 4x - 4x {}^{3} e { }^{ - x {}^{4} } [/tex]

[tex] \frac{dy}{dx} = 4x(1 - x {}^{2} e {}^{ - x {}^{4} } )[/tex]

[tex] \frac{dy}{dx} = 0 \\ 4x = 0 \: \: \: \: \: 1 - x {}^{2} e {}^{ - x {}^{4} } = 0[/tex]

[tex]4x = 0 \\ x = 0 \: \: \: \: \: \: y(0) = 0 + e {}^{0} = 1[/tex]

Answer:

  18x

Step-by-step explanation:

The HCF is the highest factor common to both expressions. We can find it by factoring both expressions and identifying common factors.

  36(x³+x²y-12xy²) = 18x·2·(x +4y)(x -3y)

  54(x³y+4x³y²-21xy) = 18x·3y·(x +7)(x -3)

The highest common factor is 18x.

Answer: I prefer to find the area of the base first and then multiply by the height

The correct answer is D

The mean is also called average. The arithmetic mean is found by adding the numbers in the list and dividing the sum by the number of numbers in the list.

Mean = (Sum of all the observations/Total number of observations)

Darryl's credit scores are Experian score is 719, Equifax's score is 741, and Transunion's score is 780.

To find the mean we have to add the given values i.e., the sum of Experian score, Equifax's score, and Transunion's score.

719+741+780

there are a total of 3 scores so we have to divide them by 3.

[tex]\frac{719+741+780}{3}[/tex]

≅[tex]746.67[/tex]

The nearest whole point is 747.

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Answer:

a = sqrt(c^2 - b^2)

see image.

Step-by-step explanation:

If you square both sides of the equation, you get the Pythagorean Theorem back again.

c = sqrt(a^2 + b^2)

c^2 = a^2 + b^2

Solve for a means to get a all by itself on one side of the equation. Subtract b^2 and then take the square root. See image.

The combinations of ordered triplets that are possible is equal to: A. 18.

Since the common difference (d) between the consecutive terms in an arithmetic progression (AP) is constant, we have:

d = y - x = z - y

Rearranging the terms, we have:

x = y - d

z = y + d

From the given equation:

x + y + z = 30

y - d + y + y + d = 30

3y = 30

y = 30/3

y = 10.

So the arithmetic progression (AP) is given by:

(10 - d), 10, (10 + d)....

Since the common difference (d) is a natural number which would be from 1 to 9, the combinations of ordered triplets that are possible is as follows:

Number of possibilities = 9 × 2

Number of possibilities = 18.

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Complete Question:

If x, y, z are three natural numbers in A.P. such that x + y + z = 30 then the possible number of ordered triplet (x, y, z) is?

A. 18

B. 19

C. 20

D. 21

Answer:

[tex]f=-18a^2+3[/tex]

Step-by-step explanation:

Given equation:

[tex]f=3-2(3a)^2[/tex]

[tex]\textsf{Apply exponent rule} \quad (ab)^n=a^n b^n:[/tex]

[tex]\implies f=3-2(3^2a^2)[/tex]

Rewrite 3² as 9:

[tex]\implies f=3-2(9a^2)[/tex]

Remove the parentheses:

[tex]\implies f=3-2 \cdot 9a^2[/tex]

Multiply the numbers 2 · 9 = 18 :

[tex]\implies f=3-18a^2[/tex]

Rearrange to make the variable the first term:

[tex]\implies f=-18a^2+3[/tex]

In a playing card there are 52 cards. (i) '2' of spades: Number of favourable outcomes i.e. '2' of spades is 1 out of 52 cards.

Answer:

you can use the spreadsheet

Answer:

The perimeter of the garden is [tex]12a[/tex].

Step-by-step explanation:

[tex]3a+14+4(a-6)+10+5a=12a[/tex]