Which function in vertex form is equivalent to f(x) = x2 + 8 – 16x?
A. f(x) = (x – 8)2 – 56
B. f(x) = (x – 4)2 + 0
C. f(x) = (x + 8)2 – 72
D.f(x) = (x + 4)2 – 32

The rate of change and the initial value are given as -5 and 27 respectively.

To represent a straight line on a graph consider two points namely x and y intercepts of the line. To find x-intercept put y = 0 and for y-intercept put x = 0. Then draw a line passing through these two points.

Given that,

The points for the given linear function are given as (0, 27) and (5, -8) respectively.

The rate of change is equivalent to the slope of the straight line and is given as,

(-8 - 27)/(5 - 0) = -35/7 = -5

Now, the equation of the function can be written as,

(y - 27)/(x - 0) = -5

=> y - 27 = -5x

=> y = -5x + 27

The initial value is given at x = 0 as below,

y = -5 × 0 + 27

  = 27

Hence, the rate of change for the function is -5 and its initial value is 27.

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Given:

Elizabeth has $252.00 in her account on Sunday and a table is given of her deposits and purchases over the following week.

To Find:

The day Elizabeth first gets charged an overdraft fee.

Answer:

Thursday's expenditure of $57.40 exceeds her bank balance at the time of transaction, and therefore Thursday is the day Elizabeth would first get charged with an overdraft fee.

Step-by-step explanation:

Elizabeth would first get charged with an overdraft fee when her expenditure exceeds her bank balance, i.e., the amount that is debited is more than what is in her bank account at that particular time.

So we analyse her transactions one by one.

On Sunday, we are given that she has $252.

On Monday, $114.60 is debited. So the remaining balance would be

$252 - $114.60 = $137.4

But on the same day, $150 is also credited to her account. So her balance would then be

$137.4 + $150 = $287.4

Next, on Tuesday, $79.68 is debited so the balance she is left with would be

$287.4 - $79.68 = $207.72

On Wednesday, $161.39 is debited so the balance she is left with would be

$207.72 - $161.39 = $46.33

On Thursday, $57.40 is debited so the balance she is left with would be

$46.33 - $57.40 = -$11.07

Thus, we see that Thursday's expenditure of $57.40 exceeds her bank balance and therefore Elizabeth would be charged with an overdraft fee as her bank balance after the transaction would be -$11.07.

Answer:

open circle on -3 and all numbers to the right shaded

To find the y-intercept of y = 5x - 21, identify the constant term of the equation, which is -21, indicating the y-intercept is -21.

The y-intercept of the line given by the equation y = 5x - 21 can be found by analyzing the equation in the form y = mx + b, where m is the slope and b is the y-intercept.

Answer:

[tex]\displaystyle \frac{dy}{dt} = \frac{-3}{2(4t- 3)^2\sqrt{\frac{t}{4t - 3}}}[/tex]

General Formulas and Concepts:

Calculus

Differentiation

Derivative Property [Multiplied Constant]:                                                           [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]

Derivative Property [Addition/Subtraction]:                                                         [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]

Basic Power Rule:

Derivative Rule [Quotient Rule]:                                                                           [tex]\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}[/tex]

Derivative Rule [Chain Rule]:                                                                                 [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]

Step-by-step explanation:

Step 1: Define

Identify

[tex]\displaystyle y = \sqrt{\frac{t}{4t - 3}}[/tex]

Step 2: Differentiate

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation

The required, there are 108 cups of water in the pail as a pail hold 6 3/4 gallon of water.

Unit of measurement is defined as every entity having its measures in dimensions, weight, and time. Such as for length we have a meter, for liquid we have liters, and so on.

Here,
To convert 6 3/4 gallons to cups, we can use the fact that there are 16 cups in one gallon.

So, we can start by multiplying 6 (the number of whole gallons) by 16 to get the number of cups in those gallons:

6 gallons × 16 cups/gallon = 96 cups

Next, we need to convert the 3/4 of a gallon to cups. To do this, we can multiply the fraction by the number of cups in one gallon:

3/4 gallon × 16 cups/gallon = 12 cups

So, the total amount of water in the pail is:

96 cups + 12 cups = 108 cups

Therefore, there are 108 cups of water in the pail.

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Final answer:

The snail moved a total distance of 2 1/9 yards after converting 76 inches to yards using the conversion factor of 1 yard equals 36 inches.

Explanation:

To convert the distance a snail moved from inches to yards, we use the conversion factor that 1 yard equals 36 inches. Given that the snail moved 76 inches, we divide this number by 36 to find the distance in yards.

76 inches ÷ 36 inches/yard = 2 yards and 4 inches.

Since there are 36 inches in a yard, the remaining 4 inches must be converted to a fraction of a yard by dividing by 36:

4 inches ÷ 36 = 1/9 yard.

Therefore, the snail moved 2 yards and 1/9 yard, or 2 1/9 yards in total.

if both fractions are less than  1/2 then the sum has to be less than 1

1/8 + 3/8 = 4/8 = 1/2 , less than 1

7/16 + 1/4 = 11/16, less than 1

3/16 + 1/16 = 4/16 = 1/4, less than 1

The student's question is related to dividing 17 by 4520. This is performed by a straight division resulting in a small decimal quotient, not a product. The calculation is best done using a calculator and rounding to the proper number of significant figures.

The equation given, 17 by 4520, is asking for the result of dividing 17 by 4520. To compute this, we simply perform the division, recognizing that since 4520 is much larger than 17, the answer will be a very small decimal value. The term 'product' in the student's question seems to be a mistake, as they are likely referring to the quotient obtained from the division.

When attempting to perform the computation mentally and simplify the process, one might attempt to find numbers that are more easily divisible or apply scientific notation to simplify calculations, much like re-expressing numbers to make them more manageable. However, for this division, no mental math trick simplifies it effectively, and using a calculator might be the best option.

When going through this process, it is also important to consider significant figures when giving the final answer, rounding as necessary based on the precision of the data provided.

The limit of the expression as (x, y) approaches (0, 0) does not exist.

To find the limit as (x, y) approaches (0, 0) of the expression \(\frac{x^4 - 20y^2}{x^2 + 10y^2}\), we can try approaching the limit along different paths. Let's consider approaching along the x-axis (y=0) and along the y-axis (x=0) separately.

Approaching along the x-axis: \(\lim_{{(x, 0) \to (0, 0)}} \frac{{x^4 - 20(0)^2}}{{x^2 + 10(0)^2}} = \lim_{{(x, 0) \to (0, 0)}} \frac{{x^4}}{{x^2}} = \lim_{{(x, 0) \to (0, 0)}} x^2 = 0.

Approaching along the y-axis: \(\lim_{{(0, y) \to (0, 0)}} \frac{{(0)^4 - 20y^2}}{{(0)^2 + 10y^2}} = \lim_{{(0, y) \to (0, 0)}} \frac{{-20y^2}}{{10y^2}} = \lim_{{(0, y) \to (0, 0)}} -2 = -2.

Since the limit approaches different values along different paths, the limit does not exist.

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Answer:

$189.78

Step-by-step explanation:

"The computation of the interest on January 20 is shown below:

= Principal × interest rate × number of days ÷ total number of days in a year

= $7,000 × 8% × 122 days ÷ 360 days

= $7,000 × 8% × 0.338

= $189.78

The 122 days are calculated below:

September - 10 days

October - 31 days

November - 30 days

December - 31 days

January  - 20 days

Total    - 122 days

And we assume the 360 days in a year"

Answer:

y=5x+42

Where y is the money he earns and x is the number of person he serves.

Step-by-step explanation:

Let the number of person he serves =x

It is given that he earns $5 as tip for each person he serves.

Hence total tip earned for serving x person=5x

His fixed wages are $7 per hour and he works for 6 hours a day. Hence his wages per day are $7*6 =$42

Hence total money earned for a day , say y can be represented as

y=5x+42

Final answer:

To factor the expression 10x-25y, we find the greatest common factor, which is 5, and write the expression as 5(2x - 5y).

Explanation:

To factor the expression 10x−25y, we need to find the greatest common factor (GCF) that both terms share. In this case, both terms are divisible by 5. When we factor out the GCF, we get:

Identify the GCF of 10x and 25y, which is 5.

Divide both terms by the GCF: 10x/5 = 2x and 25y/5 = 5y.

Write the factored expression: 5(2x - 5y).

Thus, the expression 10x−25y factored is 5(2x - 5y)

The equivalent expression to [tex]x^2 + 2x - 24 / (3x + 18([/tex] is (x - 4) / 3, after factoring and simplifying the common terms.

The question is asking for an expression equivalent to [tex]x^2 + 2x - 24 / (3x + 18[/tex].

This expression can be factored and simplified.

First, notice that the quadratic expression in the numerator can be factored into (x + 6)(x - 4).

Similarly, the denominator can be factored as 3(x + 6). After factoring, we have:

= (x + 6)(x - 4) / 3(x + 6)

We can now simplify the expression by canceling out the common factor of (x + 6):

= (x + 6)(x - 4) / 3(x + 6) = (x - 4) / 3

So the equivalent expression is (x - 4) / 3.