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Linear Functions: Rate of Change and initial value
-Quiz-Level H
Joaquin's car has a full tank of gas at the start of a trip. The graph shows the amount of gas in the
tank, y, as a function of the distance traveled, x.
What is the initial value of the function?

The linear equation that represents the point (-2,3) and 3 more points with a slope of -1/2 is y = -1/2x + 2

The given parameters are

Slope, m = -1/2

Point, (x1, y1) = (-2, 3)

The linear equation is calculated as:

y = m(x - x1) + y1

Substitute the known values in the above equation

y = -1/2(x + 2) + 3

Expand

y = -1/2x - 1 + 3

Evaluate the sum

y = -1/2x + 2

This means that the linear equation that represents the point (-2,3) and 3 more points with a slope of -1/2 is y = -1/2x + 2

See attachment for the graph of the point and the three other points

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Answer:

[tex]\textsf{Increasing function}: \quad \left[0, \dfrac{1}{16}\right)[/tex]

[tex]\textsf{Decreasing function}: \quad \left(\dfrac{1}{16}, \infty\right)[/tex]

Step-by-step explanation:

Given function:

[tex]f(x)=\sqrt{x}-2x[/tex]

As a negative number cannot be square rooted, the domain of the function is restricted to [0, ∞).

[tex]\textsf{A function is \textbf{increasing} when the \underline{gradient is positive}}\implies f'(x) > 0[/tex]

[tex]\textsf{A function is \textbf{decreasing} when the \underline{gradient is negative}} \implies f'(x) < 0[/tex]

Differentiating produces an algebraic expression for the gradient as a function of x.  Therefore, differentiate the given function:

[tex]\begin{aligned}f(x) & = \sqrt{x}-2x\\& = x^{\frac{1}{2}}-2x\\\implies f'(x) & = \dfrac{1}{2}x^{\left(\frac{1}{2}-1\right)}-2x^{(1-1)}\\ & = \dfrac{1}{2}x^{-\frac{1}{2}}-2x^0\\ & = \dfrac{1}{2\sqrt{x}}-2\end{aligned}[/tex]

Increasing function

To find the interval where f(x) is increasing, set the differentiated function to more than zero and solve for x:

[tex]\begin{aligned}f'(x) & > 0\\\implies \dfrac{1}{2\sqrt{x}}-2 & > 0\\\dfrac{1}{2\sqrt{x}} & > 2\\\ \dfrac{1}{2} & > 2\sqrt{x}\\\dfrac{1}{2 \cdot 2} & > \sqrt{x}\\ \dfrac{1}{4} & > \sqrt{x}\\ \sqrt{x} & < \dfrac{1}{4}\\\left(\sqrt{x}\right)^2 & < \left(\dfrac{1}{4}\right)^2\\ x & < \dfrac{1}{16}\end{aligned}[/tex]

As the domain is restricted, the function is increasing when:

[tex]\textsf{Solution}: \quad 0 \leq x < \dfrac{1}{16}[/tex]

[tex]\textsf{Interval notation}: \quad \left[0, \dfrac{1}{16}\right)[/tex]

Decreasing function

To find the interval where f(x) is decreasing, set the differentiated function to less than zero and solve for x:

[tex]\begin{aligned}f'(x) & < 0\\\implies \dfrac{1}{2\sqrt{x}}-2 & < 0\\\dfrac{1}{2\sqrt{x}} & < 2\\\ \dfrac{1}{2} & < 2\sqrt{x}\\\dfrac{1}{2 \cdot 2} & < \sqrt{x}\\ \dfrac{1}{4} & < \sqrt{x}\\ \sqrt{x} & > \dfrac{1}{4}\\\left(\sqrt{x}\right)^2 & > \left(\dfrac{1}{4}\right)^2\\ x & > \dfrac{1}{16}\end{aligned}[/tex]

Therefore, the function is decreasing when:

[tex]\textsf{Solution}: \quad x > \dfrac{1}{16}[/tex]

[tex]\textsf{Interval notation}: \quad \left(\dfrac{1}{16}, \infty\right)[/tex]

Note: The answer quoted in the original question is incorrect for the quoted function (please refer to the attached graph for proof).

Differentiation Rules

[tex]\boxed{\begin{minipage}{4.8 cm}\underline{Differentiating $ax^n$}\\\\If $y=ax^n$, then $\dfrac{\text{d}y}{\text{d}x}=nax^{n-1}$\\\end{minipage}}[/tex]

To have $600,000 for retirement in 30 years with 6% interest , we need to deposit $498.29 in the account each month.

As given in the question,

Final amount 'A'= $600,000

Time 't' = 30years

Rate of interest 'r'= 6%

Month =12

Let 'P' be the amount deposit in the account each month

[tex]A = \frac{(P(1+\frac{r}{m})^{mt}-1 )}{\frac{r}{m} }\\\\\implies 600,000 = \frac{(P(1+\frac{6}{12\times 100})^{12\times 30}-1 )}{\frac{6}{12\times 100} }\\\\\implies 600,000 \times \frac{1}{200}= P (\frac{12.06}{12})^{360}-1 \\\\\implies 3000 +1 = P(1.005)^{360} \\\\\implies P = \frac{3001}{6.02258} \\\\\implies P = \$498.29[/tex]

Therefore, to have $600,000 for retirement in 30 years with 6% interest , we need to deposit $498.29 in the account each month.

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Answer:

-15/4 = x

Step-by-step explanation:

17 = -13 -8x

Solving for x

Add 13 to each side

17 +13 = -13+13-8x

30 = -8x

Divide each side by -8

30/-8 = -8x/-8

Simplify by dividing the top and bottom by 2

-15/4 = x

Answer:

10-18i

Step-by-step explanation:

1(1-9i)+9(1-9i)

1-9i+9-9i

1+9-9i-9i

10-18i

A triangle in which one of the interior angles is 90° is called a right triangle. The longest side opposite to the right angle is the hypotenuse and the two arms of the right angle are the height and the base.

The area of each triangle is 200 square inches.

A triangle in which one of the interior angles is 90° is called a right triangle. The longest side opposite to the right angle is the hypotenuse and the two arms of the right angle are the height and the base.

We have,

The area of a right triangle is given by:

= 1/2 x base x height

We have from the figure,

Base = 20 inches

Height = 20 inches

Area = 1/2 x 20 x 20

Area = 10 x 20

Area = 200 square inches

Thus the area of each triangle is 200 square inches.

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The complete question is:

You work as an assistant to a carpenter who designed the tabletop below. He tells you that each shape is a right triangle, and each is the same size. You now need to calculate the area of one triangle so that you can begin building the tabletop. What is the area of each triangle in square inches?

A. 200 square inches

B. 400 square inches

C. 570 square inches

D. 2,800 square inches

E. 5,600 square inches

The price of an adult ticket is x= $10

What is mathematical expression?

A mathematical expression is defined as the collection of the numbers variables and functions by using signs like addition, subtraction, multiplication, and division.

Let the adult ticket be x

Let the children ticket be y

first day,

1*x + 6*y = 52

x + 6y = 52   equation 1

second day,

1*x + 8*y = 66

x + 8y = 66   equation 2

Solving both the equations

Subtracting equation 2 -equation 1, we get

x + 8y - x - 6y = 66 - 52

2y = 14

y = 7

putting the value of y in equation 1, we get

x + 6*7 = 52

x = 52 - 42

x = 10

Therefore, price of an adult ticket is x= $10

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Answer:

Reflection over y- axis

Answer:
No solution

Step-by-step explanation:
Let's solve your system by elimination.−2x+14y=−29;x−7y=19Multiply the second equation by 2, then add the equations together.(−2x+14y=−29)2(x−7y=19)
Becomes

−2x+14y=−292x−14y=38Add these equations to eliminate x:0=9

The size of the file that has been downloaded is 50 megabytes.

Given total size of the file is 244 megabytes

The percentage that completed the download till now = 18.5%

We have to determine the size of the file downloaded till now can be written as

= 18.5 % of 244

= 18.5 / 100 * 244

= 45.14 megabytes

≈ 50 megabytes

The size of the file that has been downloaded is 50 megabytes.

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it will be 7:8

Step-by-step explanation:

28/4 32/4

7 8

Answer:
(-10,-10)Step-by-step

explanation:

Yes 9x-9y=03x-4y=10In

elimination, we want both equations to have the same form and like terms to be lined up.  We have that.  We also need one of the columns with variables to contain opposites or same terms. Neither one of our columns with the variables contain this.  We can do a multiplication to the second equation so that the first terms of each are either opposites or sames. It doesn't matter which.  I like opposites because you just add the equations together. So I'm going to multiply the second equation by -3.I will rewrite the system with that manipulation:9x-9y=0-9x+12y=-30----------------------Add them up!0+3y=-30     3y=-30       y=-10So now once you find a variable, plug into either equation to find the other one.I'm going to use 9x-9y=0 where y=-10.So we are going to solve for x now.9x-9y=0 where y=-10.9x-9(-10)=0  where I plugged in -10 for y.9x+90=0  where I simplified -9(-10) as +90.9x     =-90 where I subtracted 90 on both sides.x=       -10 where I divided both sides by 9.The solution is (x,y)=(-10,-10)

Answer:

514 grams is equal to 0.514 kilograms

The lateral area of the square pyramid that has a base of 10ft and height of 10 ft is calculated as: 200 ft²

A square pyramid is a solid that has a square base and four triangular lateral faces.

The lateral area of the square pyramid is therefore the total area of the four triangular lateral faces of the square pyramid.

Area of one triangular lateral face = 1/2(base)(height)

Lateral area of the square pyramid = Total area of the four triangular lateral faces = 4 × 1/2(base)(height) = 2(base)(height)

We know the following:

Base = 10 ft

Height = 10 ft

Therefore:

Lateral area of the square pyramid = 2(10)(10)

Lateral area of the square pyramid = 200 ft²

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Answer:

24/15

Step-by-step explanation:

120 : 75 =

120/5 : 75/5 =

24 : 15 =

24/15

Answer:

8 : 5 or 8/5

Step-by-step explanation:

starts as 120 : 75

divide both by 5

= 24 : 15

divide both by 3

= 8 : 5 or 8/5

you just need to get them down to the smallest they can. in this case it is 8 and 5. for every 8 $10 bills there are, you have 5 $20 bills.

Answer:

Price per pound = $0.64

which means one pound of banana costs 0.64$.

so,If 1 pound costs 0.64$ then 6.7 pounds would cost:

1 pound = $0.64

6.7 pounds= 0.64× 6.7

that is equal to $4.288.

Step-by-step explanation:

Price per pound = $0.64

which means one pound of banana costs 0.64$.

so,If 1 pound costs 0.64$ then 6.7 pounds would cost:

1 pound = $0.64

6.7 pounds= 0.64× 6.7

that is equal to $4.288.

Answer:

1/4 gallon

Step-by-step explanation:

Just multiply the numerator and denominator of the given rate by 2:

2(1/8) gallon 1/4 gallon

------------------ = ---------------- = 1/4 per gallon

2(1/2) hour 1 hour

Answer:

2

Step-by-step explanation:

-3/3 + (7-2^2) = _

-1+3=2

The value of the given expression when simplified is equal to 318.

In mathematics, an expression is defined as a set of numbers, variables, and mathematical operations formed according to rules dependent on the context.

The given expression can be simplified using the PEMDAS as shown below.

-7 + 5 - 40 × -8

As per the PEMDAS multiplication should be solved first,

= -7 + 5 + 320

Further, solve addition,

= -7 + 5 + 320

= -7 +325

= 318

Hence, the value of the given expression when simplified is equal to 318.

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Answer:

B, J, C.

Collinear points means the points are on the same straight line, which means they have the same gradient. B, J, C are the only points on the sane straight line.

The transformed function can be seen in the image below, and the rule is:

g(x)= x/3 + 7

Here we have the parent function, and we want to apply a translation of 7 units up and stretch it vertically by a scale factor of 3

First, we need to apply the translation, remember that a vertical translation of N units is written as:

g(x) = f(x) + N

If we want the translation to be upwards, then we need to use a positive value of N, so in this case we wil have:

g(x) = f(x) + 7

And now we apply a stretch of scale factor 3, this means that we need to divide the variable inside of the function by 3, so we get:

g(x) = f(x/3) + 7

This will give:

g(x)= x/3 + 7

Where we use the fact that f(x) = x

The graph of this function can be seen below.

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The wages earned depends on the time worked is the correct relationship between the quantities on the given graph that means option C is correct

The graph given in the question shows the time worked on x axis and wages earned on y axis and a ray staring from the origin and moving upward

from the graph we can say that when time worked is zero then wages earned is zero. As the time worked is increasing the wages earned is also increasing .

Hence, it can be said that The wages earned depends on the time worked is the correct relationship between the quantities on the given graph that means option C is correct

The correct question is -Determine the relationship between the quantities on the given graph.

Total Wages

Wages Earned ($)

Time Worked (hours)

OA. The hourly rate depends on the wages earned.

OB. The time worked depends on the wages earned.

OC. The wages earned depends on the time worked.

OD. The time worked depends on the hourly rate.

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Answer:2

Step-by-step explanation: x+1+2x=3x+1        3x+1=7        3x=7-1       x=6/3   x=2

hope it helps

Answer:

3

Step-by-step explanation:

x+1+2x=7

variables should be on the left and the whole number on the right.

something like this: x+2x=7-1

simplify the equation: 3x=6

find x: x= 6/3=2

so u know x is 2

then substitute the 2 into x. so 2+1 =3#

Your answer is three.

The true statement is that (a) in approximately 50% of the months he sold less than 9 cars.

The given parameter is the box and whisker plot

As mentioned above, the box and whisker plot is a great tool to show the five-number summary.

The five-number summaries are:

On the box and whisker plot, we have the following five-number summaries:

The difference between the quartlles is

Difference = Upper quartile - Lower quartile

So, we have

Difference = 12 - 4

Evaluate

Difference = 8

This means that in approximately 50% of the months he sold 8 cars

8 is less than 9

This means that in approximately 50% of the months he sold less than 9 cars.

Hence, the true statement is that (a) in approximately 50% of the months he sold less than 9 cars.

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For x = -2,-6, and -10 the value of the polynomial will be satisfied.

Roots of an equation are the solution of that equation since an equation consists of hidden values of the variable to determine them by different processes and then the resultant is called roots.

Given the polynomial,

x³ + 18x² + 92x + 120 = 0

By hit trial method let's put x = -2

-8 + 72 - 184 + 120 = 0

-192 + 192 = 0

0 = 0

Now,divide x³ + 18x² + 92x + 120 = 0 by x + 2

(x + 2)(x²+ 16x + 60) = 0

(x + 2)(x + 6)(x + 10) = 0

x = -2,-6 and -10.

Hence "For x = -2,-6, and -10 the value of the polynomial will be satisfied".

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