Compute the missing data in the table for the following exponential function

The identified equations are linear as they conform to the standard linear format y = mx + b. They encompass scenarios like flu cases over years and predicted total hours for given square footage, all exhibiting linear relationships. Substituting particular x-values verifies solutions through identities.

Identifying Linear Equations

From the information provided, the task is to identify which equations are linear. A linear equation will generally have a format of y = mx + b, where m represents the slope and b represents the y-intercept. According to Practice Test 4 solutions, all three equations listed in option 1 (y = -3x, y = 0.2 +0.74x, y=-9.4 - 2x) are linear because they fit this model.

Additionally, the use of x and y in the context of independent and dependent variables is consistent with linear relationships. In the case of flu cases depending on the year, the year is independent and the number of cases is dependent. Similarly, relationships like the total number of hours required depending on the square footage, or the total payment based on the number of students, fit the linear model with an equation of the form y = mx + b.

Moreover, speaking of identities, when specific values of x are substituted in an equation resulting in an obvious equation such as 6 = 6, they confirm the solutions to the linear equation being correct.

The range of the function f(x) is:

                           { f(x) | –∞ < f(x) ≤ 4}

By looking at the graph of the function f(x) we see that in the interval :

           (-∞,0]

The function f(x) takes a constant value as:  f(x)=4

and after that i.e. for x≥0 , the function f(x) is decreasing continuously.

Hence, we could say that the function f(x) takes all the real values which are less than and equal to 4.

          Hence, the range of the function is:

              { f(x)| –∞ < f(x) ≤ 4}

Answer:

(A)The systolic pressure of a person of age 44 is 135.1 mm Hg

(B) If a person's systolic pressure is 133.36 mm Hg, their age is 41 years.

Step-by-step explanation:

Given : P = 0.007 y² - 0.01 y +122

where P is systolic pressure and y is age of a person

(A) Here age of the person, y =44

So, P =0.007 (44²) -0.01 (44) +122 = 13.552 -0.44 +122 = 135.112 =135.1 mm

∴ The systolic pressure of a person of age 44 is 135.1 mm Hg

(B) Here P = 133.36 mm Hg

So,

133.36 = 0.007 y² - 0.01 y +122

=>0.007 y² -0.01 y -11.36 =0

=> 7 y² -10 y -11360 =0

Solving the above quadratic equation using quadratic formula, we have

[tex]y = \frac{5+\sqrt{79545} }{7}[/tex]

or [tex]y = \frac{5-\sqrt{79545} }{7}[/tex]

y = 41 or y = -39.57

Since age cannot be negative, y= 41

∴ If a person's systolic pressure is 133.36 mm Hg, their age is 41 years.

The systolic pressure of a 44-year-old person is approximately 132.8 mm Hg, and a person with a systolic pressure of 133.36 mm Hg is roughly 46 years old.

To answer this question, we will be using the given equation P = 0.007y^2 - 0.01y + 122, where P represents a person's systolic blood pressure and y represents their age.

A) To find the systolic pressure for a person aged 44 years, we substitute y=44 into the equation. This gives us P = 0.007 * (44)^2 - 0.01 * 44 + 122 = 132.8 mm Hg.

B) To find the age of a person with a systolic pressure of 133.36 mm Hg, we set P=133.36 and solve the equation for y. This can be done using methods such as factoring, completing the square, or using the quadratic formula. Upon solving, we find y roughly equals 46, so the person is approximately 46 years old when rounded to the nearest whole year.

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Answer:

h = –1.5

Step-by-step explanation:

we know that

In the function  

[tex]f\left(x\right)=\left|x-h\right|+k[/tex]

The point (h,k) is the vertex of the function

where

h is the x-coordinate of the vertex

k is the y-coordinate of the vertex

In this problem the vertex is the point (-1.5,-3.5)

therefore

h=-1.5

the function  is

[tex]f\left(x\right)=\left|x+1.5\right|-3.5[/tex]

see the graph in the attached figure

Answer:

There were 275 computers

Step-by-step explanation:

Computers replaced = total computers * percent replaced

What do we know?

The percent replaced is 20  = .2

55 computers were replaced.

Substitute this in

55 = total computers * .2

Divide each side by .2

55/.2 = total computers *.2 /.2

275 = total computers

There were 275 computers

To find the total number of computers in the school, you set up the equation 0.20 * x = 55, where x is the total number of computers. Solving for x gives us x = 275, meaning there are 275 computers in the school.

The question is asking us to find the total number of computers in the school knowing that 20% of them were replaced and knowing that 55 computers were replaced. To find the total number of computers, we need to understand that the 55 computers represent the 20% that were replaced. So, we set up a proportion where 20% (0.20) of the total number of computers (which we will call x) equals to 55. The equation will look like this: 0.20 * x = 55.

We divide both sides of the equation by 0.20 to solve for x:

x = 55 / 0.20

x = 275

Thus, the total number of computers in the school is 275.

Answer:

[tex]y=(x+3)^{2} -1[/tex]

Step-by-step explanation:

General equation of a parabola that opens up is [tex]y=a(x-h)^{2} +k[/tex] , a>0

So equation becomes [tex]y=(x-h)^{2} +k[/tex] which implies option A and option B can never be true for this parabola.

Also this parabola's vertex lies in 3rd quadrant where coordinates of both x and y will be negative.

So, the equation of parabola will be of the form [tex]y=(x+h)^{2} - k[/tex]

Hence, option 4 that is [tex]y=(x+3)^{2} -1[/tex] is correct.

The correct equation of the graph is: x² + 6x - 35y + 8 = 0.

The equation of the graph shown, which is a parabola that opens upwards and crosses the x-axis at -2 and -4, can be determined by examining its key features.

The fact that the parabola opens upward indicates that the coefficient of the squared term (x²) in the equation must be positive.

The parabola crosses the x-axis at -2 and -4. This means that the two x-intercepts are -2 and -4.

Putting this information together, we can form the equation:

y = a * (x - r1) * (x - r2)

where 'a' is a positive constant (coefficient of x²), and r1 and r2 are the x-intercepts.

Since the x-intercepts are -2 and -4, the equation becomes:

y = a * (x - (-2)) * (x - (-4))

y = a * (x + 2) * (x + 4)

Now, let's find the value of 'a.' We are given that the parabola passes through the point (3, 1). We can substitute these coordinates into the equation to find 'a':

1 = a * (3 + 2) * (3 + 4)

1 = a * 5 * 7

a = 1 / (5 * 7)

a = 1 / 35

Now that we have the value of 'a,' we can finalize the equation:

y = (1/35) * (x + 2) * (x + 4)

Expanding the equation:

y = (1/35) * (x² + 6x + 8)

To put the equation in standard form, we can multiply both sides by 35 to eliminate the fraction:

35y = x^2 + 6x + 8

Finally, rearrange to get the equation in standard form:

x² + 6x - 35y + 8 = 0

So, the correct equation of the graph is: x² + 6x - 35y + 8 = 0.

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Answer:

The percentage of left-handed 6th graders is 20%.

Step-by-step explanation:

To find this, start by finding the total number of lefties.

19 girls + 11 boys = 30 students.

Next find the total number of students

90 girls + 60 boys = 150 students

Now divide the lefties by the total number.

30/150 = 20%

Answer:

[tex]y+1=-2(x-5)[/tex]

Step-by-step explanation:

We can write the equation of a line in 3 different forms including slope intercept, point-slope, and standard depending on the information we have. We have a point given and a slope from the equation. We will chose point-slope since we have a point and the slope.  

We will substitute [tex]m=-2[/tex] and [tex]x_1=5\\y_1=-1[/tex].

[tex]y-(-1)=-2 (x-5)[/tex]

[tex]y+1=-2(x-5)[/tex]

This is the equation of the line with slope -2 that passes through (5,-1).

ANSWER

g(x) > h(x) for x = -1 is TRUE

For positive values of x, g(x) > h(x) is TRUE

For negative values of x, g(x) > h(x) is also TRUE

EXPLANATION

The given functions are

[tex]g(x)={x}^{2}[/tex]

and

[tex]h(x)=-{x}^{2} [/tex]

If

[tex]x=0[/tex]

[tex]g(0)={0}^{2}=0[/tex]

[tex]h(0)=-({0})^{2}=0[/tex]

Based on this options A and B are FALSE.

When

[tex]x=-1[/tex]

[tex]g(-1)={( - 1)}^{2}=1[/tex]

[tex]h(-1)=-{(-1)}^{2}=-1[/tex]

[tex]g( - 1)>\:h(-1)[/tex]

for x=-1 is True.

When x=3,

[tex]g(3)={3}^{2}=9[/tex]

and

[tex]h(3)=-{3}^{2}=-9[/tex]

[tex]g(3)>\: h( 3)[/tex]

g(x) < h(x) for x = 3 is a FALSE statement.

For positive values of x, g(x) > h(x) is TRUE

See graph.

For negative values of x, g(x) > h(x) is also TRUE

See graph

The statements that are true for the functions g(x) = x^2 and h(x) = -x^2 are: h(x) will always be greater than g(x), g(x) < h(x) for x = 3, and for positive values of x, g(x) > h(x).

The statements that are true for the functions g(x) = x^2 and h(x) = -x^2 are:

These statements can be verified by plugging in values for x and comparing the outputs of the functions.

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Answer: The answer is B...........

Answer:

The correct option is A. The median-median line regression line is a better prediction.

Step-by-step explanation:

The given median-median line for a dataset is

[tex]y=1.133x+0.489[/tex]

The least-squares regression line for the same dataset is

[tex]y=1.068x+0.731[/tex]

The given point is (50,60).

Substitute x=50 in each given equation.

[tex]y=1.133(50)+0.489=57.139[/tex]

[tex]y=1.068(50)+0.731=54.131[/tex]

Since the value of median-median line at x=50 is near to 60 than the value of least-squares regression at x=50.

The median-median line regression line is a better prediction. Therefore the correct option is A.

Answer: D

Step-by-step explanation:

                        David - f(x)           Ronald - g(x)

f(x) = [tex]\sqrt[3]{x-1}[/tex]

f(2) = [tex]\sqrt[3]{2-1}[/tex]   = 1.00                      1.26                

f(4) = [tex]\sqrt[3]{4-1}[/tex]    = 1.44                      1.59

f(6) = [tex]\sqrt[3]{6-1}[/tex]    = 1.71                       1.82

f(10) = [tex]\sqrt[3]{10-1}[/tex]  = 2.08                    2.15

The side length for Ronald's box is greater than David's.

***********************************************************************

Answer: (0, 4)

Step-by-step explanation:

f(x) is an absolute value graph with a vertex at (0, 3).  

g(x) is a parabola reflected across the x-axis with a vertex at (4, 12).

Together: f(x) and g(x) are both increasing from 0 to 4 ⇒ (0, 4)

**********************************************************************

 0 o---------------------o 4

Question:

Approximate log base b of x, log_b(x).

Of course x can't be negative, and b > 1.

Answer:

f(x) = (-1/x + 1) / (-1/b + 1)

Step-by-step explanation:

log(1) is zero for any base.

log is strictly increasing.

log_b(b) = 1

As x descends to zero, log(x) diverges to -infinity

Graph of f(x) = (-1/x + 1)/a is reminiscent of log(x), with f(1) = 0.

Find a such that f(b) = 1

1 = f(b) = (-1/b + 1)/a

a = (-1/b + 1)

Substitute for a:

f(x) = (-1/x + 1) / (-1/b + 1)

f(1) = 0

f(b) = (-1/b + 1) / (-1/b + 1) = 1

Answer:

i believe it is c)82 but i may be wrong

Step-by-step explanation:

Set up a proportion:

The 4 foot pole casts a 5 foot shadow is written as 4/5

Let the height of the flagpole = x.

The flag pole casts a 16 foot shadow so it is written as X/16

Now set the two proportions equal to each other:

4/5 = X/16

Solve for X by cross multiplying:

5X = 64

Divide both sides by 5:

X = 64/5

X = 12.8

The flag pole is 12.8 feet tall.

Based on the information the tall of the flagpole is 12.8 feet tall .

Set up a proportion and let x = Height of the flagpole

4/5 =16/x

Solve x by cross multiplying

5x = 64

Divide both sides by 5x

x= 64/5

x=12.8 feet tall

Inconclusion the tall of the flagpole is 12.8 feet tall .

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Answer:

The equation would be y = 1/23x - 251/23

Step-by-step explanation:

To start, you need to locate the slope of the first equation. Since the slope is the coefficient of x, we know it to be -23. Now, the perpendicular slope is the opposite and reciprocal of that, which makes the new slope 1/23.

Now that we have this, we can use the point and the slope in point-slope form to get the equation.

y - y1 = m(x - x1)

y - 11 = 1/23(x - 2)

y - 11 = 1/23x - 2/23

y = 1/23x - 251/23

Answer:

Step-by-step explanation:

m<1 is 90 degrees

m<2 is 121 degrees

m<3 is 42 degrees

m<4 is 42 degrees

m<5 is 35 degrees

m<6 is 90 degrees

m<7 is 48 degrees

m<8 is 35 degrees

m<9 is 35 degrees

Answer: (D) P=24√6,  A=96√3

Step-by-step explanation:

Consider ΔABC where D is the midpoint of BC. Since ABC is an equilateral triangle, then segment AD is a perpendicular bisector with length of 12√2.   This creates ΔADC which is a 30°-60°-90° triangle.

Now you can use the rules for this special triangle to find the length of the hypotenuse.

30° ⇄ side length "a"        base - DC on ΔADC

60° ⇄ side length "a√3"   height - AD on ΔADC

90° ⇄ side length "2a"      hypotenuse - AC on ΔADC

Step 1: solve for "a"

[tex]AD: a\sqrt3=12\sqrt{12}[/tex]

       [tex]\dfrac{a\sqrt3}{\sqrt3}=\dfrac{12\sqrt2}{\sqrt3}[/tex]

       [tex]a=\dfrac{12\sqrt2}{\sqrt3}\bigg(\dfrac{\sqrt{3}}{\sqrt{3}}\bigg)[/tex]

            [tex]= \dfrac{12\sqrt6}{3}[/tex]

            [tex]=4\sqrt6[/tex]

Step 2: solve for "2a"

[tex]AC: 2a =2(4\sqrt{6})[/tex]

       [tex]=8\sqrt{6}[/tex]

Step 3: find the perimeter

The side length is equivalent for all 3 sides so

P = 3(AC)

  [tex]=3(8\sqrt{6})[/tex]

  [tex]=24\sqrt{6}[/tex]

Step 4: find the area

[tex]A=\dfrac{1}{2}b \cdot h[/tex]

    [tex]=\dfrac{1}{2}(8\sqrt6)(12\sqrt2)[/tex]

    [tex]=(48\sqrt{12})[/tex]  

    [tex]=(96\sqrt3)[/tex]  

Let X be the number of classes:

You need to multiply the cost per class by the number of classes.

For members you also need to add in the membership fee.

Members pay a total of 8 +4x

Non members pay a total of 6x

Set them to equal to solve for x, which is the number of classes taken:

8 + 4x = 6x

Subtract 4x from both sides:

8 = 2x

Divide both sides by 2:

x = 8/2 = 4

The answer is 4 classes.

Without additional context or a visual, it's impossible to determine the correct answer for the area of the shaded region using the provided options and the circle area formula A = πr². More information about the figure is needed.

The question asks to find the area of the shaded region using the area formula for a circle, which is A = πr².
However, without additional context or a diagram, it is impossible to provide a definitive answer to this question. Normally, to find the area of a shaded region involving a circle, one might calculate the area of the circle and then subtract the area of any unshaded parts that are inside the circle. However, the given answer options (a through d) suggest that the shaded region might involve an algebraic expression in terms of x. Based on typical problems, we might be dealing with a composite shape where x represents the dimension of another shape such as a square or rectangle. To find the correct answer, we would need to see the figure in question or have more information provided in the problem statement.

The area of the composite figure, which includes both the rectangle and the semicircle, is approximately [tex]\( 18.28 \)[/tex] square inches when calculated numerically The correct option is (b) is [tex]\( (8\pi + 12) \text{ in}^2 \)[/tex].

The provided image appears to show a composite figure consisting of a semicircle on top of a rectangle. To find the area of the composite figure, we need to calculate the area of the rectangle and the area of the semicircle separately, then add them together.

The formula to calculate the area of a rectangle is [tex]\( A = \text{length} \times \text{width} \)[/tex].

Given that the width of the rectangle (which is the same as the diameter of the semicircle) is 4 inches and the height (length) of the rectangle is 3 inches, the area of the rectangle is:

[tex]\[ A_{\text{rectangle}} = 4 \text{ in} \times 3 \text{ in} = 12 \text{ in}^2 \][/tex]

The formula to calculate the area of a circle is [tex]\( A = \pi r^2 \)[/tex], where [tex]\( r \)[/tex] is the radius. Since we have a semicircle, we will take half of the area of a full circle. The diameter of the semicircle is 4 inches, so the radius [tex]\( r \)[/tex] is 2 inches.

The area of the semicircle is then:

[tex]\[ A_{\text{semicircle}} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (2 \text{ in})^2 = 2 \pi \text{ in}^2 \][/tex]

Now, we'll add the areas of the rectangle and the semicircle to find the total area of the composite figure:

[tex]\[ A_{\text{total}} = A_{\text{rectangle}} + A_{\text{semicircle}} = 12 \text{ in}^2 + 2 \pi \text{ in}^2 \][/tex]

Let's calculate the total area.

The area of the composite figure, which includes both the rectangle and the semicircle, is approximately [tex]\( 18.28 \)[/tex] square inches when calculated numerically.

However, if we express the area in terms of [tex]\(\pi\)[/tex], the exact area is given by the formula:

[tex]\[ A_{\text{total}} = (8\pi + 12) \text{ in}^2 \][/tex]

Therefore, the correct answer is b. [tex]\( (8\pi + 12) \text{ in}^2 \)[/tex].

complete question given below:

What is the area of the composite figure?

(8pi+ 6) in2

(8pi+ 12) in2

(8pi+ 18) in 2

(8pi+ 24) in.2

The baseball team has won 7 out of 9 games this season, which as a fraction is represented as 7/9.

In order to find out the fraction of games the baseball team has won, we divide the number of games won by the total number of games played. In this case, the team has won 7 games out of 9 games played. Therefore, as a fraction, this is written as 7/9. This means that the team has won 7 out of every 9 games they've played so far this season.

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The baseball team has won 7 out of 9 games, so the fraction of games won is 7/9.

To calculate the fraction of games a baseball team has won, we divide the number of games won by the total number of games played. In this case, the team has won 7 games out of a total of 9 games played this season. To express this as a fraction, we write 7 as the numerator (the top number) and 9 as the denominator (the bottom number).

Therefore, the fraction of games the baseball team has won is 7/9.

Answer:

25, 12

Step-by-step explanation:

let x represent one number and y represent the other number

x+y=37    (sum is addition)

x-y=13      (difference is subtraction)

Im solving using the substitution method

x-y=13 add y to both sides to get x by itself

x=13+y

13+y+y=37 substitute x for the solution above in the other equation and simplify

2y=24

y=12

plug in y into one of the equations

x+12=37 subtract 12 from both sides

x=25

Answer:

The equation of this line would be y + 8 = 4(x - 3)

Step-by-step explanation:

In order to get this we can start with the base form of point-slope.

y - y1 = m(x - x1)

Now that we've got this, we can plug in the slope for m. We can also plug in the point for (x1, y1).

y + 8 = 4(x - 3)

Answer:

11 cans cost $9.57

Step-by-step explanation:

There are two ways of doing this, one being the longer and the other being the shorter.

The shorter one is simply multiplying the number by the new number over 5:

[tex]4.35 * \frac{11}{5}[/tex]

This works because this finds the amount the 5 has increased from 5 to 11 and muliplies the number by this.

[tex]4.35 * \frac{11}{5}[/tex] = 9.57

So 11 cans cost $9.57

Answer:

1 time

Step-by-step explanation:

f(x) = |x(x^2 − 1)|

The mean value theorem states

f'(c) = f(b) -f(a)

         -------------

           b-a

b = .75

a = .5

f(b) = abs(.75 * (.75^2 -1)) = abs (.75*(-.4375))=abs(-.328125)

                                        = .328125

f(a) = abs(.5 * (.5^2 -1)) = abs(.5*(-.75))=abs(-.375) = .375

                       .328125- .375

f'(c)  =       -------------------------------------------------

                          .75-.5

f'(c) = -.1875

Answer:

9 - 2/x

Step-by-step explanation:

Answer:

y = 3500(3/7 ) ^ 1/3

y = 239.23

Step-by-step explanation:

For exponential decay

y =a b^x

a = 3500

b<1

We can use the point (3, 1500)

1500 = 3500 * b^3

Divide each by 3500

1500/3500 = 3500/3500 b^3

15/35 = b^3

3/7 = b^3

Take the cube root of each side

(3/7) ^ 1/3 = b^3 ^ 1/3

b = (3/7 ) ^ 1/3

The function is

y = 3500 (3/7) ^ 1/3 x

We want to find y when x=9.5

y = 3500 (3/7) ^ (9.5 /3)

y = 3500 ( 3/7) ^ 3. 166666666

y = 3500 (.06835)

y = 239.23

Lets check a point and see if we a close

(7,500)

y = 3500 (3/7)^ (1/3*7)

y = 3500 (3/7) ^ (7/3)

y = 484.68

That is pretty close.  We do not know the exact value since we are reading from a graph.

Answer:

The cost of each DVD is $11.8 .

Step-by-step explanation:

Let us assume that the cost of one DVD be x.

As given

Hector spent $25.75 for 2 DVDs that cost the same amount.

The sales tax on his purchase was $3.15.

Hector also used a coupon for $1.00 off his purchase.

Than the equation

Total Hector Spents for two DVDs = 2 × cost of one DVD + Sales tax + Coupon cost .

Putting the value

25.75 = 2x + 3.15 - 1.00

25.75 = 2x +2.15

25.75 - 2.15 = 2x

23.6 = 2x

[tex]x = \frac{23.6}{2}[/tex]

x = $11.8

Therefore the cost of each DVD is $11.8 .