1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Lady_Fox [76]
3 years ago
11

Let f(x)=a(2)^x. This exponential function is intersected twice by the line g(x)=x+b. One intersection is at point P(-2,2). Find

the values of a and b. The other intersection occurs at point Q(r,s), where r and s are both integers. Find the values of r and s. Show algerbraically that r is a solution to the equation f(x)=g(x).
Mathematics
1 answer:
mr_godi [17]3 years ago
7 0

Answer:

See below

Step-by-step explanation:

To find the values of a and b, we use the intersection point P.

Since the graph of f is intersected by P=(-2,2), f(-2)=2, that is, a(2)^(-2)=2. Then a=2(2)²=8.

Since the graph of g is intersected by P=(-2,2), g(-2)=2, that is, -2+b=2. Then b=2+2=4.

Hence a=8, b=4.

To find the other point of intersection Q, we need to find a number x such that f(x)=g(x), that is, 8(2)^(x)=x+4. Rewrite this equation as follows:

f(x)=8(2^x)=2^3(2^x)=2^{3+x}=g(x)=x+4

It is not easy to solve this equation analitically. Instead, you can apply a quantitative method. First, the left hand side is an exponential function, so it is always positive. Then the solution will be a number in which the right side (x+4) is positive, that is, x > -4. The solution is an integer, then the possible values are x=-3,-2,-1,0,1,2,...

Now, the RHS must be a power of two, so the possible integers are x=-3,-2,0,4,12,28,... However, the LHS grows very quickly, while the LHS grows slowly. Then, it is only necessary to test small integer values (we already know that x=-2 is a solution).

If we test with x=-1, we get 2²=4=3, then x=-1 is not a solution. Any x>-1 will not be a solution, because f increases much faster than g, and then they won't intersect. So our possibilities reduce to x=-3,-2. Taking x=-3, we get f(-3)=2⁰=1=-3+4=g(-3). Thus, (-3,0) is the other point of intersection Q. (r=-3, s=0).  

You might be interested in
Louis bought a laptop for $720. It was marked $90 off because it was an open product. He also got a 15% discount, which was take
Mekhanik [1.2K]
The original price is $952.94
3 0
3 years ago
Read 2 more answers
Convert the equation y + 1 = -3/5 (x+5)
cestrela7 [59]
If you simplify each side of the equation you get this  y= -4 - 3x/5
4 0
3 years ago
The divisor of 0.004 is almost zero. What does this tell you about the quotient?
nasty-shy [4]

Answer:

  Its magnitude will be larger than 0.004.

Step-by-step explanation:

When a divisor is less than 1, the quotient will be greater than the dividend.

When the divisor is "almost zero", the quotient will be much greater than the dividend. Here, the dividend may be considered to be "almost zero", so we cannot say anything about the actual quotient except to say its magnitude will be greater than the dividend.

_____

The dividend is positive, so the quotient will have the same sign as the divisor. (Negative divisors can be "almost zero," too.)

3 0
3 years ago
Many high school students take the AP tests in different subject areas. In 2007, of the 144,796 students who took the biology ex
Nataly [62]

Answer:

(0.582-0.485) - 1.64 \sqrt{\frac{0.582(1-0.582)}{144796} +\frac{0.485(1-0.485)}{211693}}=0.0942  

(0.582-0.485) + 1.64 \sqrt{\frac{0.582(1-0.582)}{144796} +\frac{0.485(1-0.485)}{211693}}=0.09978  

And the 90% confidence interval would be given (0.0942;0.09978).  

We are confident at 90% that the difference between the two proportions is between 0.0942 \leq p_A -p_B \leq 0.09978

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".  

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

p_A represent the real population proportion female for Biology

\hat p_A =\frac{84199}{144796}=0.582 represent the estimated proportion female for biology

n_A=144796 is the sample size for A

p_B represent the real population proportion female for calculus AB

\hat p_B =\frac{102598}{211693}=0.485 represent the estimated proportion female for Calculus AB

n_B=211693 is the sample size required for B

z represent the critical value for the margin of error  

The population proportion have the following distribution  

p \sim N(p,\sqrt{\frac{p(1-p)}{n}})  

The confidence interval for the difference of two proportions would be given by this formula  

(\hat p_A -\hat p_B) \pm z_{\alpha/2} \sqrt{\frac{\hat p_A(1-\hat p_A)}{n_A} +\frac{\hat p_B (1-\hat p_B)}{n_B}}  

For the 90% confidence interval the value of \alpha=1-0.90=0.1 and \alpha/2=0.05, with that value we can find the quantile required for the interval in the normal standard distribution.  

z_{\alpha/2}=1.64  

And replacing into the confidence interval formula we got:  

(0.582-0.485) - 1.64 \sqrt{\frac{0.582(1-0.582)}{144796} +\frac{0.485(1-0.485)}{211693}}=0.0942  

(0.582-0.485) + 1.64 \sqrt{\frac{0.582(1-0.582)}{144796} +\frac{0.485(1-0.485)}{211693}}=0.09978  

And the 90% confidence interval would be given (0.0942;0.09978).  

We are confident at 90% that the difference between the two proportions is between 0.0942 \leq p_A -p_B \leq 0.09978

5 0
3 years ago
HELP ON A TEST
Strike441 [17]
First three. A, b, c.
The triangle is a right triangle.
The triangle is a Pythagorean triple.
The triangles hypotenuse is 39 inches.
7 0
3 years ago
Other questions:
  • The average rainfall in Phoenix is 8.29 inches per year. The table shows recent data on the difference in annual rainfall from t
    6·2 answers
  • Find the measure of angle K
    6·1 answer
  • $43,000 at 7% for 8 years​
    5·1 answer
  • The Finest Hour department store wants to run a special ad this week in The Country Times. It wants to run a half-page ad on Fri
    10·1 answer
  • a candle supply store sells votive wax and low-shrink wax. how many pounds of low-shrink wax should be mixed with 8 pounds of vo
    13·1 answer
  • What is the range of possible sizes for side x?<br><br><br> 8.8
    12·1 answer
  • 550 calories in a 100 gram bar of chocolate. Write this rate as a simplified fraction
    15·1 answer
  • WILL GIVE BRAINLIEST BUT PLEASE ANSWER NOWWWWWWW MUST BE CORRECT
    11·2 answers
  • Hal, Zelda, Maya, and Jason each recorded the height and age of five classmates. They used the data to create the tables below.
    12·1 answer
  • Jeans cost $38. They are on sale for 45% off. What is the sale price?
    15·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!