9B

Decide whether the following statement makes sense​ (or is clearly​ true) or does not make sense​ (or is clearly​ false). Explain your reasoning. When I graphed the linear​ function, it turned out to be a wavy curve. Choose the correct answer below.

The statement does not make sense because a linear function has a​ straight-line graph.

Decide whether the following statement makes sense​ (or is clearly​ true) or does not make sense​ (or is clearly​ false). Explain your reasoning. After 100​ years, a population growing at a rate of​ 2% per year will have grown by twice as many people as a population growing at a rate of​ 1% per year.

The statement does not make sense because both populations grow​ exponentially, not linearly.

Determine whether the following statement makes sense​ (or is clearly​ true) or does not make sense​ (or is clearly​ false). Explain your reasoning. We can use the fact that radioactive materials decay exponentially to determine the ages of ancient bones from archaeological sites.

The statement makes sense because if the original amount of material is known and the​ half-life is​ known, then the amount of time can be found using the exponential model.

Decide whether the following statement makes sense​ (or is clearly​ true) or does not make sense​ (or is clearly​ false). Explain your reasoning. My freeway speed is the rate of change in my distance with respect to time.

The statement makes sense because the slope of a linear function is the change in the dependent variable divided by the change in the independent variable.

What is general shape of an exponential decay​ function?

A falling curve that​ approaches, but never​ reaches, the horizontal axis

What is the general shape of an exponential growth​ function?

A steeply rising curve

Describe how you can graph an exponential function with the help of doubling time or​ half-life. What is the general shape of an exponential growth​ function? What is the shape of an exponential decay​ function?

Start at the point left parenthesis 0 comma Upper Q 0 right parenthesis0,Q0​, where Upper Q 0Q0 is the initial value of the function. For an exponentially growing​ quantity, the value Q of the function is 2Upper Q 0Q0 after one doubling​ time, 4Upper Q 0Q0 after two doubling​ times, 8Upper Q 0Q0 after three doubling​ times, and so on. Fit a curve between these points.

Explain how the function is used for exponential growth and decay. Choose the correct answer below. Select all that apply.

The function is used for exponential growth if rgreater than>0. The function is used for exponential decay if rless than<0.

Describe the general equation for a linear function. How is it related to the standard algebraic form y=mx+​b?

The general equation for a linear function is dependent variableequals=initial valueplus+​(rate of changetimes×independent ​variable). In the standard algebraic form of a linear​ function, y is the dependent​ variable, x is the independent​ variable, m is the rate of​ change, and b is the initial value.

How is the rate of change of a linear function related to the slope of its​ graph?

The smaller the rate of​ change, the shallower the graph. the greater the rate of​ change, the steeper the graph.

Decide whether the following statement makes sense​ (or is clearly​ true) or does not make sense​ (or is clearly​ false). Explain your reasoning. I graphed two linear​ functions, and the one with the greater rate of change had the greater slope.

This makes sense. According to the definition of linear​ functions, the rate of change is equal to the slope of the graph and the greater the rate of​ change, the greater the slope.

What does it mean to say that a function is​ linear?

a. The function can be described by an equation of the form y=mx+b. Your answer is correct. B. The function has a​ straight-line graph. c. The function has a constant rate of change.

Briefly explain how to find the doubling time and​ half-life from the exponential equation. Choose the correct answer below. Substitute Qequals=2Upper Q 0Q0 and tequals=Upper T Subscript doubleTdouble or Qequals=one half12Upper Q 0Q0 and tequals=Upper T Subscript halfThalf into Upper Q equals Upper Q 0 left parenthesis 1 plus r right parenthesis Superscript tQ=Q0(1+r)t. ​Next, divide both sides by Upper Q 0Q0​, and solve for ​(1plus+​r) by raising both sides of the equation to the power StartFraction 1 Over Upper T Subscript double EndFraction or StartFraction 1 Over Upper T Subscript half EndFraction .1Tdouble or 1Thalf. Substitute this expression for left parenthesis 1 plus r right parenthesis(1+r) to get the alternative exponential equation form in terms of the doubling time or​ half-life.

divide 1/Tdouble or 1/Thalf (1+r)

Describe the meanings of all the variables in the exponential function Upper Q equals Upper Q 0 left parenthesis 1 plus r right parenthesis Superscript tQ=Q0(1+r)t. Explain how the function is used for exponential growth and decay.

t=time This is the correct answer. B. requals=fractional growth rate for the quantity​ (or decay​ rate)