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### 4.1 Linear Functions

### 4.2 Modeling with Linear Functions

ⓐ $C(x)=0.25x+25,000$

ⓑ The *y*-intercept is $(0,25,000)$. If the company does not produce a single doughnut, they still incur a cost of $25,000.

### 4.1 Section Exercises

The point of intersection is $\left(a,\phantom{\rule{0.5em}{0ex}}\text{}a\right).$ This is because for the horizontal line, all of the $y$ coordinates are $a$ and for the vertical line, all of the $x$ coordinates are $a.$ The point of intersection is on both lines and therefore will have these two characteristics.

$\begin{array}{l}f(0)=-(0)+2\\ f(0)=2\\ y-\mathrm{int}:(0,2)\\ 0=-x+2\\ x-\mathrm{int}:(2,0)\end{array}$

$\begin{array}{l}h(0)=3(0)-5\\ h(0)=-5\\ y-\mathrm{int}:(0,-5)\\ 0=3x-5\\ x-\mathrm{int}:\left(\frac{5}{3},0\right)\end{array}$

$\begin{array}{l}-2x+5y=20\\ -2(0)+5y=20\\ 5y=20\\ y=4\\ y-\mathrm{int}:(0,4)\\ -2x+5(0)=20\\ x=-10\\ x-\mathrm{int}:(-10,0)\end{array}$

The rate of change is 0.1. For every additional minute talked, the monthly charge increases by $0.1 or 10 cents. The initial value is 24. When there are no minutes talked, initially the charge is $24.

The slope is –400. this means for every year between 1960 and 1989, the population dropped by 400 per year in the city.

### 4.2 Section Exercises

(–30, 0) Thirty years before the start of this model, the town had no citizens. (0, 75,000) Initially, the town had a population of 75,000.

$\left(-15,\text{}0\right)$ : The *x*-intercept is not a plausible set of data for this model because it means the baby weighed 0 pounds 15 months prior to birth. $\left(0,\text{7}.\text{5}\right)$ : The baby weighed 7.5 pounds at birth.

$\left(\text{58}.\text{7},\text{}0\right):$ In roughly 59 years, the number of people inflicted with the common cold would be 0. $\left(0,\text{12},0\text{25}\right)$ Initially there were 12,025 people afflicted by the common cold.

- ⓐC(x) = 0.15x + 10
- ⓑThe flat monthly fee is $10 and there is a $0.15 fee for each additional minute used
- ⓒ$113.05

### 4.3 Section Exercises

When our model no longer applies, after some value in the domain, the model itself doesn’t hold.

The closer the number is to 1, the less scattered the data, the closer the number is to 0, the more scattered the data.

Yes, trend appears linear because $r=0.\text{985}$ and will exceed 12,000 near midyear, 2016, 24.6 years since 1992.

$(\mathrm{-2},\mathrm{-6}),(1,\text{\u221212}),(5,\mathrm{-20}),(6,\text{\u221222}),(9,\text{\u221228});$ Yes, the function is a good fit.