Broadband Internet Connections The number of households...

Broadband Internet Connections The number of households using broadband Internet connections, such as cable and DSL, increased from 6 million in 2000 to 30 million in 2004 . (Source: eMarketer.) (a) Find a linear function given by $$ B(x)=m\left(x-x_{1}\right)+y_{1} $$ that models these data, where $x$ is the year. (b) Use $B(x)$ to estimate the years when the number of households using broadband Intemet connections was 24 million or more. Assume that the domain of $B$ is 2000 to 2006

Broadband Internet Connections The number of
households using broadband Internet connections, such as cable and DSL, increased from 6 million in 2000 to 30 million in 2004 . (Source: eMarketer.)
(a) Find a linear function given by
$$
B(x)=m\left(x-x_{1}\right)+y_{1}
$$
that models these data, where $x$ is the year.
(b) Use $B(x)$ to estimate the years when the number of households using broadband Intemet connections was 24 million or more. Assume that the domain of $B$ is 2000 to 2006

Key Concepts

Domain Restriction

Domain restriction limits the set of input values for which the model is considered valid. In the context of modeling, it is important to define the appropriate domain to ensure that predictions and analyses remain accurate and relevant to the observed or intended data range.

Estimation Using Linear Models

Estimation using linear models involves substituting values into the constructed linear function to predict outcomes at specific points. This is useful for forecasting or interpolating data within the established trend, and it relies on the assumption that the linear relationship persists over the interval of interest.

Point-Slope Form

The point-slope form is a way to express the equation of a line when you know one point on the line and the slope. This provides a convenient formulation to construct the linear model directly from given data, facilitating easier problem-solving and analysis.

Slope Calculation

The slope of a line quantifies its steepness and is calculated as the ratio of the change in the dependent variable to the change in the independent variable. This fundamental concept in mathematics is critical in determining how one variable changes in response to another within linear models.

Linear Function Modeling

Linear function modeling is the process of representing the relationship between two variables with a straight line. This approach is used to simplify and predict behaviors of data where the change between variables is uniform, making it easier to understand trends and interpolate values within a given range.

Transcript

00:01 Okay, so what i have here is i have a word problem.

00:02 I have a story, right? and i'm trying to figure out first, you know, how to fill in the rest of this function.

00:10 Okay, but let's first make sure we understand the story.

00:14 So we have a number of households using broadband internet connection.

00:19 I'm increased from $6 million in 2000, so $6 million in 2000 to $30 million in 2004.

00:28 Now, my x value represents the year, and my y value, or in this case, my b value, is going to represent the number of households.

00:39 And from my two values, i get two points, right? my first x, y point is going to be the year 2000, and it's going to be $6 million, so just six.

00:51 And my second value is going to be in 2004, i get $30 million.

00:58 And we're in millions.

01:05 Okay.

01:07 So from this, i could kind of fill in the rest of this, right? i have an x one and a y one, right? i could just call this x one and y one and plug them in there.

01:16 But how do i figure out my slope, right? what is slope? well, it's changing y over change in x so i can subtract my y values, 30 minus 6 over my x values, 2004 minus 2000.

01:29 This will just be 24 divided by 4.

01:33 Or six will be my rate of change.

01:36 So plugging these all in, i can get the linear equation 6 times x minus 2000 plus 6.

01:46 Awesome...

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