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When we talk about exponential growth, we refer to a situation where a quantity increases at a rate proportional to its current value. The key characteristic of exponential growth is that the growth speed accelerates over time. This is because each increase is a percentage of the entire amount, which means as the amount grows, the actual increase does as well.

For example, let's look at Andrew and Meghan's rent situation. If the rent increases by 11.1% each year, we would say the rent is experiencing exponential growth. The formula to calculate the rent after a certain number of years is given by the equation: \( y=2700(1+0.111)^x \), where \( y \) is the rent after \( x \) years. Note that 11.1% translates to 0.111 as a decimal in the formula, and the initial rent is \( 2700 \) dollars. With each passing year, the rent increases by 11.1% of the previous year's rent, leading to a faster increase as time goes on.

In contrast to exponential growth, a linear equation describes a constant rate of increase. It can be represented by the equation of a straight line: \( y=mx+b \). Here, \( m \) is the slope of the line, representing the fixed rate of increase, and \( b \) is the y-intercept, representing the starting value. The variable \( x \) would typically represent time.

Using again the example of Andrew and Meghan, if their rent increases by a constant amount of \( 200 \) dollars every year, the rent could be described by the linear equation: \( y = 200x + 2700 \). This means that no matter how many years pass, the amount added to the rent is always \( 200 \) dollars, creating a straight line on a graph when we plot rent against time.

Graphing functions is a powerful way to visualize mathematical relationships. By plotting a function, we can quickly see how changes in one variable affect another. To graph exponential and linear functions, we choose a range of \( x \) values (which might represent time, as in our rent example) and calculate the corresponding \( y \) values (such as rent amounts).

When we graph the exponential function \( y=2700(1+0.111)^x \) and the linear function \( y=200x+2700 \) for Andrew and Meghan's rent, we will see two distinctly different shapes. The exponential graph curves upwards, becoming steeper as \( x \) increases, while the linear graph is a straight line with a consistent slope. Recognizing these shapes helps to understand the behavior of the rent over time.

The rate of increase is crucial in understanding how quantities change over time. With a linear rate of increase, such as in the rent model described by \( y=200x+2700 \), the rent rises by a fixed amount each year. It’s predictable and steady, and over any interval of time, the increase is the same, which is seen as a straight line on a graph.

In contrast, an exponential rate of increase means that the quantity goes up by a fixed percentage each year. It gets larger as the base amount gets larger. So for Andrew and Meghan's rent under an exponential model, not only does the rent go up every year, but the actual dollar increase each year is larger than the year before. This is a critical distinction from the linear model and contributes to the exponential curve seen on the graph.