6a77469e-9381-4500-a839-ce4c573f3f8d.html

<h4>Connecting Exponential Situations and Graphs</h4>
<p>Match each graph representation to the description of the situation characterized by exponential change.</p>
<p>Graph 1</p>
<p><img alt="GRAPH 1 SHOWS AN INCREASING EXPONENTIAL FUNCTION WITH A \(y\)-intercepts OF 100 AND PASSING THROUGH THE POINTS (2, 400) AND (3, 800). " height="270" src="https://k12.openstax.org/contents/raise/resources/958249736641840d0ebaa8c3a767661b98b06c7c" width="270">&nbsp; &nbsp; </p>
<p>Graph 2<br>
  <img alt="GRAPH 2 SHOWS AN INCREASING EXPONENTIAL FUNCTION WITH A \(y\)-intercepts OF 1 AND PASSING THROUGH THE POINTS (1, 5) AND (2, 25)." height="256" src="https://k12.openstax.org/contents/raise/resources/b182bac20ed4149d755cc6dae5629a1b0b156532" width="256">
</p>
<p>Situation 1: A dangerous bacterial compound forms in a closed environment but is immediately detected. This bacteria is known to double in concentration in a closed environment every hour and can be modeled by the function \(P(t) = 100 \cdot 2^t\), where \(t\)<em>&nbsp;</em>is measured in hours.</p>
<p>Situation 2: Loggerhead turtles reproduce every 2 to 4 years, laying approximately 120 eggs in a clutch. Using this information, we can derive an approximate equation to model the turtle population. As is often the case in biological studies, we will count only the female turtles. If we start with a population of one female turtle in a protected area and assume that all turtles survive, we can roughly approximate the population of female turtles by \(P(t) = 5^t\).&nbsp;</p>
<p>Since each graph represents exponential growth, and they both have the same labels, you have to look more closely at each. The biggest difference is the \(y\)-intercept or initial value. Graph 1 has an initial value of 100, and Graph 2 has a very small initial value. When you reread the situations, you can see that Situation 1 starts with 100 bacteria, and Situation 2 starts with 1 female turtle. This helps you see that Situation 1 belongs to Graph 1 and Situation 2 belongs to Graph 2.</p>
<h4>Try It: Connecting Exponential Situations and Graphs</h4>
<p><img alt="Graph of function." src="https://k12.openstax.org/contents/raise/resources/7cb598f229f63245610c8a599f14aa3097af3a00">&nbsp; &nbsp;<img alt="Graph of a decreasing exponential function, origin O. hours after taking medicine and mg of medicine in the body.&Acirc;&nbsp;" src="https://k12.openstax.org/contents/raise/resources/55a7e419b404f2d13a9d3dfa7f758af3130bc093"></p>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal1" data-schema-version="1.0">
  <div class="os-raise-ib-cta-content">
    <p>Write a possible situation for each graph. Make sure to include vocabulary about exponential functions.</p>
  </div>
  <div class="os-raise-ib-cta-prompt">
    <p>Write down your answer, then select the <strong>solution</strong> button to compare your work. </p>
  </div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal1">
  <p>Compare your answer:</p>
  <p>Here is how to write an exponential situation when given a graph:</p>
  <p>First, determine if the function is growth or decay. If any points are provided, you can use them to determine the growth factor or initial value. Be sure to include the labels of the graph. Here are sample answers.</p>
  <p>First, graph the number of people infected growing exponentially each week.</p>
  <p>Second, graph the amount of medicine in the body starting at 80 mg and decaying by a factor of \(\frac14\) each hour.</p>
</div>