Field Trip Model 172218

1. **Stating the problem** The task is to make a **real-life model** using a **rational function**, then analyze its domain, range, intercepts, zeros, asymptotes, and any holes, and finally write a reflection that you can submit to your teacher. 2. **Choosing a real-life situation** A simple and realistic situation is **cost per person for a field trip**. Suppose a group pays a fixed travel fee of $2400$ plus a snack fee of $150$ per person. If $x$ is the number of participants, then the total cost is $2400+150x$. The **cost per person** is the total cost divided by the number of participants, so the rational function is $$C(x)=\frac{2400+150x}{x}$$ This can also be written as $$C(x)=\frac{2400}{x}+150$$ 3. **Concept organizer: rational expression, rational equation, and rational function** - A **rational expression** is a fraction with polynomials, such as $$\frac{2400}{x}$$. - A **rational equation** is an equation that contains a rational expression, such as $$\frac{2400}{x}+150=300$$. - A **rational function** is a function written using a rational expression, such as $$C(x)=\frac{2400}{x}+150$$. Important rule: the denominator can never be $0$, because division by zero is undefined. 4. **Domain** The domain is all allowed values of $x$. Since $x$ is the number of participants, it must be positive. Also, $x\neq 0$ because the formula has $x$ in the denominator. So the domain is $$x>0$$ or in interval form, $$(0,\infty)$$ Meaning in context: there must be at least one participant, and the number of participants should be a positive whole number in real life. 5. **Range** To find the range, look at $$C(x)=\frac{2400}{x}+150$$ Since $\frac{2400}{x}>0$ for every positive $x$, the cost per person is always greater than $150$. As $x$ gets larger, $\frac{2400}{x}$ gets smaller, so the cost approaches $150$ but never reaches it. Therefore, the range is $$C(x)>150$$ or in interval form, $$(150,\infty)$$ Meaning in context: the cost per person is always more than $150$ because everyone still shares the fixed fee. 6. **x-intercept** The x-intercept happens when $C(x)=0$. Set the function equal to zero: $$\frac{2400}{x}+150=0$$ Subtract $150$ from both sides: $$\frac{2400}{x}=-150$$ Multiply both sides by $x$: $$\cancel{x}\cdot\frac{2400}{\cancel{x}}=-150x$$ $$2400=-150x$$ Divide both sides by $-150$: $$\frac{2400}{-150}=x$$ $$x=-16$$ But $x=-16$ is not allowed in the context because the number of participants cannot be negative. So there is **no valid x-intercept** for the real-life situation. 7. **y-intercept** The y-intercept happens when $x=0$. But $x=0$ is not allowed because it makes the denominator zero. So there is **no y-intercept**. Meaning in context: the model does not apply when there are no participants. 8. **Zero(s)** Zeros are the values of $x$ that make the function equal to $0$. From Step 6, the algebra gives $x=-16$, but this is not valid in context. So the function has **no real-life zero**. 9. **Vertical asymptote** A vertical asymptote occurs where the denominator is zero. For $$C(x)=\frac{2400}{x}+150$$ the denominator is $x$. Set it equal to zero: $$x=0$$ So the vertical asymptote is $$x=0$$ Meaning in context: the model breaks down when there are no participants. 10. **Horizontal asymptote** To find the horizontal asymptote, look at the end behavior. As $x$ becomes very large, $$\frac{2400}{x}\to 0$$ so $$C(x)\to 150$$ Therefore, the horizontal asymptote is $$y=150$$ Meaning in context: when many people join the trip, the cost per person gets closer and closer to $150$. 11. **Holes** A hole appears when a factor cancels out from the numerator and denominator. The function $$C(x)=\frac{2400+150x}{x}$$ does not have a common factor that cancels with $x$. So there are **no holes**. 12. **Graph description** The graph is a rational-function curve with two branches. For positive $x$, the graph starts very high near $x=0$ and decreases toward $y=150$. The curve shows that the more participants join, the lower the cost per person becomes. Key features to label on the graph: - Vertical asymptote: $$x=0$$ - Horizontal asymptote: $$y=150$$ - No x-intercept - No y-intercept 13. **Sample table of values for graphing** Here are some points you can plot: $$\begin{aligned} C(1)&=\frac{2400}{1}+150=2550 \\ C(2)&=\frac{2400}{2}+150=1350 \\ C(3)&=\frac{2400}{3}+150=950 \\ C(4)&=\frac{2400}{4}+150=750 \\ C(5)&=\frac{2400}{5}+150=630 \\ C(10)&=\frac{2400}{10}+150=390 \\ C(20)&=\frac{2400}{20}+150=270 \\ C(30)&=\frac{2400}{30}+150=230 \end{aligned}$$ These values show the cost getting closer to $150$ as $x$ increases. 14. **Solving a related rational equation** If the teacher asks when the cost per person is $300$, solve: $$\frac{2400}{x}+150=300$$ Subtract $150$ from both sides: $$\frac{2400}{x}=150$$ Multiply both sides by $x$: $$\cancel{x}\cdot\frac{2400}{\cancel{x}}=150x$$ $$2400=150x$$ Divide both sides by $150$: $$\frac{2400}{150}=x$$ $$x=16$$ So if there are $16$ participants, the cost per person is $300$. 15. **Real-life interpretation** This model helps the group see that sharing the fixed cost among more people makes the cost per person smaller. It is useful for planning a field trip because the class can estimate how much each participant must pay. The function also shows why organizers want more participants: the cost becomes more affordable. 16. **Reflection** Our group learned that rational functions are useful in describing real-life situations where one quantity depends on another by division. We also learned that the domain, range, intercepts, and asymptotes help explain the behavior of the function. Mathematics models real life by turning a situation into a formula that we can analyze and graph. One challenge was understanding why the function has no y-intercept and why the cost approaches but never reaches $150$. Our group worked together by sharing ideas, checking calculations, and explaining the meaning of each result. 17. **Final answer you can submit** **Real-life rational function model:** $$C(x)=\frac{2400}{x}+150$$ **Domain:** $$x>0$$ **Range:** $$C(x)>150$$ **x-intercept:** none in context **y-intercept:** none **Zero(s):** none in context **Vertical asymptote:** $$x=0$$ **Horizontal asymptote:** $$y=150$$ **Holes:** none **Meaning:** The cost per person decreases as the number of participants increases, but it will always stay above $150$ because of the fixed fee. **Reflection:** Our group learned that rational functions are helpful in solving everyday problems like field trip budgeting. They show how costs change when more people share the expense. We also learned that graphs, asymptotes, and domain restrictions help us understand the situation clearly. Working together made the task easier because each member could help check the model, calculations, and explanation.