The table shows the cost y of an event when x people attend. What type of function can you use to model the data? How many people attend the event when it costs $600?
Understand the Problem
The question describes a relationship between the number of people attending an event (x) and the cost of the event (y), as shown in a table. The question asks us to determine what type of function can be used to model the data, and how many people attend the event if the cost is $600. We can model the function with a linear function, and interpolate the data in the table to get to the answer.
Answer
Linear Function; $x=50$
Answer for screen readers
The function is linear. When the cost is $600, 50$ people attend the event.
Steps to Solve
- Determine the type of function
First, we need to check if the function is linear. We can do this by checking if the rate of change between the points is constant. The change in $x$ is consistently 15. Let's find the change in $y$: $460 - 355 = 105$ $565 - 460 = 105$ $670 - 565 = 105$ $775 - 670 = 105$ Since the rate of change is constant, the function is linear.
- Determine the slope
The slope $m$ is the change in $y$ divided by the change in $x$. In this case, the change in $x$ is 15, and the change in $y$ is 105. Therefore, the slope $m$ is:
$m = \frac{105}{15} = 7$
- Determine the y-intercept
The y-intercept is the value of $y$ when $x = 0$. We can use the point-slope form of a linear equation $y = mx + b$ and one of the points from the table to find the y-intercept $b$. Let's use the point (15, 355):
$355 = 7(15) + b$ $355 = 105 + b$ $b = 355 - 105 = 250$
- Write the equation of the line
Now we have the slope $m = 7$ and the y-intercept $b = 250$, so we can write the equation of the line: $y = 7x + 250$
- Solve for x when y = 600
We want to find the value of $x$ when $y = 600$. Plug $y = 600$ into the equation and solve for $x$:
$600 = 7x + 250$ $600 - 250 = 7x$ $350 = 7x$ $x = \frac{350}{7} = 50$
The function is linear. When the cost is $600, 50$ people attend the event.
More Information
The slope of the linear function represents the cost per person attending the event, which is $7. The y-intercept represents a fixed cost of $250, perhaps for renting the venue or other fixed expenses.
Tips
A common mistake is to assume the relationship is non-linear without checking the rate of change. Another common mistake is in the arithmetic when calculating the slope or y-intercept. Finally, mistakes can occur when solving for $x$ after plugging in the value for $y$.
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