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The number of cards in Brian's collection can be represented by a linear function. Identify the rate of change and initial value of this function. Explain what these values represent in terms of the situation.
To identify the rate of change and initial value of the linear function representing the number of cards in Brian's collection, we need more information. Specifically, we need data points that relate the number of cards to another variable, such as time or money. With this information, we can calculate the slope (rate of change) and y-intercept (initial value) of the linear function.
For example, if we have data points such as:
- At time t=0, Brian had 50 cards
- At time t=1 year, Brian had 100 cards
- At time t=2 years, Brian had 150 cards
We can calculate the rate of change by finding the difference in the number of cards over the difference in time. In this case, the rate of change is (150-50)/(2-0) = 100/2 = 50 cards per year.
The initial value can be determined by observing the number of cards when t=0, which in this case is 50 cards.
In this scenario, the rate of change of 50 cards per year represents how Brian's card collection is increasing. The initial value of 50 cards represents the starting point or the number of cards Brian had initially.
To identify the rate of change and initial value of the linear function representing the number of cards in Brian's collection, we need more information about the situation. Specifically, we need two points on the graph of the linear function.
For example, let's say we have the following information:
- At the beginning, Brian had 5 cards in his collection.
- After one month, Brian had 15 cards in his collection.
We can use these two points to find the rate of change and initial value.
To find the rate of change (also known as the slope), we can use the formula:
Rate of change = (Change in y) / (Change in x)
In this case, the change in y is 15 - 5 = 10 (the increase in the number of cards) and the change in x is 1 month (the time interval). Therefore, the rate of change is:
Rate of change = 10 / 1 = 10 cards per month.
The rate of change represents the average increase in the number of cards per month in Brian's collection.
To find the initial value (also known as the y-intercept), we can use one of the points given. In this case, the initial value is 5 (the number of cards at the beginning).
The initial value represents the number of cards in Brian's collection at the starting point (in this case, when he began collecting cards).
Therefore, in this situation, the rate of change is 10 cards per month, indicating the average increase in the number of cards each month, and the initial value is 5 cards, representing the number of cards in Brian's collection at the beginning.
To identify the rate of change and initial value of a linear function, we need two pieces of information: any two points on the line.
Let's assume we have two points to work with. Let's say at time t = 0, Brian had 10 cards, and at t = 2, he had 30 cards. We can now calculate the rate of change and initial value using this information.
The rate of change, also known as the slope, can be calculated by dividing the change in the output (cards) by the change in the input (time):
Rate of change (slope) = (change in cards)/(change in time)
Here, the change in cards = 30 - 10 = 20, and the change in time = 2 - 0 = 2.
So, the rate of change (slope) of the linear function would be:
Rate of change = (20)/(2) = 10
Now, let's find the initial value, which is the value of the function when the input (time) is 0. In this case, the initial value represents the starting number of cards Brian had in his collection.
By using the point (0,10), we see that when t = 0 (time = 0), Brian's collection had 10 cards. Therefore, the initial value of the linear function is 10.
So, in terms of the situation, the rate of change is 10, indicating that Brian's collection grows by 10 cards over a unit increase in time. The initial value of 10 represents the starting point of the collection, the number of cards he had when he began collecting.