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Complete this assessment to review what you’ve learned. It will not count toward your grade.

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Use the image to answer the question.

An L-shaped curve is plotted on a coordinate plane where the x-axis ranges from negative 10 to 20 and the y-axis ranges from 0 to 30, both by 2-unit increments. The curve passes through 3 marked and labeled points.

A boat washes up onto an island, introducing 25 rabbits into the environment. The rabbits multiply at a rate of 50 percent every month. Does this graph accurately model the situation? Enter 1 for yes or 2 for no.

(1 point)
Remaining Attempts : 3
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5 answers

To determine whether the graph accurately models the situation of rabbits multiplying at a rate of 50 percent every month, we need to consider the nature of exponential growth.

A growth rate of 50 percent per month can be modeled with an exponential function, specifically:

\[ P(t) = P_0 \times (1 + r)^t \]

Where:

  • \( P(t) \) is the population at time \( t \),
  • \( P_0 \) is the initial population (25 rabbits in this case),
  • \( r \) is the growth rate (0.50 for 50%),
  • \( t \) is the number of months.

If the L-shaped curve in the graph represents exponential growth (which typically shows a smooth, upward-curving line), then it could accurately model the rabbit population growth. If it appears to have flat sections or sudden jumps, it might not accurately reflect continuous exponential growth.

Since I cannot see the graph, I suggest assessing the curvature of the L-shaped curve. If it resembles an exponential growth graph, then the answer would be 1 (yes); if not, then it would be 2 (no).

Given the details, if you believe the curve does not reflect exponential growth, then you should choose 2 (no). Otherwise, choose 1 (yes).

Use the image to answer the question.

A concave-up, increasing curve is plotted on a coordinate plane where the x-axis ranges from negative 2 to 4 in increments of 2 and the y-axis ranges from 0 to 20 in increments of 5. The curve passes through 3 marked points.

Use the graph to fill in the missing elements of the exponential growth equation.

(1 point)
y=
(
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Remaining Attempts : 3

To fill in the missing elements of the exponential growth equation, which generally follows the form:

\[ y = a \cdot b^x \]

Where:

  • \( y \) is the population at time \( x \),
  • \( a \) is the initial population size (the value of \( y \) when \( x = 0 \)),
  • \( b \) is the growth factor (how the population increases, typically a value greater than 1 for growth).

To fill in the equation, please identify the following from the graph:

  1. Value of \( a \): Look for the value of \( y \) when \( x = 0 \) (if marked on the graph). This will be your \( a \).

  2. Value of \( b \): Observe how much \( y \) increases for each unit increase in \( x \). This requires determining the ratio of \( y \) values for other points on the graph (e.g., if \( y \) doubles or increases by 50% as \( x \) increases by 1).

Once you identify \( a \) and \( b \), you would fill in the blanks in the equation as follows:

If, for example, \( a = 5 \) and \( b = 2 \), the equation would be:

\[ y = 5 \cdot 2^x \]

If you provide specific values seen on the graph or more context, I can help you fill in the equation more accurately.

Use the image to answer the question.

A coordinate plane's x-axis ranges from 0 to 5 by 0.25 unit increments but labeled at every 1-unit intervals. The y-axis ranges from 0 to 10 by increments of 0.5, but labeled at intervals of 2 units. A curve is plotted passing through 4 marked and labeled points.

What is the common ratio of this sequence?

(1 point)
Remaining Attempts : 3

To find the common ratio of a sequence, you need to identify the values of the dependent variable (typically \( y \)) at corresponding points of the independent variable (typically \( x \)) both at the same increments or intervals.

The common ratio in a geometric sequence is found by taking a pair of consecutive \( y \) values and dividing the second one by the first one. Mathematically, it's represented as:

\[ \text{Common Ratio} = \frac{y_n}{y_{n-1}} \]

Where:

  • \( y_n \) is the current term,
  • \( y_{n-1} \) is the previous term.

Here's how to determine the common ratio from the graph:

  1. Identify the \( y \) values at consecutive \( x \) positions.
  2. Record the values of \( y \) at two consecutive intervals.
  3. Divide the second \( y \) value by the first \( y \) value you recorded.
  4. Repeat this for pairs of \( y \) values until you establish consistency (the same ratio for all pairs indicates a common ratio).

For example, if the \( y \) values are:

  • \( y_1 = 2 \) at \( x_1 \)
  • \( y_2 = 4 \) at \( x_2 \)

The common ratio \( r \) would be:

\[ r = \frac{4}{2} = 2 \]

If you provide the specific \( y \) values from the points on the curve, I can help calculate the common ratio for you!