Finland's annual electricity consumption in gigawatt hours (GWh) in 1970−2014 is described by the function 𝑠⁡(𝑥)=−1,3⁢𝑥3+58,8⁢𝑥2+1349,4⁢𝑥+21 583, where x is the time since 1970 in years.

b) Use a geometry program to draw the graph of the function s in the interval 0 ≤  x  ≤ 44.
c) Using the graph, determine in which year the electricity consumption decreased.

Let's break down the steps of the problem.

### Part (b) - Drawing the Graph of the Function

The given function representing Finland's annual electricity consumption is:

\[
s(x) = -1.3x^3 + 58.8x^2 + 1349.4x + 21,583
\]

where \( x \) is the number of years since 1970, and \( s(x) \) gives the electricity consumption in GWh.

We are asked to draw the graph of this function in the interval \( 0 \leq x \leq 44 \). This interval corresponds to the years from 1970 to 2014 because \( x = 44 \) represents the year 2014 (since 1970 + 44 = 2014).

If you're using a geometry or graphing program (such as GeoGebra, Desmos, or any graphing calculator), here’s how to proceed:

1. **Input the function** \( s(x) = -1.3x^3 + 58.8x^2 + 1349.4x + 21,583 \).
2. Set the graph window to cover the interval from \( x = 0 \) to \( x = 44 \).
3. Ensure that the \( y \)-axis (representing electricity consumption) has an appropriate range to show the values of \( s(x) \). You may need to adjust the scale based on the output of the function.

### Part (c) - Determining the Year When Consumption Decreased

Once the graph is plotted, visually inspect it to find where the consumption (the value of \( s(x) \)) starts to decrease. A decrease in consumption corresponds to a downward slope of the graph.

Look for the highest point (local maximum) on the graph, as this indicates the peak of electricity consumption. After this point, the graph will begin to slope downward, indicating that consumption has decreased.

To determine the exact year:
1. Identify the \( x \)-coordinate where the decrease begins (i.e., where the slope changes from positive to negative).
2. Add that value of \( x \) to 1970 to find the corresponding year.

For example:
- If the consumption starts decreasing at \( x = 34 \), this would correspond to the year \( 1970 + 34 = 2004 \).

Would you like me to help you plot the graph numerically or further explain how to find the maximum point where the consumption decreases?

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Here are five questions that expand on this problem:

1. What does the derivative of the function \( s(x) \) tell you about the changes in electricity consumption?
2. How would you determine the exact year of peak consumption algebraically?
3. Could the function \( s(x) \) be used to predict future electricity consumption, and how reliable would that be?
4. What factors might cause the shape of the graph to shift in real-world electricity consumption data?
5. How can we use the integral of \( s(x) \) to find total electricity consumption over a given period?

**Tip:** The derivative of a function gives you the rate of change, which is useful to find when a function reaches its maximum or minimum.

Finland's annual electricity consumption in gigawatt hours (GWh) in 1970−2014 is described by the function s(x) = -1.3x^3 + 58.8x^2 + 1349.4x + 21,583, where x is the time since 1970 in years.
b) Use a geometry program to draw the graph of the function s in the interval 0 ≤ x ≤ 44.
c) Using the graph, determine in which year the electricity consumption decreased.

Learn how to graph Finland's electricity consumption from 1970 to 2014 using the function s(x) = -1.3x^3 + 58.8x^2 + 1349.4x + 21,583. Explore the graph and find the year when electricity consumption started to decrease by analyzing the function's critical points. Suitable for high school students in Grades 11-12.

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