Content Partner
Grades 6-8, 9-12
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Students will interpret the United States business cycle in terms of a piecewise function. They will analyze specific points, slopes, and relative extrema of different sections of the cycle and formulate equations that represent the United States’ real GDP from historical data. Students will then relate the equations created from data given with the learned knowledge of economic reasoning related to real GDP.
80 minutes.
PRIOR KNOWLEDGE
For this lesson, students will need to be able to:
describe in words what these functions look like and give a scenario in the economy that matches what they represent.
(f(x) is a downward facing quadratic function whereas g(x) is an upward facing quadratic function; this can be seen by the negative and positive coefficient on the first term of each function respectively. A recession of any sort due to decreased production could be a scenario where f(x) is seen; while, g(x) could be seen as new businesses emerge.)
Why are business cycles usually quadratic and not linear? (A typical pattern for a business cycle is that an economy begins to expand and then spending and production build upon themselves to accelerate the expansion. Some change occurs to slow the growth. A leveling off occurs and the economy may begin to shrink. That shrinkage may accelerate and then begin to slow as the economy approaches a trough.)
ANSWER
Three pieces; the first piece would be an upward facing quadratic function, the second piece would be a wider downward facing quadratic function, and the third piece would be a positively sloped linear function.
In the past 100 years, moments in history like the Great Depression or more recently the housing boom and crash has heightened our awareness of the peaks and troughs in our economy. Economically speaking, these highs (peaks) and lows (troughs) are telling of the Real GDP which many economists use to measure the successfulness of a countries economy and well-being. Students can use their knowledge of mathematics in using the vertex of a quadratic function, applying properties of linear equations, and analyzing non-linear functions to quantitatively describe and compare different cycles in the economy. Students can then compare present situations with those in the past.
Scenario 1 |
Eating out at your favorite local restaurant |
Real GDP |
Scenario 2 |
Service station attendant changes your flat tire |
Real GDP |
Scenario 3 |
Dad helping his child change a flat tire |
NOT |
Scenario 4 |
An Italian vacationing in New York City |
Real GDP |
Scenario 5 |
An American vacationing in Italy |
NOT |
Scenario 6 |
Buying a bracelet while vacationing in Japan |
NOT |
Scenario 7 |
Buying a chair from a factory in the US |
Real GDP |
Scenario 8 |
Bread sold to a family in Kansas |
Real GDP |
Scenario 9 |
The bread sold to McDonald’s to use for hamburgers in Kansas |
NOT |
Scenario 10 |
Buying your prom dress |
Real GDP |
Scenario 11 |
Catering for your prom |
Real GDP |
Scenario 12 |
Your first brand new car |
Real GDP |
Scenario 13 |
Tires installed on a car at the factory |
NOT |
Scenario 14 |
Going to the movie theater |
Real GDP |
Scenario 15 |
Getting snacks at the movie theater |
Real GDP |
y = a(x – h)^{2} + k
30 = a(2 – 3)^{2} + 40
30 = a(-1)^{2} + 40
30 = 1a + 40
-10 = 1a
-10 = a
With the a just found and the given vertex, write the quadratic function for all points on the curve.
y = -10(x – 3)^{2} + 40
The equation could also be simplified to be represented in standard form.
y = -10(x^{2} – 3)(x – 3) + 40
y = -10(x^{2} – 3x – 3x + 9) + 40
y = -10(x^{2} – 6x + 9) + 40
y = -10x^{2} + 60x – 90 + 40
y = -10x^{2} + 60x – 50
With the a just found and the given vertex, write the quadratic function for all points on the curve.
The equation could also be simplified to be represented in standard form.
Given the points (11, 30) and (14, 70) on the line, find slope:
Use one of the points and the found slope to solve for the y-intercept (b).
Write the linear function for all points on the line.
Content Partner
Grades 6-8, 9-12
Grades 9-12
Grades 3-5, 6-8, 9-12
Grades 6-8, 9-12