Exponential Slope Investigation

Last updated almost 8 years ago

21 questions

Note from the author:

This is a guided discovery investigation that leads students to the number 'e' via slope patterns in functions. Embedded Geogebra labs create a place for students to play! Let the contest begin!

Exponential Slope Investigation

Exponential Slope Investigation

The slope of the linear function above depends on the value of x at which you measure the slope.

What is the slope of the curve above when x = 3.7 ?

The slope of the quadratic function above depends on the value of x at which you measure the slope.

Can you identify a pattern between the slope of the curve and x-value at which the slope is measured? Describe the pattern you observe.

What will the slope of the above quadratic equation be when x = 50?

At what x-value does the curve have a slope of 5?

The slope of the cubic function above depends on the value of x at which you measure the slope.

Can you identify a pattern between the slope of the curve and x-value at which the slope is measured? Describe the pattern you observe.

What will the slope of the above quadratic equation be when x = 20?

At what x-value does the graph have a slope of 5? You may want to use a calculator to help make your prediction! Give your answer to THREE decimal places.

Make an observation about the slope of the curve above. Based on the slope pattern, can you predict what the equation of the function is? Explain how you might know.

At what x-value is the slope of the curve equal to 10? You may need to use a calculator! Round your answer to three decimal places.

Which statement best describes the slope of the exponential function?

Clicking the "Interesting Fact" button shows the ratio of the slope and the current y-value. Explain what this fact seems to imply.

Use the interesting fact and a calculator to estimate the slope of this curve when the y-value is 20. Give your answer to one decimal place.

Move the purple slider to x = 0. What is the slope of the curve at this point?

Move the black slider above and show the "interesting fact". What value on the black slider creates the exact same exponential function from the previous set of questions.

Change the base of the exponential function to 5 and then slide the purple slider. What is the constant ratio of the slope and the y-value for this curve?

Play with various values for the base. Based on what you observe, which of the statement(s) are accurate?

Play with the black slider until you make the ratio equal to 1. What value of the base achieves this?

The slope of the linear function above depends on the value of x at which you measure the slope.

What is the slope of the curve above when x = 3.7 ?

The slope of the quadratic function above depends on the value of x at which you measure the slope.

Can you identify a pattern between the slope of the curve and x-value at which the slope is measured? Describe the pattern you observe.

What will the slope of the above quadratic equation be when x = 50?

At what x-value does the curve have a slope of 5?

The slope of the cubic function above depends on the value of x at which you measure the slope.

Can you identify a pattern between the slope of the curve and x-value at which the slope is measured? Describe the pattern you observe.

What will the slope of the above quadratic equation be when x = 20?

At what x-value does the graph have a slope of 5? You may want to use a calculator to help make your prediction! Give your answer to THREE decimal places.

Make an observation about the slope of the curve above. Based on the slope pattern, can you predict what the equation of the function is? Explain how you might know.

At what x-value is the slope of the curve equal to 10? You may need to use a calculator! Round your answer to three decimal places.

Which statement best describes the slope of the exponential function?

Clicking the "Interesting Fact" button shows the ratio of the slope and the current y-value. Explain what this fact seems to imply.

Use the interesting fact and a calculator to estimate the slope of this curve when the y-value is 20. Give your answer to one decimal place.

Move the purple slider to x = 0. What is the slope of the curve at this point?

Move the black slider above and show the "interesting fact". What value on the black slider creates the exact same exponential function from the previous set of questions.

Change the base of the exponential function to 5 and then slide the purple slider. What is the constant ratio of the slope and the y-value for this curve?

Play with various values for the base. Based on what you observe, which of the statement(s) are accurate?

Play with the black slider until you make the ratio equal to 1. What value of the base achieves this?