From a Pair of Points to a Graph Any point on a two-dimensional graph can be represented by two numbers, which are usually written in the in the form x, ywhere x defines the horizontal distance from the origin and y represents the vertical distance.
Provide additional examples of the graphs of exponential functions and model writing the equation using well-chosen points on a graph. How would you describe this graph? Examples of Student Work at this Level The student attempts to write a linear function or an exponential expression.
Why Exponential Functions Are Important Many important systems follow exponential patterns of growth and decay.
Encourage the student to relate features of the graph to the parameters in the equation. An Example from the Real World Sincehuman population growth has been exponential, and by plotting a growth curve, scientists are in a better position to predict and plan for the future.
Inthe world population was 1. Taking as the starting point, this gives the pair of points 0, 1. The student does not understand the basic form of an exponential function. Questions Eliciting Thinking What do the values in the equation tell you about the exponential equation and its graph? This yields the following pair of equations: Got It The student provides complete and correct responses to all components of the task.
What happens if you substitute one for x in your function? Is unable to solve the equation to find the growth factor. Provide additional opportunities to write exponential functions given their graphs. Uses function notation incorrectly or not at all.
Provide opportunities for the student to explore and investigate exponential functions given in context. Did you write the function? Can you determine the rate of growth from either your equation or the graph?
What does function notation look like?
Questions Eliciting Thinking What does the value you calculated,tell you about this graph? Henochmath walks us through an easy example to clarify this procedure. Which one did you find? Include examples of both growth and decay.
Neither Point on the X-axis If neither x-value is zero, solving the pair of equations is slightly more cumbersome. If needed, review function notation and guide the student to use function notation when writing functions. In general, you have to solve this pair of equations: You can substitute this value for b in either equation to get a.
Discuss with the student differences in the rates of increase of the two functions and relate these differences to the forms of the equations. What is the difference between f and f x?
By taking data and plotting a curve, scientists are in a better position to make predictions. If neither point has a zero x-value, the process for solving for x and y is a tad more complicated.Jul 08, · How to Write an Exponential Function Given a Rate and an Initial Value Two Methods: Using the Rate as the Base Using "e" as the base Community Q&A Exponential functions can model the rate of change of many situations, including population growth, radioactive decay, bacterial growth, compound interest, and much %(1).
Finding an Exponential Equation with Two Points and an Asymptote Find an exponential function whose asymptote is y=0 and passes through the points.
Provide additional opportunities to write exponential functions given their graphs. Consider implementing other MFAS exponential tasks Writing an Exponential Function From a Table (F-LE), Challenge the student to write an exponential function that contains the points (1, 10) and (2, 20).
Graphs up to two functions with tracing to explore points of intersection. Polar Coordinates. Graphs an exponential function using coefficients generated from two data points.
Conic Sections. Graphs conic sections from the standard form equation, exploring translations and rotations. Plugging this value, along with those of the second point, into the general exponential equation produces = bwhich gives the value of b as the hundredth root of / or So the equation becomes y = (hundredth root of ) x.
Concept Write Exponential Equations Assessment Writing a Exponential Growth Function given a table of Values (ask Mr. Sieling for login info) An explanation of how to write an exponential equation from a table 3. QUIZ (Level 2) What point do all three lines have in common?
d. Which line decreases the fastest?Download