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the inputs. Students also write and interpret exponential functions with fractional exponents in context (they studied the meaning of such expressions in a previous unit). For example, if represents the area in square meters of a pond covered
by algae weeks after an algae-control treatment is applied, then
represents the area covered after 1 day ( of a week). They see that, aside from using
graphs, they can use properties of exponents to estimate or Snd the value of a function when the input is a fraction.
This work allows students to Snd growth factors over fractional intervals of input, for example, to Snd the annual growth factor of a population given its growth factor every decade. It also enables them to write expressions to highlight diVerent aspects of the
same situation. For instance, if is time in days,
starts at 7 and triples every 5 days. The growth factor per day is
approximately 1.25. Since an equivalent expression is , we can conclude that it is growing by roughly 25% each day.
Next, students encounter the constant . They learn that it is irrational, its value
is approximately 2.7, and it is used in many exponential functions that model real-life situations where a percent change is applied continuously. Students make sense of exponential functions of the form , interpret them in context, and graph them. (Students are not expected to build exponential functions with base in this course.)
In the second half of the unit, students learn about logarithms as a way to express the exponent of an expression. For example, if the expression has a value of 32, we can reason that the exponent is 5, but we can also write to express the value of . They see that the solution to the equation can be written as , and that these two equations are equivalent. Students then learn to solve exponential equations using logarithms, including natural logarithms, working mainly with base 2, 10, and .
The last couple of lessons expose students to logarithmic functions in base 2 and base 10. Students analyze the graphs, interpret them in context, and use them to answer questions about real-life situations such as population growth, acidity of substances, and intensity of earthquakes. Logarithmic functions are not studied in depth in this course.
F6 Transformations of Functions
Prior to this unit, students have worked with a variety of function types, such as polynomial, radical, and exponential. The purpose of this unit is for students to consider
shows a quantity that
, which is
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Course Guide Algebra