Grade 9 · Mathematics
Quarter 3 · Week 6 · TEKS
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This lesson introduces Grade 9 students to writing exponential functions in the form f(x) = ab^x, where b is a rational number, to model real-world and mathematical situations involving growth and decay. Students will learn to identify key features of these functions, including the y-intercept and asymptote, and graph them to visualize their behavior. The lesson aligns with TEKS standards for Algebra I and emphasizes applying mathematical process standards to interpret and represent exponential relationships.
Students should understand basic function notation, the concept of exponents with integer and rational exponents, and have experience graphing simple functions such as linear and quadratic functions.
An exponential function is a function where the variable appears in the exponent. The general form is f(x) = ab^x, where 'a' is the initial value or coefficient, 'b' is the base which is a positive rational number, and 'x' is the exponent. The base 'b' determines whether the function models growth or decay: if b > 1, the function models exponential growth; if 0 < b < 1, it models exponential decay.
The coefficient 'a' represents the initial amount or starting value of the function when x = 0, because f(0) = ab^0 = a*1 = a. The base 'b' controls the rate of growth or decay. For example, if b = 2, the function doubles each time x increases by 1 (growth). If b = 0.5, the function halves each time x increases by 1 (decay). Understanding these parameters helps in writing functions from real-world situations.
When graphing f(x) = ab^x, key features to identify include: the y-intercept at (0, a), since f(0) = a; and the horizontal asymptote, which is the line y = 0, because as x approaches negative or positive infinity, the function approaches zero but never touches it. For growth functions (b > 1), the graph rises from left to right; for decay functions (0 < b < 1), the graph falls from left to right.
Exponential functions model many real-world phenomena: population growth, compound interest, radioactive decay, and cooling processes. For example, a population that doubles every year can be modeled by f(x) = a*2^x, where 'a' is the initial population. Radioactive decay can be modeled by f(x) = a*(1/2)^x, where the function halves over each time period. Writing the function requires understanding the initial value and the growth/decay rate.
A common misconception is confusing the coefficient 'a' with the base 'b'. Remember, 'a' is the starting value (y-intercept), and 'b' is the growth or decay factor. Another misconception is thinking the asymptote is crossed; exponential functions approach but never cross the horizontal asymptote y=0. Also, students may confuse exponential growth with linear growth; exponential growth increases faster as x increases.
Given a bacteria population that starts with 500 bacteria and doubles every hour, write the exponential function f(x) = ab^x that models this growth.
The initial amount a is 500 because that is the starting population at time x=0. The population doubles every hour, so the base b is 2. Therefore, the function is f(x) = 500 * 2^x.
f(x) = 500 * 2^x
A radioactive substance has a mass of 100 grams and loses half its mass every hour. Write the exponential decay function that models this situation.
The initial mass a is 100 grams. Since the substance loses half its mass every hour, the base b is 1/2 (or 0.5). The function is f(x) = 100 * (1/2)^x.
f(x) = 100 * (1/2)^x
Graph the function f(x) = 3 * (1.5)^x. Identify the y-intercept and the asymptote.
The y-intercept is at (0, 3) because f(0) = 3 * (1.5)^0 = 3. The horizontal asymptote is y = 0, which the graph approaches but never touches. Since 1.5 > 1, the graph shows exponential growth, rising from left to right.
Y-intercept at (0, 3); asymptote y = 0; graph rises from left to right.
Present a real-world scenario such as a population of bacteria doubling every hour. Ask students to predict how many bacteria there will be after several hours and discuss how the number changes over time.
Students discuss their predictions and reasoning about how the population changes over time.
This scenario introduces the concept of exponential growth, setting the stage for writing functions that model such situations.
Provide students with data from a real-world growth or decay situation (e.g., a radioactive substance losing half its mass every hour). Guide them to identify the initial value and the rate of change.
Students analyze the data, calculate ratios between successive values, and attempt to write an exponential function that fits the data.
By exploring data, students see how exponential functions describe growth and decay with consistent multiplicative change.
Explain the form f(x) = ab^x, defining each parameter and how it relates to the data or problem context. Demonstrate graphing examples of growth and decay functions, highlighting the y-intercept and asymptote.
Students take notes, ask questions, and practice writing functions from given scenarios.
Clear explanation of function form and graph features helps students connect symbolic and graphical representations.
Present new problems involving exponential growth and decay. Have students write functions, graph them, and interpret the meaning of key features in context.
Students work individually or in pairs to solve problems, graph functions, and explain their reasoning.
Applying knowledge to varied problems deepens understanding and builds fluency.
Assign an exit ticket where students write and graph an exponential function from a given real-world problem and identify key features.
Students complete the exit ticket independently, demonstrating their understanding.
This assessment checks students' ability to write, graph, and interpret exponential functions as required by the standards.
Students analyze the data, calculate ratios between successive values, and attempt to write an exponential function that fits the data.
What is the value of f(0) in the exponential function f(x) = 7 * (0.8)^x?
f(0) = 7.
Write an exponential function for a population that starts at 200 and triples every year.
f(x) = 200 * 3^x.
A car's value decreases by 20% each year. If the car is worth $15,000 now, write an exponential function to model its value after x years.
f(x) = 15000 * (0.8)^x.
Given the function f(x) = 5 * (0.6)^x, describe the key features of its graph and explain what they mean in a real-world decay context.
Y-intercept at (0, 5), base 0.6 indicates decay, graph decreases as x increases, asymptote y=0.
A bacteria culture grows according to the function f(x) = a * b^x. After 2 hours, the population is 800, and after 5 hours, it is 6400. Find the values of a and b.
b = 2, a = 200.
Provide step-by-step guided examples with visual aids. Use graphing calculators or software to help visualize functions. Offer sentence frames to help explain function parameters.
Use visuals and real-world examples to explain vocabulary. Provide bilingual glossaries for key terms like 'growth', 'decay', 'exponent'. Encourage peer discussion to practice explaining concepts orally.
Challenge students to write functions with fractional exponents. Explore transformations of exponential functions, such as vertical shifts or reflections. Investigate compound interest problems involving continuous growth.
Ask students to find examples of exponential growth or decay in their home or community, such as savings account interest, population changes, or cooling of hot drinks, and write a function to model one example.
Students learned to write exponential functions in the form f(x) = ab^x to model growth and decay in real-world and mathematical contexts. They identified key features of these functions, including the y-intercept and horizontal asymptote, and practiced graphing them. Understanding the roles of 'a' and 'b' helps interpret how quantities change over time, connecting algebraic expressions to real-world phenomena.
Use the exact standards alignment and retrieved-source provenance stored with this enrichment.