Topic 10 : Exponential and Logarithmic Functions Exponential Models: Geometric sequences and series

An infinite sequence is a function whose domain is the set of positive integers. a 1, a 2, a 3, a 4,..., a n,... Definition of Sequence The first three terms of the sequence a n = 2n 2 are a 1 = 2(1) 2 = 2 a 2 = 2(2) 2 = 8 a 3 = 2(3) 2 = 18. finite sequence terms Geometric Sequences and Series

A sequence is geometric if the ratios of consecutive terms are the same. 2, 8, 32, 128, 512,... Definition of Geometric Sequence geometric sequence The common ratio, r, is 4.

The nth term of a geometric sequence has the form a n = a 1 r n - 1 where r is the common ratio of consecutive terms of the sequence. The nth Term of a Geometric Sequence 15, 75, 375, 1875,... a 1 = 15 The nth term is 15(5 n-1 ). a 2 = 15(5) a 3 = 15(5) 2 a 4 = 15(5) 3

Example: Find the 9th term of the geometric sequence 7, 21, 63,... Example: Finding the nth Term a 1 = 7 The 9th term is 45,927. a n = a 1 r n – 1 = 7(3) n – 1 a 9 = 7(3) 9 – 1 = 7(3) 8 = 7(6561) = 45,927

Evaluate (1.08) (0.95) (1 – 0.02) ( ) –10 ≈ ≈ ≈ ≈ 46.32

You can model growth or decay by a constant percent increase or decrease with the following formula: In the formula, the base of the exponential expression, 1 + r, is called the growth factor. Similarly, 1 – r is the decay factor.

In 1981, the Australian humpback whale population was 350 and increased at a rate of 14% each year since then. Write a function to model population growth. Use a graph to predict when the population will reach 20,000. P(t) = a(1 + r) t Substitute 350 for a and 0.14 for r. P(t) = 350( ) t P(t) = 350(1.14) t Simplify. Exponential growth function. Check It Out! Example 2

A motor scooter purchased for $1000 depreciates at an annual rate of 15%. Write an exponential function and graph the function. Use the graph to predict when the value will fall below $100. f(t) = a(1 – r) t Substitute 1,000 for a and 0.15 for r. f(t) = 1000(1 – 0.15) t f(t) = 1000(0.85) t Simplify. Exponential decay function. Check It Out! Example 3

Identifying from an equation: Linear Has an x with no exponent. y = 5x + 1 y = ½x 2x + 3y = 6 Quadratic Has an x 2 in the equation. y = 2x 2 + 3x – 5 y = x x 2 + 4y = 7 Exponential Has an x as the exponent. y = 3 x + 1 y = 5 2x 4 x + y = 13

Identifying from a graph: Linear Makes a straight line Quadratic Makes a parabola Exponential Rises or falls quickly in one direction

LINEAR, QUADRATIC or EXPONENTIAL? a)b) c)d)

Is the table linear, quadratic or exponential? Quadratic See same y more than once. 2 nd difference is the same Linear Never see the same y value twice. 1 st difference is the same Exponential y changes more quickly than x. Never see the same y value twice. Common ratio

Is the table linear, quadratic or exponential? xy xy xy

Write an equation for a function EXAMPLE 3 Tell whether the table of values represents a linear function, an exponential function, or a quadratic function. Then write an equation for the function. x –2 – y

SOLUTION Write an equation for a function EXAMPLE 3 STEP 1 Determine which type of function the table of values represents. x –2 – y First differences: –1.5 – Second differences: 1 1 1

Concept

Example 1 A. Graph the ordered pairs. Determine whether the ordered pairs represent a linear, quadratic, or exponential function. (1, 2), (2, 5), (3, 6), (4, 5), (5, 2) Answer: The ordered pairs appear to represent a quadratic equation.

Example 1 Answer: The ordered pairs appear to represent an exponential function. B. Graph the ordered pairs. Determine whether the ordered pairs represent a linear, quadratic, or exponential function. (–1, 6), (0, 2),

Example 1 A.linear B.quadratic C.exponential A. Graph the set of ordered pairs. Determine whether the ordered pairs represent a linear, quadratic, or exponential function. (–2, –6), (0, –3), (2, 0), (4, 3)

Example 1 A.linear B.quadratic C.exponential B. Graph the set of ordered pairs. Determine whether the ordered pairs represent a linear, quadratic, or exponential function. (–2, 0), (–1, –3), (0, –4), (1, –3), (2, 0)

Example 2 A. Look for a pattern in the table of values to determine which kind of model best describes the data. – First differences: Answer: Since the first differences are all equal, the table of values represents a linear function. 2

Example 2 B. Look for a pattern in the table of values to determine which kind of model best describes the data. –24–8 First differences: The first differences are not all equal. So, the table of values does not represent a linear function. Find the second differences and compare __ –2 __ –

Example 2 16 First differences: The second differences are not all equal. So, the table of values does not represent a quadratic function. Find the ratios of the y-values and compare. 1 __ Second differences: 364 __ __ Ratios: __ –24–8–2 __ –

Example 2 The ratios of successive y-values are equal. Answer: The table of values can be modeled by an exponential function.

Example 2 A.linear B.quadratic C.exponential D.none of the above A. Look for a pattern in the table of values to determine which kind of model best describes the data.

Example 2 A.linear B.quadratic C.exponential D.none of the above B. Look for a pattern in the table of values to determine which kind of model best describes the data.

Example 3 Write an Equation Determine which kind of model best describes the data. Then write an equation for the function that models the data. Step 1Determine which model fits the data. –1 –8 –64 –512 –4096 –7–56–448–3584 First differences:

Example 3 Write an Equation –7–56–448–3584 First differences: –49–392–3136 Second differences: × 8 The table of values can be modeled by an exponential function. –1–8–64Ratios:–512–4096 × 8

Example 3 Write an Equation Step 2Write an equation for the function that models the data. The equation has the form y = ab x. Find the value of a by choosing one of the ordered pairs from the table of values. Let’s use (1, –8). y= ab x Equation for exponential function –8= a(8) 1 x = 1, y = –8, b = 8 –8= a(8)Simplify. –1= aAn equation that models the data is y = –(8) x. Answer: y = –(8) x

Example 3 A.quadratic; y = 3x 2 B.linear; y = 6x C.exponential; y = 3 x D.linear; y = 3x Determine which model best describes the data. Then write an equation for the function that models the data.

Example 4 Write an Equation for a Real-World Situation KARATE The table shows the number of children enrolled in a beginner’s karate class for four consecutive years. Determine which model best represents the data. Then write a function that models that data.

Example 4 Write an Equation for a Real- World Situation UnderstandWe need to find a model for the data, and then write a function. PlanFind a pattern using successive differences or ratios. Then use the general form of the equation to write a function. SolveThe first differences are all 3. A linear function of the form y = mx + b models the data.

Example 4 Write an Equation for a Real- World Situation y= mx + bEquation for linear function 8= 3(0) + bx = 0, y = 8, and m = 3 b= 8Simplify. Answer:The equation that models the data is y = 3x + 8. CheckYou used (0, 8) to write the function. Verify that every other ordered pair satisfies the function.

Example 4 A.linear; y = 4x + 4 B.quadratic; y = 8x 2 C.exponential; y = 2 ● 4 x D.exponential; y = 4 ● 2 x WILDLIFE The table shows the growth of prairie dogs in a colony over the years. Determine which model best represents the data. Then write a function that models the data.

Identify functions using differences or ratios EXAMPLE 2 ANSWER The table of values represents a linear function. x – 2– y – Differences: b.

Identify functions using differences or ratios EXAMPLE 2 Use differences or ratios to tell whether the table of values represents a linear function, an exponential function, or a quadratic function. ANSWER The table of values represents a quadratic function. x–2–1012 y–6 –406 First differences: Second differences: a.

GUIDED PRACTICE for Examples 1 and 2 2. Tell whether the table of values represents a linear function, an exponential function, or a quadratic function. ANSWER exponential function 0 y 2 x – 2–

Time 2 Exponential Practice