United Arab Emirates Ministry of Education Alain ...

United Arab Emirates Ministry of Education Alain Educational Office

Math Department Grade 10 Term 2

Revision Sheet For

Final Exam

Lesson (4-1) a

Exponential Functions, Growth, and Decay

The base of an exponential function indicates whether the function shows growth or decay. Exponential function: f (x) abx ? a is a constant. ? b is the base. The base is a constant. If 0 b 1, the function shows decay. If b 1, the function shows growth. ? x is an exponent.

f(x) 1.2x a 1 b 1.2 b 1, so the function shows exponential growth.

g(x) 10(0.6)x a 10 b 0.6 0 b 1, so the function shows exponential decay.

Tell whether each function shows growth or decay. Then graph.

1. h(x) 0.8(1.6)x

2. p(x) 12(0.7)x

a

b

a ___________ b ___________

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Lesson (4-1) b

Exponential Functions, Growth, and Decay (continued)

When an initial amount, a, increases or decreases by a constant rate, r, over a number of time

periods, t, this formula shows the final amount, A (t).

A (t) , the final

A (t) a (1 r)t

Time, t, is measured in years.

amount, is a function

of time, t.

a is the initial amount.

The rate, r, usually is a percent.

An initial amount of $15,000 increases by 12% per year. In how many years will the amount reach $25,000?

Step 1 Step 2

Identify values for a and r. a $15,000 r 12 0.12 Substitute values for a and r into the formula. f (t) a (1 r)t

Remember:

Ornth1e2%graph0,.1x2

corresponds to t and y corresponds to f (t).

f (t) 15,000 (1 0.12 )t

f (t) 15,000 (1.12)t Simplify.

Step 3 Graph the function using a graphing calculator. Modify the scales: [0, 10] and [0, 30,000].

Step 4 Use the graph and the [TRACE] feature on the calculator to find f (t) 25,000.

Step 5 Use the graph to approximate the value of t when f (t) 25,000.

t 4.5 when f (t) 25,000

The amount will reach $25,000 in about 4.5 years.

Write an exponential function and graph the function to solve.

3. An initial amount of $40,000 increases by 8% per year. In how many years will the amount reach $60,000?

a. a b. r c. f (t) ______________________ d. Approximate t when f (t) 60,000

t ____________

Lesson (4-2) a

Inverses of Relations and Functions

To graph an inverse relation, reflect each point across the line yx. Or you can switch the x- and y-values in each ordered pair of the relation to find the ordered pairs of the inverse.

Remember, a relation is a set of ordered pairs.

To write the inverse of the relation, switch the places of x and y in each ordered pair.

x 0 1 4 6 10 y 3 6 9 10 11

x3 6 y0 1

9 10 11 4 6 10

The domain is all possible values of x: {x | 0 x 10}.

The range is all possible values of y: {y | 3 y 11}.

The domain of the inverse corresponds to the range of the original relation: {x | 3 x 11}. The range of the inverse corresponds to the domain of the original relation: {y | 3 y 10}.

Complete the table to find the ordered pairs of the inverse. Graph the relation and its inverse. Identify the domain and range of each relation.

1. Relation x 0 2 5 8 10 y 6 10 12 13 13

Inverse x 6 y 0

Relation: Domain: _______________ Range: _______________

Inverse: Domain: _______________ Range: _______________

Lesson (4-2) b

Inverses of Relations and Functions (continued)

Inverse operations undo each other, like addition and subtraction, or multiplication and division. In a similar way, inverse functions undo each other. The inverse of a function f (x) is denoted f1(x). Use inverse operations to write inverse functions.

Function: f(x) x 8

Function: f(x) 5x

Subtraction is the opposite of addition. Use subtraction to write the inverse.

Inverse: f1(x) x 8

Division is the opposite of multiplication. Use division to write the inverse.

Inverse: f 1(x) x 5

Choose a value for x to check in the original function. Try x 1.

f(x) x 8 f(1) 1 8 9

Substitute 9, into f1 (x). The output of the inverse should be 1.

f1 (x) x 8 f 1 (9) 9 8 1

Choose a value for x to check in the original function. Try x 2.

f(x) 5x f (2) 5 (2) 10

Substitute 10 into f1 (x). The output of the inverse should be 2.

f 1(x) x f 1(10) 10 2

5

5

Think: (1, 9) in the original function should Think: (2, 10) in the original function should

be (9, 1) in the inverse.

be (10, 2) in the inverse.

Use inverse operations to write the inverse of each function.

2. f (x) x 4

3. f (x) x 6

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Use x 5 to check.

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Use x 12 to check.

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4. f (x ) x 3

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5. f (x) 14x

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