# Student Activity: a Generic Function

**Topics:**Derivative, Mathematical analysis, Continuous function

**Pages:**8 (572 words)

**Published:**February 5, 2013

A Generic Function

Use the generic graph of f(x) with domain [–6, –3] and [–2, 6] to answer the questions below. 7

Y

6

5

4

3

2

1

X

-7

-6 -5

-4 -3

-2 -1 0

-1

1

2

3

4

5

6

7

-2

-3

-4

-5

-6

-7

1.

What is the range of f(x)?

2.

What is the domain?

3.

On what intervals is f(x) decreasing?

4.

On what intervals will the following statements be true?

a)

As x increases, y increases.

b)

As x increases, y is constant.

c)

As x increases, y increases at a constant rate.

5.

For what values of x is f(x) > x?

6.

What is the absolute maximum value for f(x)?

7.

Give the coordinates of the point where the global minimum value of f(x) occurs.

8.

What is the absolute maximum value over the interval −6 ≤ x ≤ −3 ?

9.

For what values of x, x > 0, is f(x) concave down?

35

Student Activity

10.

Let

a)

b)

c)

g ( x) = f (− x)

Find g(–4) and g(2).

Determine the value of x where the maximum value of g(x) occurs. Describe the transformation of the graph.

11.

Let

a)

b)

c)

g ( x) = − f ( x)

Find g(–2.5) and g(4).

Determine the minimum value of g(x).

Describe the transformation of the graph.

12.

Let

a)

b)

c)

g ( x ) = f (2 x )

Find g(–2) and g(1).

Determine the slope of g(x) on the interval [0, 2].

Sketch a graph of g(x).

7

6

5

4

3

2

1

-7 -6 -5 -4 -3 -2 -1

-1

-2

-3

-4

-5

-6

-7

y

1234567

36

x

Student Activity

13.

Let

a)

b)

c)

g ( x ) = f ( x − 1)

Find g(0) and g(1).

Determine the intervals where g(x) is increasing.

Sketch a graph of g(x).

7

6

5

4

3

2

1

-7 -6 -5 -4 -3 -2 -1

-1

-2

-3

-4

-5

-6

-7

y

1234567

x

14.

Let

a)

b)

c)

g ( x ) = f ( x + 3)

Find g(–2) and g(3).

If x > 0, determine where g(x) is concave down.

Describe the transformation of the graph.

15.

Let

a)

b)

c)

g ( x) = f ( x) + 3

Find g(–2) and g(3).

Determine the y-intercept of g(x).

Describe the transformation of the graph.

16.

Write a piecewise function to describe f(x) over the domain [–6, –3] [–2, 6] The portion of the graph on the interval (–6, –4) should be modeled as a cosine function with its maximum at (–4, 3) and its minimum at (–6, –2).

The section on (–4, –3) should be modeled as a parabola with its vertex at (–3, –2). The section on (5, 6] is a parabola with its vertex at (5, 5).

37

Student Activity

17.

What are the roots of f(x)?

18.

Let h ( x ) = f ( | x | ) .

a) On the grid provided, graph h(x).

b) Write a piecewise function to describe

h(x) using as few equations as possible.

Hint: Use absolute values to decrease

the number of required equations.

7

Y

6

5

4

3

2

1

-7 -6 -5 -4 -3 -2 -1 0

-1

-2

-3

-4

-5

-6

-7

38

X

1

2

3

4

5

6

7

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