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Answer 1

16)a)The rational function  f (x ) = 1/x ²

The graph of rational functions can be recognised by the fact two or more parts.

1) Find the intercpts of function y = 1/x ²

To find  y  intercept  x = 0 in the function.

y = 1/(0) ²

Not defined.

To find x  intercept y = 0 in the function.

0 = 1/x ²

In this case there is no x , y  intercepts.

2) Vertical asymptotes can be found by making denominator = 0.

x ² = 0

Vertical asymptote is x = 0.

3) To find horizontal asymptote, first find the degree of the numarator and  the degree of denominator.

Degree of the numarator = 0 and the degree of denominator = 2.

Since the degree of the numerator is less than to the degree of the denominator, the line y = 0 is the horizontal asymptote.

4) To graph the function, we need some points to make more accurate graph.

x	y = 1/x 2	(x, y )
-3	y = 1/(-3)2 = 0.11	(- 3, 0.11)
-2	y = 1/(-2)2 = 0.25	(-2, 0.25)
-1	y = 1/(-1)2 = 1	(-1, 1)
1	y = 1/(1)2 = 1	(1, 1)
2	y = 1/(-2)2 = 0.25	(2, 0.25)
3	y = 1/(-3)2 = 0.11	(3, 0.11)

 

Answer 2

Contd....

Graph

1) Draw the coordinate plane.

2) Next dash the horizontal and vertical asymptotes.

3) Plot the coordinate pairs found in the table and connect the plotted points .

When you draw your graph, use smooth curves complete the graph.

 

Answer 3

16)b)The cubic function is y = x³ + 2

1) Find the Zeros

y = x³ + 2 using zeros to graph the polynomial.

To find x intercept substitute y = 0 in the function.

0 = x³ + 2

x³ = - 2

x = -1.25

x intercept is (-1.25, 0).

To find y intercept substitute x = 0 in the function.

y = 0 + 2

y intercept is (0, 2).

2)Test points

Make the table of values for the polnomial.

Choose  random values for x and find the corresponding values for y.

x	y = x3 + 2	(x, y )
-1.5	y = (-1.5)3+2	(-1.5, -3.375)
-1	y = (-1)3+2	(-1, 1)
-0.5	y = (-0.5)3 + 2	(-0.5, -0.125)
1.5	y = (1.5)3+2	(1.5, 1.375)
1	y = (1)3+2	(1, 3)
0.5	y = (0.5)3+2	(0.5, 0.125)

 

Answer 4

Contd...

3) End behavior y = x³ + 2

Degree of the polynomial is 3 and leading coefficient 1.

The graph of a polynomial function is always a smooth curve; that is, it has no breaks or corners.

All odd degree polynomials behave on their ends like cubics.

All odd degree polynomials  have ends that head off in opposite directions.depending on whether the polynomial has, respectively, a positive or negative leading coefficient.

The above polynomial odd degree  polynomial with a positive leading coefficient .

So the graph falls to the left and rises to the right.

4)Graph

1.Draw a coordinate plane.

2.Plot the x, y  intercepts and coordinate points found in the table.

3.Then sketch the graph, connecting the points with a smooth curve.