Page 340 #52

By:Hangny Dao

Zoilita AgredaGuerline Guerrier

the Radicals present:

Don’t worry, we’ll break it down for you.

Factoring

f(x) = (x+√3)2(x-2)4

The problem was present to us in that format.

Since it is already considered ‘factored,’ the equation will be kept like that throughout the whole process.

X-Intercept

To find the x-intercept(s), set y equal to zero.

0 = (x+√3)2(x-2)4

0 = (x+√3)(x+√3)(x-2)(x-2)(x-2)(x-2)0 = x+√3 0 = x-2X = -√3 x = +2

x-intercepts: (-√3,0) (2,0)*Note: Though the equation had two factors of (x+√3) and four factors

of (x-2), there was no need to include that results from that since it would have been the same answers.

Y-Intercept

To find the y-intercept, set x equal to zero.

f(0) = (x+√3)2(x-2)4

f(0) = (0+√3)2(0-2)4

f(0) = (√3)2(-2)4

*Note: according to the rules of basic algebra, when a square root is raised to the power of two, the squares cancel each other, leaving only the number under the square root.

f(0) = (3)(16) = 48

y-intercept: (0,48)

Multiplicity

List of Zeros Multiplicity Touches/Crosses

(-√3,0) 2 Touches

(2,0) 4 Touches

The multiplicity of an equation is determined by the power in which the factors are raised by.

The rules of multiplicity follows:•If the factor is raised to an even power, the line of the graph touches the zero of that factor.•If the factor is raised to an odd power, the line of the graph crosses the zero of that factor.• The zeros in these cases, would be the x-intercepts.

The original equation again was: f(x) = (x+√3)2(x-2)4

The Degree of the Graph

The degree of the graph is used to determine the basic shape of the graph.

If the total degree of the graph is even and positive, the two end points will both be pointing up. If negative, then the tails would both point down.

If the total degree of the graph is odd and positive/negative, the two tails will be pointing in opposite directions.

To determine the final degree of the graph, we must look at all the degrees of the factors. The equation again was f(x) = (x+√3)2(x-2)4, there are two factors raised to a different power. The rules of exponents states that when two factors are multiplied, the exponents of the factors are added.

Thus, the degree of f(x) = (x+√3)2(x-2)4, is x6.

Additional Points

The additional points will help to determine the graph since so far the we only have three points.

X f(x) (x,y)

-2 18.38 (-2,18.38)

-1 43.41 (-1,43.41)

-0.5 59.30 (-0.5,59.30)

1 7.464 (1,7.464)

3 22.39 (3,22.39)

4 525.7 (4,525.7

GraphTo determine what this graph would look like, we shall be using the previous information that we had found out.

• x-intercepts are at (-√3,0) and (2,0)

• y-intercept is at (0,48)• Multiplicity told us that the

graph will touch at (-√3,0) and (2,0). Since those are the only x-intercepts, the range of the graph would be y≥0

• The degree of the graph was an even number and the original equation was positive, so the two tails would be continuing in the positive y direction.

• Lastly, the use of the additional points to determine how the graph would curve.

So, the graph would look like this:

Thank you Zoilita Agreda for providing the information on factoring, x-intercept and y-intercept used in this powerpoint.

Thank you Guerline Guerrier for providing the information on multiplicity and the degree of the graph used in this powerpoint.

Thank you Hangny Dao for providing the graph with the use of Winplot and the template used in this powerpoint.

Music:Break Down – TaeyangThank You – 2PM

Credits