Question

The graph of the function *f *is shown in the *xy*-plane above, where y =f . Which of the following functions could define *f *?

Hint:

**Hint:**

First, we carefully observe the graph to find the zeroes of the function, that is, we find the points at which is 0. From there, we get the factors that appear in the equation of the graph. Then we check the multiplicity of each of these factors to get the final equation. By multiplicity, we mean the number of times each factor appears in the equation or we can say, multiplicity of a factor is the power of the factor in the equation. Also, the degree of the equation is one greater than the number of turning points in the graph. (Turning point means the point on the graph where the function turns from increasing to decreasing or vice-versa.)

## The correct answer is:

### By observing the graph, we see that the curve cuts the x-axis at three points. These points are given as

So is 0 at each of these three points.

Hence, the factors which appear in the equation are

Next, we find the multiplicity of each of these factors.

We know that if a graph bounces off the x-axis at a point, then the multiplicity of the factor associated with the point is even.

And if the graph crosses the x-axis through a point, then the multiplicity of the factor associated with the point is odd.

So, the multiplicity of ( x + 3) is odd and the multiplicity of ( x + 1 ) and ( x - 2 ) is even.

Clearly, there are four turning points on the graph, so the degree of the equation is 5.

Clearly, there are four turning points on the graph, so the degree of the equation is 5.

Thus, a possible function which defines the curve is given by

Here, the degree of the equation is 5(=1+2+2). The multiplicity of (x +3 ) is 1, which is odd and the multiplicity of (x + 1) and ( x- 2) is 2, which is even.

The correct option is C).

.

**Note:**

We need to know properties about factors, multiplicity, leading co-efficients and turning points to find out the function from a graph.

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**Note: **

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