Question

Identify the zeros for the function and sketch a graph.

y=x(x-2)(X+3)

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the zeros of the function, we set y equal to zero and solve for x:

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

Setting each factor equal to zero, we have three possible solutions:

x = 0, x - 2 = 0, x + 3 = 0

Simplifying each equation, we find:

x = 0, x = 2, x = -3

Therefore, the zeros of the function are x = 0, x = 2, and x = -3.

Now, let's sketch the graph of the function. To do this, we can use the zeros we found to determine the behavior of the graph.

Since the factors are all linear, the graph will have a zero at x = 0 (where the graph crosses the x-axis) and will be shifted upward or downward depending on the sign of each factor.

The factor (x - 2) indicates that the graph will cross the x-axis at x = 2, while the factor (x + 3) indicates that the graph will cross the x-axis at x = -3.

Since the first factor (x) has a power of 1, and the other two factors are linear, the graph will have multiplicity 1 at each zero, meaning the graph will cross the x-axis at each zero.

To sketch the graph, we can plot the zeros at x = 0, x = 2, and x = -3, and then observe the behavior of the graph between each zero.

At x = 0, the graph crosses the x-axis.
At x = 2, the graph crosses the x-axis.
At x = -3, the graph crosses the x-axis.

The graph will resemble a "U" shape, with the vertex located between x = 0 and x = 2. We can also determine the concavity by checking the sign of the coefficient of the quadratic term.

Since there is no quadratic term in the given function, the graph will be a simple linear graph without any concavity.

Here is a rough sketch of the graph:

^
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--------------|--------------
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---------------------------->
-3 0 2

Note: The scale and precise shape of the graph may not be perfectly represented in this text-based sketch, but it should give a general idea of the graph's behavior.