refer to the polynomial function f(x)= -x(x-1)(x+2) in anwering the folloowing questions.
1.what is the y intercept?
2.what is the behavior of the left end of its graph?
3.will the graph pass through the origin?
4,what is the behavior of the left end of this graph?
5how many turning points does the graph will have?
6.what is the degree.
7.is there a point where the graph is tangent to the x axis?
choises to the questions.
a.even
b.odd
c.negative.
d.positive
e.rising
f.falling
f.0
g.2
h.yes
i.no
j.-2
k.1
make a table of signs and sketch the graph of the dunction D.
2/SKETCH THE GRAPH OF THE FUNCTION IN C.
Tell you what. I'll do the graph, and you answer the questions.
http://www.wolframalpha.com/input/?i=-x(x-1)(x%2B2)
1. The y-intercept of the polynomial function f(x) = -x(x-1)(x+2) can be found by substituting x = 0 into the function. Therefore, the y-intercept is f(0) = -0(0-1)(0+2) = 0.
2. The behavior of the left end of the graph can be determined by evaluating the limit as x approaches negative infinity. You can calculate the limit of f(x) as x approaches negative infinity. If the limit is positive infinity, the behavior is rising. If the limit is negative infinity, the behavior is falling. Let's evaluate the limit:
lim(x→-∞) f(x) = lim(x→-∞) -x(x-1)(x+2) = -∞
Since the limit is negative infinity, the behavior of the left end of the graph is falling.
3. To determine if the graph passes through the origin, we need to check if f(0) = 0. We already found that f(0) = 0, so the graph does pass through the origin.
4. The behavior on the right end of the graph can be determined by evaluating the limit as x approaches positive infinity. Let's calculate the limit:
lim(x→∞) f(x) = lim(x→∞) -x(x-1)(x+2) = -∞
Since the limit is negative infinity, the behavior on the right end of the graph is falling.
5. The number of turning points can be determined by counting the number of distinct real roots of the polynomial function. We can see that the function has three distinct real roots: x = 0, x = 1, and x = -2. Therefore, the graph will have a total of 3 turning points.
6. The degree of the polynomial function f(x) = -x(x-1)(x+2) is determined by the highest power of x in the function. In this case, the highest power of x is x³, so the degree of the polynomial function is 3.
7. A graph is tangent to the x-axis at a point if the function has a double root at that point. Checking the function, we can see that it does not have any double roots. Therefore, there is no point where the graph is tangent to the x-axis.
Here is a table of signs for the function f(x) = -x(x-1)(x+2):
x | -∞ | -2 | 0 | 1 | +∞
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f(x) | - | + | 0 | - | -
And here is a sketch of the graph of the function:
f(x)
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- - - -| - - - - - - - -
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x-axis
To answer the questions about the polynomial function f(x) = -x(x-1)(x+2), we can break down the steps:
1. Y-intercept:
To find the y-intercept, we substitute x = 0 into the function. So, f(0) = -0(0-1)(0+2) = 0. Therefore, the y-intercept is 0. (Answer: f)
2. Behavior of the left end of the graph:
To determine the behavior of the left end of the graph, we look at the leading term of the polynomial, which is -x^3. Since the degree of the polynomial is odd (3), and the leading coefficient is negative (-1), the behavior of the left end of the graph is downward (falling). (Answer: f)
3. Does the graph pass through the origin (0,0)?
To check if the graph passes through the origin, we substitute x = 0 into the function. f(0) = -0(0-1)(0+2) = 0. Since f(0) = 0, the graph does pass through the origin. (Answer: h)
4. Behavior of the right end of the graph:
To determine the behavior of the right end of the graph, we look at the leading term of the polynomial, which is -x^3. Since the degree of the polynomial is odd (3), and the leading coefficient is negative (-1), the behavior of the right end of the graph is upward (rising). (Answer: e)
5. Number of turning points:
The number of turning points in a polynomial function can be found by counting the number of possible sign changes in the coefficients. In this case, we have three factors -x, (x-1), and (x+2). So, there can be a maximum of three sign changes, which implies the graph can have a maximum of three turning points. (Answer: g)
6. Degree of the polynomial:
The degree of a polynomial is determined by the highest power of x. In this case, the largest power of x is x^3, so the degree of the polynomial is 3. (Answer: g)
7. Is there a point where the graph is tangent to the x-axis?
A polynomial graph can only be tangent to the x-axis if it has a double root, which means one of the factors is squared. In this function, none of the factors are squared, so the graph does not have a point where it is tangent to the x-axis. (Answer: i)
To make a table of signs, we need to determine the signs of the function for different intervals of x. This can be done by considering the intervals where each factor is positive or negative. Once we have the signs, we can sketch the graph based on the sign changes and behavior determined above.
Unfortunately, as a text-based AI, I am unable to create visual sketches of graphs. It would be best to use a graphing calculator or software to sketch the graph of the function based on the information provided.