Given the following function:
[tex]k(x)=x^3-5x^2[/tex]
We will find the end behavior of the function.
the given function has a degree = 3 (odd)
And the leading coefficient is positive
the end behavior will be as follows:
[tex]\begin{gathered} x\to-\infty\Rightarrow k(x)\to-\infty \\ x\to\infty\Rightarrow k(x)\to\infty \end{gathered}[/tex]
So, the answer will be:
The end behavior of the function is down to the left and up to the right.
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Part (2), we will find the y-intercepts
The y-intercept is the value of y when x = 0
So, we will substitute x = 0 and then solve y
[tex]y=0^3-5(0^2)=0[/tex]
So, the answer will be:
y-intercept = (0, 0)
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Part 3: we will find the zeros of k(x)
The zeros of the function are the values of x which make k(x) = 0
So, we will write the equation k(x) = 0 and then solve it for x.
[tex]\begin{gathered} x^3-5x^2=0 \\ x^2(x-5)=0 \\ x^2=0\to x=0 \\ x-5=0\to x=5 \end{gathered}[/tex]
So, the answer will be:
Zeros of k: 0,5
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Part 4: we will find the graph of k(x)
From the previous parts, we can conclude that
The graph of the function will be as shown in option D
