f(x) = x4 - 6x3 + 9x2 a. Use the Leading Coefficient Test to determine the graph’s end behavior.b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept.c. Find the y-intercept.d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither.e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly.
Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
f(x) = x4 - 6x3 + 9x2
a. Use the Leading Coefficient Test to determine the graph’s end behavior.
b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept.
c. Find the y-intercept.
d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither.
e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly.
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Given:
Solution:
a) Use the leading coefficient test to determine the graph's end behavior:
Since,
The leading coefficient here is positive and of even degree
Hence, The highest degree is 4 (even) and polynomial is a positive function.
Therefore,
The graph rises to the left and right;
b) To determine:
The x-intercept=?
Solution:
The x-intercept of the graph is that point at which the graph crosses or touches the x-axis.
Substituting y=0 and finding the values for x=?
So,
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