3 This problem concerns the polynonianctionx 2ra B a Does t

3. This problem concerns the polynonia|nction,(x) =-2ra + B\' a. Does this function have yas symmetry, origin symmetry, or neither? How can you tell without looking at the graph? b. Find the Kintercepts State whether the graph orosses the w-axis, or touches the w-axis and tuns around, at each intercept. Explain how you can tell without looking at the graph Give an example of one polynomial function with arthese features below. Tom ake grade, give your answer in factored form easy for me to . The end behavior is the same as the end behavior of- There are only two zeros: 7 and -1 The graph crosses the x-axis at-1 The graph touches the -axis at -7 and tures around Page 1 3

Solution

a) End behaviour is the same as x^(-5)

i.e. as x tends to infinity f(x) tends to -infinity faster.

So the polynomial is of degree 5.

Two zeroes are there 7 and -1

Since touches at x=7,

f(x) =- K(x) (x-7)²(x+1) where K(x) is a polynomial of degree 2 with positive leading coefficient.

Since f(x) is of degree 5, K(x) a quadratic has imaginary roots.

i.e. K(x) = ax²+bx+c, where b²-4ac <0

An example is

x²+2x+2 for K(x)

Hence an example for f(x)

f(x) = -(x²+2x+2)(x-7)²(x+1)

with point of inflection at x=7 and zeroes at 7,7, -1

end behaviour going to -infty as x goes to infty.