Question

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What are the zeros of the function?
Use the zeros to find all of the linear factors of the polynomial function.
Write the equation of the graphed function f(x), where a is the leading coefficient. Use the factors found in the previous question. Express the function as the product of its leading coefficient and the expanded form of the equation in standard form.
Use the y-intercept of the graph and your equation from part E to calculate the value of a.
Given what you found in all of the previous parts, write the equation for the function shown in the graph.

Answer 1

Answer:

Step-by-step explanation:

The zeros of this cubing function are easily read from the graph:  {-20, -5, 15}.

The factors of this polynomial are therefore (x + 20), (x + 5) and (x - 15).

The y-intercept is (0, 1).

The function is thus f(x) = a(x + 20)(x + 5)(x - 15).

According to the y-intercept, if x = 0, y = 1.

Thus, y = 1 = f(0) = a(20)(5)(-15), or 1 = a(100)(-15), or 1 = -1500a.

Then a = -1/1500, and the function is:

f(x) = (-1/1500)(x^3 + .. +  .. + ... ).  We must multiply out (x + 20)(x + 5)(x - 15) to obtain f(x) in finished form.

Answer 2

Answer:

zeros: (-20, 0), (-5, 0) and (15, 0)

factors: (x + 20), (x + 5) and (x - 15)

f(x) = a*(x + 20)*(x + 5)*(x - 15)

a = -1/1500

f(x) = -1/1500*(x + 20)*(x + 5)*(x - 15)

f(x) = -1/1500*x^3 - 1/100*x^2 + 11/60x + 1

Step-by-step explanation:

The zeros of the function are those points where the function intercepts the x-axis. They are: (-20, 0), (-5, 0) and (15, 0)

The zeros of a polynomial are expressed as factors as follows: (x - a) where a is a zero. Then, for this case, the factors are (x + 20), (x + 5) and (x - 15)

The equation of f(x) use the factors and the leading coefficient as follows: f(x) = a*(x + 20)*(x + 5)*(x - 15)

Applying the distributive property of multiplication, we get the expanded form:

f(x) = a*(x + 20)*(x + 5)*(x - 15)

(x + 20)*(x + 5) = x^2 + 5x + 20x + 20*5 = x^2 + 25x  + 100

f(x) = a*(x^2 + 25x  + 100)*(x - 15)

(x^2 + 25x  + 100)*(x - 15) = x^3 - 15x^2 + 25x^2 - 15*25x + 100x - 100*15 = x^3  + 15x^2 - 275x - 1500

f(x) = a*(x^3 + 15x^2 - 275x - 1500)

The y-intercept of the graph is (0, 1). Replacing this point into the function equation:

1 = a*(0 + 20)*(0 + 5)*(0 - 15)

1 = a*20*5*(-15)

1 = a*(-1500)

a = -1/1500

Replacing this value  into the function equation:

f(x) = (-1/1500)*(x^3 + 15x^2 - 275x - 1500)

f(x) = -1/1500*x^3 - 1/100*x^2 + 11/60x + 1