Question

Hi! I didn’t do so well on one of my assignments recently & was wondering if someone could help me learn how to properly solve for the problems I missed. Exercise #1

Answer 1

Exercise 1

Step-by-step explanation

We see that the graph is the plot of a polynomial function.

This polynomial has the following properties:

- Zeros:

• x₁ = -1 with multiplicity m₁ = 2, because the curve bounces the x-axis,

,

• x₂ = 2 with multiplicity m₂ = 1, because the curve crosses the x-axis,

,

• x₃ = 5 with multiplicity m₃ = 1, because the curve crosses the x-axis.

- y-intercept:

• y₀ = -4.

(a) The general form of this polynomial is:

`f(x)=a\cdot(x-x_1)^(m_1)\cdot(x-x_2)^(m_2)\cdot(x-x_3)^(m_3).`

Where:

• x₁, x₂ and x₃ are zeros of the polynomial,

,

• m₁, m₂ and m₃ are the multiplicities,

,

• a is a constant factor.

Replacing the values from above, we get:

`y=f(x)=a\cdot(x+1)^2\cdot(x-2)\cdot(x-5).`

(b) Replacing the data of the y-intercept point (x, y) = (0, -4), we have:

`\begin{gathered} -4=a\cdot(0+1)^2\cdot(0-2)\cdot(0-5), \\ -4=a\cdot1\cdot(-2)\cdot(-5), \\ 10a=-4. \end{gathered}`

Solving for a, we get:

`a=-(4)/(10)=-0.4.`

Replacing the value a = -0.4 in the equation of the polynomial, we get:

`y=-0.4\cdot(x+1)^2\cdot(x-2)\cdot(x-5).`Answer

(a) The factored form of the graph is:

`y=a\cdot(x+1)^2\cdot(x-2)\cdot(x-5)`

(b) The value of the constant a is:

`a=-0.4`

The complete equation of the curve is:

`y=-0.4\cdot(x+1)^2\cdot(x-2)\cdot(x-5)`