Exercise 1
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:
[tex]f(x)=a\cdot(x-x_1)^{m_1}\cdot(x-x_2)^{m_2}\cdot(x-x_3)^{m_3}.[/tex]
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:
[tex]y=f(x)=a\cdot(x+1)^2\cdot(x-2)\cdot(x-5).[/tex]
(b) Replacing the data of the y-intercept point (x, y) = (0, -4), we have:
[tex]\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}[/tex]
Solving for a, we get:
[tex]a=-\frac{4}{10}=-0.4.[/tex]
Replacing the value a = -0.4 in the equation of the polynomial, we get:
[tex]y=-0.4\cdot(x+1)^2\cdot(x-2)\cdot(x-5).[/tex]Answer
(a) The factored form of the graph is:
[tex]y=a\cdot(x+1)^2\cdot(x-2)\cdot(x-5)[/tex]
(b) The value of the constant a is:
[tex]a=-0.4[/tex]
The complete equation of the curve is:
[tex]y=-0.4\cdot(x+1)^2\cdot(x-2)\cdot(x-5)[/tex]
