Question

Graph a parabola whose x-intercepts are at x=-3 and x=5 and whose minimum value is y=-4

Answer 1

Answer:

(See explanation for further details)

Step-by-step explanation:

The standard equation of the parabola is:

$y + 4 = C \cdot (x-k)^{2}$

The formula is now expanded into a the form of a second-order polynomial:

$y + 4 = C\cdot x^{2} -2\cdot C\cdot k \cdot x +C\cdot k^{2}$

$y = C\cdot x^{2} - (2\cdot C \cdot k) \cdot x + (C\cdot k^{2}-4)$

The general equation of the second-order polynomial is:

$x = \frac{2\cdot C \cdot k \pm \sqrt{4\cdot C^{2}\cdot k^{2}-4\cdot C\cdot (C\cdot k^{2}-4)}}{2\cdot C}$

$x = k \pm \frac{\sqrt{C^{2}\cdot k^{2}-C^{2}\cdot k^{2}+4\cdot C}}{C}$

$x = k \pm 2\cdot \frac{\sqrt{C}}{C}$

$x = k \pm \frac{2}{\sqrt{C}}$

The equations to be solved are presented herein:

$-3 = k -\frac{2}{\sqrt{C}}$

$5 = k + \frac{2}{\sqrt{C}}$

Now, the solution of the system is:

$-3 +\frac{2}{\sqrt{C}} = 5 -\frac{2}{\sqrt{C}}$

$\frac{4}{\sqrt{C}} = 8$

$\sqrt{C} = \frac{1}{2}$

$C = \frac{1}{4}$

$k = 5 - \frac{2}{\sqrt{\frac{1}{4} }}$

$k = 1$

The equation of the parabola is:

$y = \frac{1}{4}\cdot (x-1)^{2} -4$

Lastly, the graphic of the function is included as attachment.