Question

Find the equation of the parabola with the given x-intercepts and point on the graph. Use y = a(x-p)(x-q).

2. x-int: (-4,0) , (-10, 0)
P (-6,-12)

Answer 1

`\textsf{Intercept form}: \quad y=(3)/(2)(x+4)(x+10)`

`\textsf{Standard form}: \quad y=(3)/(2)x^2+21x+60`

Explanation:

`\boxed{\begin{minipage}{6 cm}\underline{Intercept form of a quadratic equation}\\\\$y=a(x-p)(x-q)$\\\\where:\\ \phantom{ww}$\bullet$ $p$ and $q$ are the $x$-intercepts. \\ \phantom{ww}$\bullet$ $a$ is some constant.\\\end{minipage}}`

If the x-intercepts are (-4, 0) and (-10, 0) then:

• p = -4
• q = -10

Substitute the values of p and q into the formula:

`\implies y=a(x-(-4))(x-(-10))`

`\implies y=a(x+4)(x+10)`

To find a, substitute the given point on the curve P (-6, -12) into the equation:

`\implies -12=a(-6+4)(-6+10)`

`\implies -12=a(-2)(4)`

`\implies -12=-8a`

`\implies a=(-12)/(-8)`

`\implies a=(3)/(2)`

Substitute the found value of a into the equation:

`\implies y=(3)/(2)(x+4)(x+10)`

Expand to write the equation in standard form:

`\implies y=(3)/(2)(x^2+14x+40)`

`\implies y=(3)/(2)x^2+21x+60`

Answer 2

• y = 1.5(x + 4)(x + 10)

=========================

Given

• x- intercepts (-4, 0) and (-10, 0),
• Point P (- 6, - 12).

Solution

The given translates as:

• p = - 4, q = - 10, x = - 6, y = - 12

Use given x - intercepts to get the equation:

• y = a(x + 4)(x + 10)

Use the coordinates of P to find the value of a:

• - 12 = a(-6 + 4)(-6 + 10)
• - 12 = a*(-2)*4
• - 12 = - 8a
• a = - 12 / - 8
• a = 1.5

The equation of this parabola is:

• y = 1.5(x + 4)(x + 10)