Question

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

3. x-int: (-4,0) , (7,0)
P (3,8)

Answer 1

`\textsf{Intercept form}: \quad y=-(2)/(7)(x+4)(x-7)`

`\textsf{Standard form}: \quad y=-(2)/(7)x^2+(6)/(7)x+8`

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 (7, 0) then:

• p = -4
• q = 7

Substitute the values of p and q into the formula:

`\implies y=a(x-(-4))(x-7)`

`\implies y=a(x+4)(x-7)`

To find a, substitute the given point on the curve P (3, 8) into the equation:

`\implies 8=a(3+4)(3-7)`

`\implies 8=a(7)(-4)`

`\implies 8=-28x`

`\implies a=(8)/(-28)`

`\implies a=-(2)/(7)`

Substitute the found value of a into the equation:

`\implies y=-(2)/(7)(x+4)(x-7)`

Expand to write the equation in standard form:

`\implies y=-(2)/(7)(x^2-3x-28)`

`\implies y=-(2)/(7)x^2+(6)/(7)x+8`

Answer 2

• y = -2/7(x + 4)(x - 7)

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

Given

• x- intercepts (-4, 0) and (7, 0),
• Point P (3, 8).

Solution

The given translates as:

• p = -4, q = 7, x = 3, y = 8

Use given x - intercepts to get the equation:

• y = a(x + 4)(x - 7)

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

• 8 = a(3 + 4)(3 - 7)
• 8 = a*7*(-4)
• 8 = - 28a
• a = 8 / - 28
• a = - 2/7

The equation of this parabola is:

• y = - 2/7(x + 4)(x - 7)