Question

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

4. x-int: (5,0) , (12, 0)
P (7, -4)

Answer 1

`\textsf{Intercept form}: \quad y=(2)/(5)(x-5)(x-12)`

`\textsf{Standard form}: \quad y=(2)/(5)x^2-(34)/(5)x+24`

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 (5, 0) and (12, 0) then:

• p = 5
• q = 12

Substitute the values of p and q into the formula:

`\implies y=a(x-5)(x-12)`

To find a, substitute the given point on the curve P (7, -4) into the equation:

`\implies -4=a(7-5)(7-12)`

`\implies -4=a(2)(-5)`

`\implies -4=-10a`

`\implies a=(-4)/(-10)`

`\implies a=(2)/(5)`

Substitute the found value of a into the equation:

`\implies y=(2)/(5)(x-5)(x-12)`

Expand to write the equation in standard form:

`\implies y=(2)/(5)(x^2-17x+60)`

`\implies y=(2)/(5)x^2-(34)/(5)x+24`

Answer 2

• y = 0.4(x - 5)(x - 12)

Explanation:

Given

• x- intercepts (5, 0) and (12, 0),
• Point P (7, - 4).

Solution

The given translates as:

• p = 5, q = 12, x = 7, y = - 4

Use given x - intercepts to get the equation:

• y = a(x - 5)(x - 12)

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

• - 4 = a(7 - 5)(7 - 12)
• - 4 = a*2*(-5)
• - 4 = - 10a
• a = -4 / - 10
• a = 0.4

The equation of this parabola is:

• y = 0.4(x - 5)(x - 12)