heavenaubry
Answered

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 :

mhanifa

Answer:

  • y = 0.4(x - 5)(x - 12)

Step-by-step 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)
semsee45

Answer:

[tex]\textsf{Intercept form}: \quad y=\dfrac{2}{5}(x-5)(x-12)[/tex]

[tex]\textsf{Standard form}: \quad y=\dfrac{2}{5}x^2-\dfrac{34}{5}x+24[/tex]

Step-by-step explanation:

[tex]\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}}[/tex]

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:

[tex]\implies y=a(x-5)(x-12)[/tex]

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

[tex]\implies -4=a(7-5)(7-12)[/tex]

[tex]\implies -4=a(2)(-5)[/tex]

[tex]\implies -4=-10a[/tex]

[tex]\implies a=\dfrac{-4}{-10}[/tex]

[tex]\implies a=\dfrac{2}{5}[/tex]

Substitute the found value of a into the equation:

[tex]\implies y=\dfrac{2}{5}(x-5)(x-12)[/tex]

Expand to write the equation in standard form:

[tex]\implies y=\dfrac{2}{5}(x^2-17x+60)[/tex]

[tex]\implies y=\dfrac{2}{5}x^2-\dfrac{34}{5}x+24[/tex]

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