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).

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



Answer :

mhanifa

Answer:

  • 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)
semsee45

Answer:

[tex]\textsf{Intercept form}: \quad y=-\dfrac{2}{7}(x+4)(x-7)[/tex]

[tex]\textsf{Standard form}: \quad y=-\dfrac{2}{7}x^2+\dfrac{6}{7}x+8[/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 (-4, 0) and (7, 0) then:

  • p = -4
  • q = 7

Substitute the values of p and q into the formula:

[tex]\implies y=a(x-(-4))(x-7)[/tex]

[tex]\implies y=a(x+4)(x-7)[/tex]

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

[tex]\implies 8=a(3+4)(3-7)[/tex]

[tex]\implies 8=a(7)(-4)[/tex]

[tex]\implies 8=-28x[/tex]

[tex]\implies a=\dfrac{8}{-28}[/tex]

[tex]\implies a=-\dfrac{2}{7}[/tex]

Substitute the found value of a into the equation:

[tex]\implies y=-\dfrac{2}{7}(x+4)(x-7)[/tex]

Expand to write the equation in standard form:

[tex]\implies y=-\dfrac{2}{7}(x^2-3x-28)[/tex]

[tex]\implies y=-\dfrac{2}{7}x^2+\dfrac{6}{7}x+8[/tex]

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