Respuesta :
Answer:
The equation of the parabola is [tex]y=\frac{x^{2}}{4} - \frac{17 x}{4} + 13[/tex].
Step-by-step explanation:
A parabola is a curve where any point is at an equal distance from a fixed point (the focus), and a fixed straight line (the directrix).
To find the formula we assume that the equation of the parabola is
[tex]y=a x^{2} + b x + c[/tex]
Since the parabola passes through the point (13, 0), then [tex]0=169 a + 13 b + c[/tex].
Since the parabola passes through the point (4, 0), then [tex]0=16 a + 4 b + c[/tex].
Since the parabola passes through the point (0,13), then [tex]13=c[/tex].
Thus, we have obtained the following system:
[tex]\begin{cases}169 a + 13 b + c=0\\16 a + 4 b + c=0\\c=13\end{cases}[/tex]
Next, we need to solve this system of equations.
[tex]\mathrm{Subsititute\:}c=13\\\\\begin{bmatrix}169a+13b+13=0\\ 16a+4b+13=0\end{bmatrix}[/tex]
[tex]\mathrm{Isolate}\:a\:\mathrm{for}\:169a+13b+13=0:\quad a=-\frac{b+1}{13}\\\\\mathrm{Subsititute\:}a=-\frac{b+1}{13}\\\\16\left(-\frac{b+1}{13}\right)+4b+13=0[/tex]
[tex]\mathrm{Isolate}\:b\:\mathrm{for}\:16\left(-\frac{b+1}{13}\right)+4b+13=0:\quad b=-\frac{17}{4}\\\\\mathrm{For\:}a=-\frac{b+1}{13}\\\\\mathrm{Subsititute\:}b=-\frac{17}{4}\\\\a=-\frac{-\frac{17}{4}+1}{13}=\frac{1}{4}\\[/tex]
[tex]\mathrm{The\:solutions\:to\:the\:system\:of\:equations\:are:}\\\\b=-\frac{17}{4},\:a=\frac{1}{4},\:c=13[/tex]
The equation of the parabola is [tex]y=\frac{x^{2}}{4} - \frac{17 x}{4} + 13[/tex].
By using the given information, we can write the quadratic equation as:
y = (1/4)*(x - 4)*(x - 13)
How to get the quadratic equation?
We know that if a quadratic equation has roots x₁ and x₂, then we can write the equation as:
y = a*(x - x₁)*(x - x₂)
Where a is the leading coefficient.
Here we know that:
x₁ = 4 and x₂ = 13, then we have:
y = a*(x - 4)*(x - 13)
And we also know that it pass through the point (0, 13), so when evaluated in x = 0 the value of y must be 13, replacing that we get:
13 = a*(0 - 4)*(0 - 13)
13 = a*(-4)*(-13)
1 = a*4
1/4 = a
So the quadratic equation is:
y = (1/4)*(x - 4)*(x - 13)
If you want to learn more about quadratic equations, you can read:
https://brainly.com/question/1214333
