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3 years ago

Write the formula for the parabola that has x − intercepts (13, 0) and (4, 0) and y − intercept (0, 13).

Mathematics

1 answer:

Radda [10]3 years ago

7 0

Answer:

The equation of the parabola is $y=\frac{x^{2}}{4} - \frac{17 x}{4} + 13$ .

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

$y=a x^{2} + b x + c$

Since the parabola passes through the point (13, 0), then $0=169 a + 13 b + c$ .

Since the parabola passes through the point (4, 0), then $0=16 a + 4 b + c$ .

Since the parabola passes through the point (0,13), then $13=c$ .

Thus, we have obtained the following system:

$\begin{cases}169 a + 13 b + c=0\\16 a + 4 b + c=0\\c=13\end{cases}$

Next, we need to solve this system of equations.

$\mathrm{Subsititute\:}c=13\\\\\begin{bmatrix}169a+13b+13=0\\ 16a+4b+13=0\end{bmatrix}$

$\mathrm{Isolate}\:a\:\mathrm{for}\:169a+13b+13=0:\quad a=-\frac{b+1}{13}\\\\\mathrm{Subsititute\:}a=-\frac{b+1}{13}\\\\16\left(-\frac{b+1}{13}\right)+4b+13=0$

$\mathrm{Isolate}\:b\:\mathrm{for}\:16\left(-\frac{b+1}{13}\right)+4b+13=0:\quad b=-\frac{17}{4}\\\\\mathrm{For\:}a=-\frac{b+1}{13}\\\\\mathrm{Subsititute\:}b=-\frac{17}{4}\\\\a=-\frac{-\frac{17}{4}+1}{13}=\frac{1}{4}\\$

$\mathrm{The\:solutions\:to\:the\:system\:of\:equations\:are:}\\\\b=-\frac{17}{4},\:a=\frac{1}{4},\:c=13$

The equation of the parabola is $y=\frac{x^{2}}{4} - \frac{17 x}{4} + 13$ .

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Answer:

Step-by-step explanation:

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In order to solve for x, we have to get it out from its position of an exponent. Do that by taking the natural log of both sides:

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Simplifying gives you

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Applying the power rule allows us to bring down the 3 in front of the ln:

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Now we can divide both sides by 3 to get

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But don't forget the substitution that we made earlier to make this easy to factor. Now we have to put it back in:

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