Question

Write the equation of the parabola that passes through the points (
–
6,
–
15), (
–
1,0), and (
–
7,0).
Write your answer in the form y=a(x–p)(x–q), where a, p, and q are integers, decimals, or simplified fractions.

Answer 1

Answer:

To write the equation of the parabola in the form \( y = a(x - p)(x - q) \), we need to find the values of \( a \), \( p \), and \( q \).

Given that the parabola passes through the points \((-6, -15)\), \((-1, 0)\), and \((-7, 0)\), we can set up the equation in the form:

\[ y = a(x - p)(x - q) \]

Since \((-1, 0)\) and \((-7, 0)\) are the x-intercepts, we have:

\[ (x + 1)(x + 7) = 0 \]

Expanding this equation, we get:

\[ x^2 + 8x + 7 = 0 \]

So, \( p + q = -8 \) and \( pq = 7 \).

Now, we can use the point \((-6, -15)\) to find the value of \( a \):

\[ -15 = a(-6 - p)(-6 - q) \]

\[ -15 = a(-6 + 1)(-6 + 7) \]

\[ -15 = a(-5)(1) \]

\[ -15 = -5a \]

\[ a = 3 \]

Now, we have \( a = 3 \) and \( p + q = -8 \) and \( pq = 7 \). To find \( p \) and \( q \), we can solve these equations simultaneously:

From \( p + q = -8 \), we can express \( p \) as \( p = -8 - q \), and substitute into \( pq = 7 \):

\[ (-8 - q)q = 7 \]

\[ -8q - q^2 = 7 \]

\[ q^2 + 8q - 7 = 0 \]

\[ (q - 1)(q + 7) = 0 \]

So, \( q = 1 \) or \( q = -7 \).

If \( q = 1 \), then \( p = -8 - 1 = -9 \).

If \( q = -7 \), then \( p = -8 - (-7) = -1 \).

So, we have two possible equations:

If \( p = -9 \) and \( q = 1 \):

\[ y = 3(x + 9)(x - 1) \]

If \( p = -1 \) and \( q = -7 \):

\[ y = 3(x + 1)(x + 7) \]

These are the two possible equations of the parabola that passes through the given points.

Answer 2

Answer:

$y=3(x+1)(x+7)$

Step-by-step explanation:

The factored form of a quadratic function is:

$y = a(x - p)(x - q)$

where a is the leading coefficient, and p and q are the zeros.

The zeros of a quadratic function are the values of x where the function equals zero, representing the points where the graph intersects the x-axis. In other words, they are the x-coordinates of the points where the y-coordinate is zero.

As the equation of the parabola passes through points (-1, 0) and (-7, 0), then the zeros of the function are x = -1 and x = -7.

Substitute these values for p and q in the factored formula:

$y = a(x - (-1))(x - (-7))\\\\\\y = a(x +1)(x +7)$

To find the value of a, substitute the other point (-6, -15) into the equation:

$-15 = a(-6 +1)(-6 +7)\\\\\\-15 = a(-5)(1)\\\\\\-15=-5a\\\\\\a=\dfrac{-15}{-5}\\\\\\a=3$

Therefore, the equation of the parabola that passes through points (-1, 0), (-6, -15) and (-7, 0) in the form y = a(x - p)(x - q) is:

$\Large\boxed{\boxed{y=3(x+1)(x+7)}}$