Question

Find the equation of the parabola that passes through
each set of points.
(0,-7),(7,-14),(-3,-19)

Answer 1

so we have the points of (0,-7),(7,-14),(-3,-19), let's plug those in the y = ax² + bx + c form, since we have three points, we'll plug each one once, thus a system of three variables, and then we'll solve it by substitution.

$\bf \begin{array}{cccllll} \stackrel{\textit{point (0,-7)}}{-7=a(0)^2+b(0)+c}& \stackrel{point (7,-14)}{-14=a(7)^2+b(7)+c}& \stackrel{point (-3,-19)}{-19=a(-3)^2+b(-3)+c}\\\\ -7=c&-14=49a+7b+c&-19=9a-3b+c \end{array}$

well, from the 1st  equation, we know what "c" is already, so let's just plug that in the 2nd equation and solve for "b".

$\bf -14=49a+7b-7\implies -7=49a+7b\implies -7-49a=7b \\\\\\ \cfrac{-7-49a}{7}=b\implies \cfrac{-7}{7}-\cfrac{49a}{7}=b\implies -1-7a=b$

well, now let's plug that "b" into our 3rd equation and solve for "a".

$\bf -19=9a-3b-7\implies -12=9a-3b\implies -12=9a-3(-1-7a) \\\\\\ -12=9a+3+21a\implies -15=9a+21a\implies -15=30a \\\\\\ -\cfrac{15}{30}=a\implies \blacktriangleright -\cfrac{1}{2}=a \blacktriangleleft \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{and since we know that}}{-1-7a=b}\implies -1-7\left( -\cfrac{1}{2} \right)=b\implies -1+\cfrac{7}{2}=b\implies \blacktriangleright \cfrac{5}{2}=b \blacktriangleleft \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill y=-\cfrac{1}{2}x^2+\cfrac{5}{2}x-7~\hfill$

Answer 2

Solving a system of equations, it is found that the equation of the parabola is:

$y = \frac{3}{10}x^2 + -\frac{31}{10}x - 7$

The standard equation of a parabola is given by:

$y = ax^2 + bx + c$

It passes through (0, -7), which means that when $x = 0, y = -7$, thus c = -7, and:

$y = ax^2 + bx - 7$

It passes through (7, -14), which means that when $x = 7, y = -14$, and thus:

$-14 = 49a + 7b - 7$

$49a + 7b = -7$

Simplifying by 7:

$7a + b = -1$

$b = -1 - 7a$

It also passes through (-3, -19), which means that when $x = -3, y = -19$, and thus:

$-19 = 9a - 3b - 7$

$9a - 3b = -12$

Since $b = -1 - 7a$:

$9a - 3(-1 - 7a) = -12$

$9a + 3 + 21a = 12$

$30a = 9$

$10a = 3$

$a = \frac{3}{10}$

$b = -1 - 7a = -1 - 7\frac{3}{10} = -\frac{10}{10} - \frac{21}{10} = -\frac{31}{10}$

Thus, the equation of the parabola is:

$y = \frac{3}{10}x^2 + -\frac{31}{10}x - 7$

A similar problem is given at https://brainly.com/question/17987697