Using the Factor Theorem to find the equation of the parabola, we have that:
[tex]a + b + c = -\frac{48}{7}[/tex]
What is the Factor Theorem?
The Factor Theorem states that a polynomial function with roots [tex]x_1, x_2, \codts, x_n[/tex] is given by:
[tex]f(x) = a(x - x_1)(x - x_2) \cdots (x - x_n)[/tex]
In which a is the leading coefficient.
In this problem, we have a horizontal parabola, defined as follows:
[tex]f(y) = a(y - y_1)(y - y_2)[/tex].
The roots are [tex]y_1 = -1, y_2 = -7[/tex], hence:
f(y) = a(y + 1)(y + 7)
f(y) = a(y² + 8y + 7).
The x-intercept is of x = -3, hence f(0) = -3, so:
[tex]-3 = 7a[/tex]
[tex]a = -\frac{3}{7}[/tex]
Then the equation is:
[tex]x = -\frac{3}{7}y^2 - \frac{24}{7}x - 3[/tex]
Then the sum of the coefficients is:
[tex]a + b + c = -\frac{3}{7} - \frac{24}{7} - \frac{21}{7} = -\frac{48}{7}[/tex]
More can be learned about the Factor Theorem at https://brainly.com/question/24380382
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