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Chapter 3: Problem 98

Write the equation of each parabola in standard form. Vertex: \((-3,-1) ;\) The graph passes through the point \((-2,-3)\)

Short Answer

Expert verified
The standard form of the equation of the parabola is \(y = -2(x + 3)^2 - 1\)

Step by step solution

01

Substitute the vertex into the standard form

We know from the problem that the vertex of the parabola is \(-3, -1\). Thus, in the standard form of the parabola equation, we can replace \(h\) with \(-3\) and \(k\) with \(-1\). This gives: \(y = a(x - (-3))^2 - 1\), or \(y = a(x + 3)^2 - 1\).
02

Substitute the known point to solve for 'a'

We can substitute the provided point \((-2, -3)\) into the 'x' and 'y' in our equation and then solve for 'a'. Substituting the 'x' with \(-2\) and 'y' with \(-3\), we get the equations: \(-3 = a(-2 + 3)^2 - 1\), which simplifies to \(-3 = a(1)^2 - 1\). Solving for 'a': we add '1' to both sides, yielding \(-2 = a\).
03

Write the final equation of the parabola

Now that we have 'a', 'h', and 'k', we can write the final standard equation of the parabola. Substituting 'a' with \(-2\), 'h' with \(-3\) and 'k' with \(-1\) into the standard form, we get the equation of the parabola as \(y = -2(x + 3)^2 - 1\) .

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