Question

tom determines that the system of equations below has two solutions, one of which is located at the vertex of the parabola

equation 1 \\((x - 3)^2 = y - 4\\)
equation 2 \\(y = -x + b\\)

in order for toms thinking to be correct, which qualifications must be met?

\\(b\\) must equal 7 and a second solution to the system must be located at the point \\((2, 5)\\)
\\(b\\) must equal 1 and a second solution to the system must be located at the point \\((4, 6)\\)
\\(b\\) must equal 7 and a second solution to the system must be located at the point \\((1, 8)\\)
\\(b\\) must equal 1 and a second solution to the system must be located at the point \\((3, 4)\\)

Explanation:

Find the vertex of the parabola

The equation of the parabola is given in vertex form:

$$ (x - 3)^2 = y - 4 \implies y = (x - 3)^2 + 4 $$

The vertex of this parabola is:

$$ (h, k) = (3, 4) $$

Determine the value of b

Since one of the solutions is located at the vertex, the point \((3, 4)\) must satisfy the linear equation \(y = -x + b\):

$$ 4 = -3 + b \implies b = 7 $$

Find the second solution to the system

Substitute \(b = 7\) into the linear equation to get \(y = -x + 7\). Set the two equations equal to find the intersection points:

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The solutions are \(x = 3\) (the vertex) and \(x = 2\).
For \(x = 2\):

$$ y = -2 + 7 = 5 $$

Thus, the second solution is \((2, 5)\).

Answer:

• (A) b must equal 7 and a second solution to the system must be located at the point (2, 5) (Correct answer)
• (B) b must equal 1 and a second solution to the system must be located at the point (4, 5)
• (C) b must equal 7 and a second solution to the system must be located at the point (1, 8)
• (D) b must equal 1 and a second solution to the system must be located at the point (3, 4)