Question

brian is solving an equation where both sides are quadratic expressions. he sets the expressions equal to \\(y\\). then he graphs the system finding that the two graphs share a common vertex, but one graph opens upward and the other opens downward. how should brian interpret this outcome?

there is exactly one intersection point.
there are exactly two intersection points.
there are exactly three intersection points.
there are infinitely many intersection points.

Explanation:

Analyze the geometric properties of the system

Using the Graphical Solution of Systems knowledge point

• The system consists of two quadratic functions (parabolas) sharing a common vertex \((h, k)\).
• One parabola opens upward: \(y = a_1(x-h)^2 + k\) with \(a_1 > 0\).
• The other parabola opens downward: \(y = a_2(x-h)^2 + k\) with \(a_2 < 0\).

Determine the intersection points

Using the Graphical Solution of Systems knowledge point

• Setting the two equations equal to find their intersection points:

$$ a_1(x-h)^2 + k = a_2(x-h)^2 + k $$

• Subtracting \(k\) from both sides and rearranging:

$$ (a_1 - a_2)(x-h)^2 = 0 $$

• Since \(a_1 > 0\) and \(a_2 < 0\), we have \(a_1 - a_2

eq 0\). Thus:

$$ (x-h)^2 = 0 \implies x = h $$

• Substituting \(x = h\) back into either equation yields \(y = k\).
• Therefore, the only intersection point is the shared vertex \((h, k)\).

Answer:

• There is exactly one intersection point. (Correct answer)
• There are exactly two intersection points.
• There are exactly three intersection points.
• There are infinitely many intersection points.