Question

question 12 (5 points)
select the graph that can be used to find the solution(s) of the system of equations.
\\(y = -|x - 2|\\)
\\(y = x^2 + 2\\)

a)
b)
c)
d)

Explanation:

Analyze the absolute value equation

Using the Absolute Value Graphing knowledge point

$$ y = -|x - 2| $$

The parent function \(y = |x|\) is reflected vertically across the \(x\)-axis and translated horizontally right by \(2\) units.
The vertex of this V-shaped graph is at \((2, 0)\), opening downwards.

Analyze the quadratic equation

Using the Parabola Vertex Form knowledge point

$$ y = x^2 + 2 $$

The parent function \(y = x^2\) is translated vertically upwards by \(2\) units.
The vertex of this U-shaped parabola is at \((0, 2)\), opening upwards.

Compare with the given options

We look for the graph that displays:

1. A downward-opening V-shaped graph with its vertex at \((2, 0)\).
2. An upward-opening parabola with its vertex at \((0, 2)\).

• Option A: The parabola has its vertex at \((0, 2)\) and opens upward. The absolute value graph has its vertex at \((2, 0)\) and opens downward. The two graphs intersect. This matches both equations perfectly.
• Option B: The parabola has its vertex shifted to the left, and the absolute value graph has its vertex shifted to the right but not at \((2, 0)\).
• Option C: The parabola has its vertex at \((0, 2)\) but the absolute value graph vertex is not at \((2, 0)\).

Therefore, Option A is the correct graph.

Answer:

• (A) Parabola vertex at (0, 2), absolute value vertex at (2, 0) (Correct answer)
• (B) Parabola vertex shifted left, absolute value vertex shifted right
• (C) Parabola vertex at (0, 2), absolute value vertex at (3, 0)