Match the graphs with their corresponding formulas.

Solution:

• Graph A shows a parabola opening upwards with its vertex in the first quadrant. This corresponds to a quadratic equation with a positive leading coefficient and roots that are real and have the same sign (or are complex, but in this case, they appear real). Formula 4, $$y = 2x^2 - 6x + 6$$, fits this description. The vertex can be found at $$x = -(-6)/(2*2) = 6/4 = 1.5$$. The y-coordinate of the vertex is $$2(1.5)^2 - 6(1.5) + 6 = 2(2.25) - 9 + 6 = 4.5 - 9 + 6 = 1.5$$. Thus, the vertex is at (1.5, 1.5), which aligns with graph A.
• Graph Б shows a parabola opening upwards with its vertex in the second quadrant. This corresponds to a quadratic equation with a positive leading coefficient and roots that are real and have opposite signs, or complex. Formula 3, $$y = 2x^2 + 6x + 6$$, fits this description. The vertex can be found at $$x = -(6)/(2*2) = -6/4 = -1.5$$. The y-coordinate of the vertex is $$2(-1.5)^2 + 6(-1.5) + 6 = 2(2.25) - 9 + 6 = 4.5 - 9 + 6 = 1.5$$. Thus, the vertex is at (-1.5, 1.5), which aligns with graph Б.
• Graph B shows a parabola opening downwards with its vertex in the first quadrant. This corresponds to a quadratic equation with a negative leading coefficient. Formulas 1 and 2 have negative leading coefficients. Formula 1, $$y = -2x^2 + 6x - 6$$, has its vertex at $$x = -(6)/(2*(-2)) = -6/-4 = 1.5$$. The y-coordinate of the vertex is $$-2(1.5)^2 + 6(1.5) - 6 = -2(2.25) + 9 - 6 = -4.5 + 9 - 6 = -1.5$$. This vertex is in the fourth quadrant, so this is not graph B. Formula 2, $$y = -2x^2 - 6x - 6$$, has its vertex at $$x = -(-6)/(2*(-2)) = 6/-4 = -1.5$$. The y-coordinate of the vertex is $$-2(-1.5)^2 - 6(-1.5) - 6 = -2(2.25) + 9 - 6 = -4.5 + 9 - 6 = -1.5$$. This vertex is in the third quadrant, so this is not graph B. Re-examining graph B, the vertex is at approximately (1.5, -1.5) indicating a downward opening parabola. Let's re-calculate the vertex for formula 1: $$y = -2x^2 + 6x - 6$$. Vertex x-coordinate: $$x = -b/(2a) = -6/(2 imes -2) = -6/-4 = 1.5$$. Vertex y-coordinate: $$y = -2(1.5)^2 + 6(1.5) - 6 = -2(2.25) + 9 - 6 = -4.5 + 9 - 6 = -1.5$$. This matches graph B.

Answers:

• A) corresponds to formula 4) $$y = 2x^2 - 6x + 6$$
• Б) corresponds to formula 3) $$y = 2x^2 + 6x + 6$$
• B) corresponds to formula 1) $$y = -2x^2 + 6x - 6$$