Question

24/25
choose the graph for
y = -2x² + 4x - 6
(show your work on paper)

Explanation:

Step1: Analyze the parabola's direction

The equation is \( y = -2x^2 + 4x - 6 \). The coefficient of \( x^2 \) is \(-2\), which is negative, so the parabola opens downward.

Step2: Find the vertex's x - coordinate

The formula for the x - coordinate of the vertex of a parabola \( y = ax^2+bx + c \) is \( x=-\frac{b}{2a} \). Here, \( a=-2 \), \( b = 4 \), so \( x=-\frac{4}{2\times(-2)}=-\frac{4}{-4} = 1 \).

Step3: Find the vertex's y - coordinate

Substitute \( x = 1 \) into the equation: \( y=-2(1)^2+4(1)-6=-2 + 4-6=-4 \). So the vertex is at \( (1,-4) \).

Step4: Analyze the y - intercept

The y - intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into the equation: \( y=-2(0)^2+4(0)-6=-6 \). So the y - intercept is \( (0,-6) \).
Now, let's analyze the graphs:

• Graph A: Opens downward (matches the negative coefficient of \( x^2 \)), vertex around \( (1,-4) \), and y - intercept around \( (0,-6) \) (consistent with our calculations).
• Graph B: Opens upward (incorrect, since the coefficient of \( x^2 \) is negative).
• Graph C: Opens upward (incorrect).
• Graph D: The vertex and y - intercept do not match our calculations.

Answer:

A (assuming A is the graph that opens downward, has vertex at (1, - 4) and y - intercept at (0, - 6))