Question

choose the graph for $y = -2x^2 + 4x - 6$ (show your work on paper)

Explanation:

Step1: Analyze the quadratic function

The given function is \( y = -2x^2 + 4x - 6 \). For a quadratic function in the form \( y = ax^2 + bx + c \), the coefficient \( a=-2 \) which is negative, so the parabola opens downward.

Step2: Find the vertex

The x - coordinate of the vertex of a quadratic function \( y = ax^2+bx + c \) is given by \( x=-\frac{b}{2a} \). Here, \( a = - 2\) and \( b = 4\), so \( x=-\frac{4}{2\times(-2)}=-\frac{4}{-4} = 1 \).
Substitute \( x = 1 \) into the function: \( y=-2(1)^2+4(1)-6=-2 + 4-6=-4 \). So the vertex is at \( (1,-4) \).

Step3: Analyze the y - intercept

The y - intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into the function: \( y=-2(0)^2+4(0)-6=-6 \). So the y - intercept is \( (0,-6) \).

Step4: Match with the graphs

We know the parabola opens downward (since \( a=-2<0 \)), has vertex at \( (1, - 4) \) and y - intercept at \( (0,-6) \). We need to check which graph has a downward - opening parabola, vertex at \( (1,-4) \) and y - intercept at \( (0, - 6) \). Looking at the options, we analyze the direction of opening, vertex, and y - intercept. The graph with the correct features (downward opening, vertex at \( (1,-4) \), y - intercept at \( (0,-6) \)) should be identified. From the visual inspection (assuming the graphs are labeled and we check the direction, vertex, and intercepts), the correct graph is the one that matches these properties. Let's assume the correct graph is the one (for example, if we consider the options, let's say the graph labeled A or C or D or B, but from the analysis, since the parabola opens downward, vertex at \( (1,-4) \) and y - intercept at \( (0,-6) \), we can match it. Let's assume the correct graph is the one with the downward opening, vertex around \( (1,-4) \) and y - intercept at \( (0,-6) \). Let's say the correct graph is the one (for the sake of this problem, we will identify it based on the analysis).

Answer:

(Assuming the correct graph is, for example, the graph labeled A (if A has downward opening, vertex at (1, - 4) and y - intercept at (0, - 6)). But since the user's image is a bit unclear, but based on the analysis, the correct graph is the one with downward opening, vertex at (1, - 4) and y - intercept at (0, - 6). Let's say the answer is the graph labeled as the one that matches these criteria. If we assume the correct graph is the one in option A (depending on the actual graph), then the answer is A. (But the exact label depends on the visual of the graphs, but the process is as above))