Which equation represents the graphed function? –3x 2 = y –x 2 = y x – 3 = y 2x – 3 = y
The Correct Answer and Explanation is:
To determine which equation represents the graphed function, we first need to understand the general form of each equation and how it relates to the graph. Each equation describes a linear or quadratic relationship between (x) and (y). Let’s analyze each option one by one:
Option 1: (-3x^2 = y)
This equation is quadratic, meaning it represents a parabolic graph. The negative coefficient ((-3)) in front of (x^2) indicates that the parabola opens downward, and the graph will be symmetric along the y-axis. The vertex of the parabola will be at the origin ((0, 0)) because there are no linear or constant terms. This type of graph typically has a “frown” shape, where the arms of the parabola open downward.
Option 2: (-x^2 = y)
Similar to the first equation, this is also a quadratic equation, but with a coefficient of (-1) instead of (-3). This means the graph will still be a downward-opening parabola, but the “width” of the parabola will be different. Specifically, the graph will be wider than the one represented by (-3x^2 = y), because the coefficient is smaller in magnitude. It will also have its vertex at the origin ((0, 0)).
Option 3: (x – 3 = y)
This is a linear equation in slope-intercept form. Rearranging it gives (y = x – 3). The equation represents a straight line with a slope of 1 and a y-intercept of (-3). This means the line will cross the y-axis at ((0, -3)) and will rise 1 unit for every 1 unit it moves to the right.
Option 4: (2x – 3 = y)
This is another linear equation in slope-intercept form. Rearranging it gives (y = 2x – 3). This equation represents a straight line with a slope of 2 and a y-intercept of (-3). The line will be steeper than the previous one (slope = 1) and will also cross the y-axis at ((0, -3)).
Conclusion
To determine which equation best represents the graph, we need to consider the shape of the graph. If the graph is parabolic (either opening upward or downward), options 1 and 2 are likely the correct choices. However, if the graph is a straight line, options 3 and 4 are more appropriate. Based on the description provided, the correct answer depends on the specific graph. If the graph is parabolic, then (-3x^2 = y) (option 1) or (-x^2 = y) (option 2) are correct, and if the graph is linear, then (x – 3 = y) (option 3) or (2x – 3 = y) (option 4) are correct.
Given the information provided, option 3 or option 4 would represent a linear graph, while option 1 or option 2 would represent a parabolic graph.