Analyzing Y=7x^2: A Step-by-Step Guide

In this article, we'll break down the quadratic equation y = 7x². Our goal is to identify the coefficients a, b, and c, determine the parabola's direction (whether it opens upwards or downwards), find the y-intercept, and then assess which table best represents these characteristics. Understanding these elements is crucial for grasping the behavior of quadratic functions.

Identifying Coefficients a, b, and c

Let's begin by expressing the given quadratic equation, y = 7x², in its standard form: y = ax² + bx + c. This standard form allows us to easily identify the coefficients a, b, and c. By comparing y = 7x² with the standard form, we can deduce the following:

• a: The coefficient of the x² term. In our equation, a = 7.
• b: The coefficient of the x term. Since there is no x term in the equation, b = 0.
• c: The constant term. Since there is no constant term in the equation, c = 0.

Therefore, we have a = 7, b = 0, and c = 0. These values are fundamental in understanding the properties of the parabola represented by the equation. Knowing these coefficients helps us determine the shape, position, and orientation of the parabola on the coordinate plane.

Determining the Direction of the Parabola

The coefficient 'a' plays a crucial role in determining whether the parabola opens upwards or downwards. In the quadratic equation y = ax² + bx + c, the sign of 'a' dictates the direction of the parabola:

• If a > 0, the parabola opens upwards, meaning it has a minimum value.
• If a < 0, the parabola opens downwards, meaning it has a maximum value.

In our equation, y = 7x², the value of a is 7, which is greater than 0. Therefore, the parabola opens upwards. This means the vertex of the parabola represents the minimum point on the graph. Understanding the direction helps visualize the overall shape and behavior of the quadratic function, especially when analyzing its graph and applications in various fields such as physics and engineering.

Finding the Y-Intercept

The y-intercept is the point where the parabola intersects the y-axis. To find the y-intercept, we set x = 0 in the equation and solve for y. In the equation y = 7x², substituting x = 0 gives us:

• y = 7(0)² = 7 * 0 = 0

Thus, the y-intercept is 0. This means the parabola passes through the origin (0, 0) on the coordinate plane. The y-intercept is a significant point on the graph as it indicates where the function's value is when the input (x) is zero. Knowing the y-intercept provides a reference point for plotting the graph and understanding the function's behavior near the y-axis.

Analyzing Table A

Let's consider a hypothetical Table A and assess if it accurately illustrates the properties we've determined for the equation y = 7x²:

Table A

In this example, Table A correctly identifies the values of a, b, and c as 7, 0, and 0, respectively. It also correctly states that the parabola opens upwards and that the y-intercept is 0. Therefore, Table A accurately represents the characteristics of the equation y = 7x². Such a table is a useful tool for summarizing the key features of a quadratic function, making it easier to analyze and compare different equations.

Conclusion

In summary, for the equation y = 7x², we found that a = 7, b = 0, and c = 0. The parabola opens upwards, and the y-intercept is 0. A table that accurately reflects these values would be the best representation of the equation's characteristics. Understanding these components helps in visualizing and analyzing quadratic functions effectively.

For further information on quadratic functions, you can visit Khan Academy's resource on quadratics.