Understanding the Graph of y2x2-8x-6: Intercepts, Line of Symmetry, and Turning Point

This article delves into the key features of the cubic function y2x2-8x-6. We will discuss the y-intercept, x-intercepts, line of symmetry, and the turning point using symbolic analysis. Understanding these features helps in accurately sketching the graph and interpreting the function's behavior. Let's begin with the basic setup of the function:

Introduction to the Function

The given cubic function is:

y 2x2 - 8x - 6

Graph Analysis: Intercepts

Y-intercept: The y-intercept is the point where the graph crosses the y-axis, which occurs when x 0. Substituting x 0 into the equation:

y 2(0)2 - 8(0) - 6 -6

Thus, the y-intercept is at (0, -6).

Graph Analysis: X-intercepts

The x-intercepts are the points where the graph crosses the x-axis, which occurs when y 0. Setting y 0 in the equation and solving for x:

0 2x2 - 8x - 6

Factoring the equation, we get:

0 2x(x - 3) - 2(x - 3) (2x - 2)(x - 3)

Solving for x:

2x - 2 0 rarr; x 1 x - 3 0 rarr; x 3

Thus, the x-intercepts are at (1, 0) and (3, 0).

Graph Analysis: Line of Symmetry and Turning Point

The line of symmetry of a quadratic function (the highest degree term is 2) is the vertical line passing through the vertex. To find the line of symmetry, we need to determine the vertex (or turning point).

Vertex Formula: For a quadratic function in the form ax2 bx c, the x-coordinate of the vertex is given by -b/(2a).

Here, a 2, b -8.

x-coordinate of the vertex: x -(-8)/(2 * 2) 8/4 2

Substituting x 2 into the function to find the y-coordinate:

y 2(2)2 - 8(2) - 6 2(4) - 8(2) - 6 8 - 16 - 6 -14

Thus, the vertex or turning point is (2, -6), and the line of symmetry is x 2.

Graph Synthesis

To summarize, the key features of the graph of y 2x2 - 8x - 6 are:

y-intercept: (0, -6) x-intercepts: (1, 0) and (3, 0) Turning point: (2, -6) Line of Symmetry: x 2

These points help in accurately sketching the graph and analyzing the behavior of the function.

Conclusion

Understanding the y-intercept, x-intercepts, turning point, and line of symmetry of a quadratic function is crucial for accurate graphing and analysis. In this article, we have provided a detailed, step-by-step guide to finding these key features of the function y 2x2 - 8x - 6.

If you have any further questions or need a more detailed explanation, feel free to ask!