Exploring the Graph of f(x) = x2 – b2: Key Aspects Revealed
The function f(x) = x2 – b2 may seem simple at first glance, but exploring its graph can reveal key aspects that help us understand its behavior.
Firstly, we notice that the graph is symmetric with respect to the y-axis. This means that for any point (x,y) on the graph, the point (-x,y) is also on the graph. This is because when we square x, the negative value is eliminated, leaving only positive values. The negative b2 term only shifts the graph vertically, not horizontally.
Secondly, we see that the graph intersects the x-axis at x = ±b. This means that when x = b or x = -b, f(x) = 0. These are known as the x-intercepts of the graph.
Thirdly, we notice that the vertex of the parabola is located at the origin (0,0). This is because when x = 0, f(x) = -b2. As we move away from the vertex in either direction, the value of f(x) increases.
Finally, we can examine the concavity of the graph by analyzing the sign of the second derivative, f”(x). We know that f”(x) = 2 for all values of x, which means that the graph is always concave up. In other words, the rate of change of the slope of the tangent lines is increasing as we move along the graph from left to right.
In summary, the graph of f(x) = x2 – b2 has several key aspects that are revealed through exploration. Its symmetry, x-intercepts, vertex, and concavity all play a role in understanding its behavior. By studying these features, we can gain a deeper appreciation for the beauty and complexity of mathematical functions.