Consider the graphs of y=x^2 and y=x^2-6
Which graph would be narrower? Why?
The graph of y = x^2 would be narrower than the graph of y = x^2 - 6.
This is because the graph of y = x^2-6 is obtained by shifting the graph of y = x^2 six units downward. Shifting a graph downward does not affect its width but shifts all the points downwards.
Therefore, the graph of y = x^2 would be narrower because it is not shifted, and its points are closer together horizontally compared to the graph of y = x^2-6.
To determine which graph is narrower, we need to understand what "narrower" means. In this context, a graph is narrower if it is stretched less horizontally.
Both equations y = x^2 and y = x^2 - 6 have the same basic shape, which is a parabola. The standard form of a parabola is y = a(x - h)^2 + k, where (h,k) is the vertex of the parabola.
By comparing the standard form of the equations y = x^2 and y = x^2 - 6, we find that the vertex of the first equation is at (0,0) since h = 0 and k = 0, and the vertex of the second equation is at (0,-6) since h = 0 and k = -6.
The vertex represents the "center" of the parabola – the point where it is most stretched or compressed. In this case, since the vertex of the second equation is farther down the y-axis (at (0,-6)), the graph of y = x^2 - 6 is narrower than y = x^2.
The reason why it is narrower is that the vertex represents the maximum or minimum point of the parabola. By shifting the vertex downward, we are effectively compressing the parabola vertically, making it narrower.
In conclusion, the graph of y = x^2 - 6 is narrower than the graph of y = x^2 because its vertex is shifted downward.
To determine which graph is narrower, we need to compare the coefficients of the x^2 terms in both equations.
The equation y=x^2 represents a standard parabola with a coefficient of 1 for the x^2 term. This means that the graph will be wider compared to a parabola with a smaller coefficient.
On the other hand, the equation y=x^2-6 represents a parabola with a coefficient of 1 for the x^2 term as well, but with the additional constant term of -6.
Since the coefficient of the x^2 term remains the same in both equations, the constant term (-6) in the second equation causes a vertical shift of the graph downwards by 6 units. However, it does not affect the width of the parabola.
Therefore, both graphs have the same width or narrowness.