1. The graph of the equation y = x2 – 3 is symmetric with respect to which of the following? The answer is the y-axis . why?
2.The graph of an odd function is symmetric with respect to which of the following? the origin is the answer, why ?
1. symmetric about the y-axis because f(-x) = f(x).
That is, if you pick a point on the graph, flip it about the y-axis, you land on another point on the graph.
2. odd function means f(-x) = -f(x)
Same thinking as above, but (x,y) is matched by (-x,-y). Join the points, and the midpoint of the line is (0,0).
1. The graph of the equation y = x^2 - 3 is symmetric with respect to the y-axis because replacing x with -x in the equation does not change the value of y.
If we substitute -x for x in the equation, we get y = (-x)^2 - 3, which simplifies to y = x^2 - 3. This means that for every point (x, y) on the graph, there is a corresponding point (-x, y) that is also on the graph.
Since the y-axis is the vertical line at x = 0, the graph is unchanged when we reflect it across the y-axis. This symmetry results in a graph that is symmetric with respect to the y-axis.
2. The graph of an odd function is symmetric with respect to the origin because replacing x with -x in the equation changes the sign of y.
In an odd function, replacing x with -x in the equation gives us a new equation with the opposite sign for y. This can be represented as f(-x) = -f(x).
For example, if we have the function f(x) = x^3, replacing x with -x gives us f(-x) = (-x)^3 = -x^3.
This means that for every point (x, y) on the graph, there is a corresponding point (-x, -y) that is also on the graph. The point (0,0) is the origin, and when we reflect the graph across the origin, the x and y values change sign, resulting in a graph that is symmetric with respect to the origin.
1. To determine the symmetry of the graph of the equation y = x^2 - 3, we can substitute -x for x in the equation and see if we get an equivalent equation.
Let's substitute -x for x in the equation:
y = (-x)^2 - 3
Simplifying, we get:
y = x^2 - 3
Since the equation remains the same, we can conclude that the graph is symmetric with respect to the y-axis. This means that if you were to fold the graph along the y-axis, the two sides would coincide.
2. An odd function is a function that satisfies the property f(-x) = -f(x) for all x in its domain.
To determine the symmetry of the graph of an odd function, we need to check if substituting -x for x in the function gives us a reflection across a specific axis.
Let's substitute -x for x in the equation:
f(-x) = -f(x)
If the equation holds true, then the graph is symmetric with respect to the y-axis. However, if the equation holds true and we can also reflect the graph across the origin (0,0), then the graph is also symmetric with respect to the origin.
For an odd function, when we substitute -x for x, we get:
f(-x) = -f(x)
This means that if we reflect the graph across the y-axis, it will look the same. But if we also reflect it across the origin (0,0), the graph will still look the same. Therefore, the graph of an odd function is symmetric with respect to the origin.