Explain how the graph of the given function is a transformation of the graph of y=x^2
*y=(x+5)^2
To understand how the graph of y = (x + 5)^2 is a transformation of the graph of y = x^2, we first need to understand the concept of transformations in graphs.
A transformation is a change made to the equation of a function that affects its graph. In this case, we have a quadratic function, y = x^2, which represents a standard parabola centered at the origin. The given function, y = (x + 5)^2, represents a modified parabola that has been shifted 5 units to the left along the x-axis.
To visualize the transformation, we can compare the key characteristics of the original function, y = x^2, with the transformed function, y = (x + 5)^2:
1. Vertex: The vertex of the standard parabola y = x^2 is at the origin (0, 0). However, the vertex of the transformed parabola y = (x + 5)^2 is shifted 5 units to the left, resulting in a new vertex at (-5, 0).
2. Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex of the parabola. For the standard parabola y = x^2, the axis of symmetry is the y-axis. However, for the transformed parabola y = (x + 5)^2, the axis of symmetry is shifted 5 units to the left, resulting in a new axis of symmetry at x = -5.
3. Y-intercept: The y-intercept is the point where the graph intersects the y-axis. For both the standard parabola y = x^2 and the transformed parabola y = (x + 5)^2, the y-intercept remains the same, which is (0, 0).
4. Shape: The shape of the standard parabola y = x^2 is a symmetric U-shape. The transformed parabola y = (x + 5)^2 retains the same U-shape, but it is shifted to the left due to the addition of the constant term (+5).
By analyzing the key characteristics mentioned above, we can see that the graph of y = (x + 5)^2 is a transformation of the graph of y = x^2 where the entire graph is shifted to the left by 5 units.