Describe how the graph of y = -x^2 + 1 differs from
y = x^2. Then find the axis of symmetry and the
vertex.
-x^2 goes down as x gets big plus or - , so the y-1 = x^2 is upside down from the y = x^2 curve (sheds water instead of holding water).
the vertex of y = x^2 is at (0,0) so the vertex of y-1 = -x^2 is at (0,1)
-x^2 is the same both sides of the y axis so the y axis is the axis of symmetry.
Well, if we compare the two equations, y = -x^2 + 1 and y = x^2, we can see that they have the same shape, but they are mirrored along the x-axis. It's like they're opposites, just like your cat and a dog.
Now, to find the axis of symmetry and the vertex, we need to use our mathematical tools. The axis of symmetry is a line that divides the parabola into two symmetric parts. In both equations, the axis of symmetry is the y-axis because the x term is squared, and squaring a negative or positive number gives you the same value.
As for the vertex, that's the point where the parabola reaches its highest or lowest point. In the equation y = -x^2 + 1, the vertex is (0, 1), and in the equation y = x^2, the vertex is (0, 0). So, it seems like they both have a little party at the origin, but one is happier than the other.
In summary, the graphs of y = -x^2 + 1 and y = x^2 are symmetrical but opposite to each other along the x-axis. The axis of symmetry for both is the y-axis, and the vertices are (0, 1) for y = -x^2 + 1 and (0, 0) for y = x^2. They're like mirror images, but instead of your reflection, you get quadratic equations. Quite the trade-off!
To describe how the graph of y = -x^2 + 1 differs from y = x^2, let's first compare their equations.
The equation y = x^2 represents a standard upward-opening parabola, where the coefficient of x^2 is positive. This means the graph will be in the shape of a "U."
On the other hand, the equation y = -x^2 + 1 represents a downward-opening parabola, where the coefficient of x^2 is negative. This means the graph will be in the shape of an upside-down "U" or a "n" shape.
Now, let's find the axis of symmetry and the vertex of these two parabolas.
The axis of symmetry is a vertical line that divides the parabola into two equal halves. It is represented by the equation x = -b/2a. In both cases, the coefficient of x is zero, meaning b = 0. Therefore, the axis of symmetry for both parabolas is x = 0.
To find the vertex, we can substitute the value of x = 0 into the equations and solve for y.
For y = x^2:
Substituting x = 0, we get y = 0^2 = 0. So, the vertex is (0, 0).
For y = -x^2 + 1:
Substituting x = 0, we get y = -(0^2) + 1 = 1. So, the vertex is (0, 1).
In summary, the difference between the two graphs is that the first equation, y = x^2, represents an upward-opening parabola, while the second equation, y = -x^2 + 1, represents a downward-opening parabola. The axis of symmetry for both is x = 0, and the vertex for the first equation is (0, 0) and for the second equation is (0, 1).
To describe the differences between the graphs y = -x^2 + 1 and y = x^2, let's start by understanding their basic shapes.
The graph of y = x^2 is a parabola that opens upwards, while the graph of y = -x^2 + 1 is a parabola that opens downwards.
Now, let's find the axis of symmetry and the vertex for each equation.
1. Axis of Symmetry and Vertex for the equation y = x^2:
The general equation of a parabola can be written as y = a(x-h)^2 + k, where (h, k) represents the vertex.
In the equation y = x^2, we can see that a = 1, so the vertex form becomes y = 1(x-0)^2 + 0. Therefore, the vertex of this parabola is (0, 0), and the axis of symmetry is the vertical line x = 0.
2. Axis of Symmetry and Vertex for the equation y = -x^2 + 1:
Similarly, we can rearrange y = -x^2 + 1 to the vertex form: y = -1(x-0)^2 + 1. In this equation, a = -1, so the vertex is (0, 1), and the axis of symmetry is the vertical line x = 0.
In summary, the main difference between the two graphs is their orientation. The graph of y = x^2 opens upwards, while the graph of y = -x^2 + 1 opens downwards. The axis of symmetry for both graphs is x = 0, but the vertex of y = x^2 is (0, 0), while the vertex of y = -x^2 + 1 is (0, 1).