To analyze and describe the equation [tex]\( y = x^2 - 1 \)[/tex], let's break it down step-by-step:
1. Form of the Equation:
The equation [tex]\( y = x^2 - 1 \)[/tex] is in the standard form of a quadratic function, which is typically written as [tex]\( y = ax^2 + bx + c \)[/tex]. In this specific case, [tex]\( a = 1 \)[/tex], [tex]\( b = 0 \)[/tex], and [tex]\( c = -1 \)[/tex].
2. Shape of the Graph:
Since this is a quadratic equation with a positive leading coefficient ([tex]\( a = 1 \)[/tex]), the graph of the equation is a parabola that opens upwards.
3. Vertex of the Parabola:
The vertex form of a parabola [tex]\( y = a(x - h)^2 + k \)[/tex] helps in identifying the vertex. For the given equation:
[tex]\[ y = x^2 - 1 \][/tex]
We can see that it is already in vertex form where [tex]\( h = 0 \)[/tex] and [tex]\( k = -1 \)[/tex]. Thus, the vertex of the parabola is at the point (0, -1).
4. X-Intercepts:
To find the x-intercepts, we set [tex]\( y \)[/tex] to 0 and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = x^2 - 1 \][/tex]
[tex]\[ x^2 - 1 = 0 \][/tex]
[tex]\[ x^2 = 1 \][/tex]
[tex]\[ x = \pm 1 \][/tex]
Therefore, the x-intercepts are [tex]\( x = -1 \)[/tex] and [tex]\( x = 1 \)[/tex].
5. Y-Intercept:
To find the y-intercept, we set [tex]\( x \)[/tex] to 0 and solve for [tex]\( y \)[/tex]:
[tex]\[ y = 0^2 - 1 \][/tex]
[tex]\[ y = -1 \][/tex]
Thus, the y-intercept is [tex]\( (0, -1) \)[/tex].
6. Symmetry:
The parabola [tex]\( y = x^2 - 1 \)[/tex] is symmetric about the y-axis. This is because for every point [tex]\((x, y)\)[/tex] on the parabola, the point [tex]\((-x, y)\)[/tex] will also lie on the parabola.
In summary, the equation [tex]\( y = x^2 - 1 \)[/tex] describes an upward-opening parabola with its vertex at (0, -1). The x-intercepts are [tex]\( (-1, 0) \)[/tex] and [tex]\( (1, 0) \)[/tex], and the y-intercept is [tex]\( (0, -1) \)[/tex]. The parabola is symmetric about the y-axis.
