Let's analyze the given parabola step by step.
The given equation of the parabola is:
[tex]\[ f(x) = 9x^2 - 1 \][/tex]
### Part A: Open Up or Down?
To determine if the parabola opens up or down, we look at the coefficient of [tex]\(x^2\)[/tex].
1. If the coefficient of [tex]\(x^2\)[/tex] (denoted as [tex]\(a\)[/tex]) is positive ([tex]\(a > 0\)[/tex]), the parabola opens up.
2. If the coefficient of [tex]\(x^2\)[/tex] (denoted as [tex]\(a\)[/tex]) is negative ([tex]\(a < 0\)[/tex]), the parabola opens down.
In our equation, [tex]\(a = 9\)[/tex], which is positive. Hence, the parabola opens up.
#### Answer:
- The graph of the resulting parabola opens up.
### Part B: The Vertex
The vertex form of a quadratic function [tex]\(f(x) = ax^2 + bx + c\)[/tex] can be determined using the formula for the vertex:
[tex]\[ x = -\frac{b}{2a} \][/tex]
To find the y-coordinate of the vertex, substitute [tex]\(x\)[/tex] back into the equation [tex]\(f(x)\)[/tex].
1. Identify coefficients: [tex]\(a = 9\)[/tex], [tex]\(b = 0\)[/tex], and [tex]\(c = -1\)[/tex].
2. Calculate the x-coordinate of the vertex:
[tex]\[ x = -\frac{b}{2a} = -\frac{0}{2 \cdot 9} = 0 \][/tex]
3. Calculate the y-coordinate by substituting [tex]\(x = 0\)[/tex] into the equation [tex]\(f(x)\)[/tex]:
[tex]\[ f(0) = 9(0)^2 - 1 = -1 \][/tex]
The vertex is at the point [tex]\((0, -1)\)[/tex].
#### Answer:
- The vertex is at the point [tex]\((0, -1)\)[/tex].
### Part C: The Y-Intercept
The y-intercept of a parabola is the point where the graph crosses the y-axis. This occurs when [tex]\(x = 0\)[/tex].
1. Substitute [tex]\(x = 0\)[/tex] in the equation [tex]\(f(x)\)[/tex]:
[tex]\[ f(0) = 9(0)^2 - 1 = -1 \][/tex]
Therefore, the y-intercept of the parabola is the point [tex]\((0, -1)\)[/tex].
#### Answer:
- The y-intercept is the point [tex]\((0, -1)\)[/tex].
### Summary:
A) The graph of the resulting parabola opens up.
B) The vertex is at the point [tex]\((0, -1)\)[/tex].
C) The y-intercept is the point [tex]\((0, -1)\)[/tex].
