Certainly! Let's work through the transformations step-by-step to find the required forms of the function [tex]\( j(x) \)[/tex].
### Given:
The original function [tex]\( f(x) = x^2 \)[/tex].
### Step 1: Reflect over the x-axis
To reflect [tex]\( f(x) = x^2 \)[/tex] over the x-axis, we multiply the function by [tex]\(-1\)[/tex]. This transformation gives us:
[tex]\[ j_1(x) = -x^2 \][/tex]
### Step 2: Translate 1 unit down
Next, we need to translate the reflected function down by 1 unit. This involves subtracting 1 from the function:
[tex]\[ j(x) = -x^2 - 1 \][/tex]
Now, we need to express this in two forms:
1. Vertex Form:
The vertex form of a quadratic equation is given by:
[tex]\[ j(x) = a(x - h)^2 + k \][/tex]
For the function [tex]\( j(x) = -x^2 - 1 \)[/tex], the vertex form will look like:
[tex]\[ j(x) = -(x - 0)^2 - 1 \][/tex]
Simplifying:
[tex]\[ j(x) = -(x^2) - 1 \][/tex]
So, the vertex form of [tex]\( j(x) \)[/tex] is:
[tex]\[ j(x) = -(x^2) - 1 \][/tex]
2. Standard Form:
The standard form of a quadratic equation is given by:
[tex]\[ j(x) = ax^2 + bx + c \][/tex]
For the function [tex]\( j(x) = -x^2 - 1 \)[/tex], it is already in this form, where [tex]\( a = -1 \)[/tex], [tex]\( b = 0 \)[/tex], and [tex]\( c = -1 \)[/tex]. Specifically:
[tex]\[ j(x) = -x^2 - 1 \][/tex]
### Summary:
- Vertex form: [tex]\( j(x) = -(x^2) - 1 \)[/tex]
- Standard form: [tex]\( j(x) = -x^2 - 1 \)[/tex]
So, we have:
[tex]\[ \text{Vertex form: } j(x) = -(x^2) - 1 \][/tex]
[tex]\[ \text{Standard form: } j(x) = -x^2 - 1 \][/tex]
