To determine one of the transformations applied to the graph of [tex]\( f(x) = x^2 \)[/tex] to produce the graph of [tex]\( g(x) = 2x^2 - 28x + 3 \)[/tex], let's go through the process of converting the equation [tex]\( g(x) \)[/tex] to its vertex form, which will help us identify the transformation.
### Step-by-Step Transformation:
1. Given function: [tex]\( g(x) = 2x^2 - 28x + 3 \)[/tex].
2. Standard form of a quadratic function:
[tex]\( g(x) = a(x - h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
3. Identifying coefficients:
From [tex]\( g(x) = 2x^2 - 28x + 3 \)[/tex], we have [tex]\( a = 2 \)[/tex], [tex]\( b = -28 \)[/tex], and [tex]\( c = 3 \)[/tex].
4. Factor out the leading coefficient [tex]\( a \)[/tex] from the first two terms:
[tex]\( g(x) = 2(x^2 - 14x) + 3 \)[/tex].
5. Complete the square inside the parenthesis:
- Take half of the coefficient of [tex]\( x \)[/tex] inside the parenthesis, square it, and add and subtract this value inside the parenthesis.
- [tex]\(\left( \frac{-14}{2} \right)^2 = (-7)^2 = 49 \)[/tex].
- Add and subtract 49 inside the parenthesis:
[tex]\[
g(x) = 2(x^2 - 14x + 49 - 49) + 3 = 2((x - 7)^2 - 49) + 3.
\][/tex]
6. Simplify the expression:
[tex]\[
g(x) = 2(x - 7)^2 - 98 + 3 = 2(x - 7)^2 - 95.
\][/tex]
### Vertex Form:
Thus, the vertex form of [tex]\( g(x) \)[/tex] is:
[tex]\[ g(x) = 2(x - 7)^2 - 95. \][/tex]
### Interpretation of Transformations:
From the vertex form [tex]\( g(x) = 2(x - 7)^2 - 95 \)[/tex], we can identify the following transformations applied to the graph of [tex]\( f(x) = x^2 \)[/tex]:
1. Horizontal shift: The term [tex]\( (x - 7) \)[/tex] indicates a shift to the right by 7 units.
2. Vertical shift: The term [tex]\(- 95\)[/tex] indicates a shift down by 95 units.
3. Vertical scaling: The coefficient [tex]\( 2 \)[/tex] indicates a vertical stretch by a factor of 2.
In this case, we are only concerned about the horizontal shift, which is one of the transformations applied. Therefore, one of the transformations is:
[tex]\[\boxed{\text{shifted right 7 units}}\][/tex]
