Transformations of Quadratic Functions Standard Form
The standard form is useful for determining how the graph is transformed from the graph of [latex]y={x}^{2}[/latex].
[latex]f\left(x\right)=a{\left(x-h\right)}^{2}+k[/latex]
where [latex]\left(h,\text{ }k\right)[/latex] is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the vertex form of a quadratic function.
graph transformations
Transforming a graph involves shifting, stretching, or flipping its shape to create a new representation.
Shift Up and Down by Changing the Value of [latex]k[/latex]
vertical shift of a parabola
You can represent a vertical (up, down) shift of the graph of [latex]f(x)=x^2[/latex] by adding or subtracting a constant, [latex]k[/latex].
[latex]f(x)=x^2 + k[/latex]
If [latex]k>0[/latex], the graph shifts upward, whereas if [latex]k<0[/latex], the graph shifts downward.
Shift left and right by changing the value of [latex]h[/latex]
horizontal shift of a parabola
You can represent a horizontal (left, right) shift of the graph of [latex]f(x)=x^2[/latex] by adding or subtracting a constant, [latex]h[/latex], to the variable [latex]x[/latex], before squaring.
[latex]f(x)=(x-h)^2[/latex]
If [latex]h>0[/latex], the graph shifts toward the right and if [latex]h<0[/latex], the graph shifts to the left.
[latex]\\[/latex]
If [latex]h>0[/latex], we have [latex]f(x)=(x-h)^2 +k[/latex]. You’ll see the negative sign, but the graph will shift right.
[latex]\\[/latex]
If [latex]h<0[/latex], we have [latex]f(x)=(x-(-h))^2 +k \rightarrow f(x)=(x+h)^2+k[/latex]. You’ll see the positive sign, but the graph will shift left.
Stretch or compress by changing the value of [latex]a[/latex].
stretch or compress a parabola
You can represent a stretch or compression (narrowing, widening) of the graph of [latex]f(x)=x^2[/latex] by multiplying the squared variable by a constant, [latex]a[/latex].
[latex]f(x)=ax^2[/latex]
The magnitude of [latex]a[/latex] indicates the stretch of the graph.
If [latex]|a|>1[/latex], the point associated with a particular [latex]x[/latex]-value shifts farther from the [latex]x[/latex]–axis, so the graph appears to become narrower, and there is a vertical stretch.
But if [latex]|a|<1[/latex], the point associated with a particular [latex]x[/latex]-value shifts closer to the [latex]x[/latex]–axis, so the graph appears to become wider, but in fact there is a vertical compression.