Quadratic Functions: Learn It 4

Transformations of Quadratic Functions Standard Form

The standard form of a quadratic function presents the function in the form

[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.

The standard form is useful for determining how the graph is transformed from the graph of [latex]y={x}^{2}[/latex].

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.

Using an online graphing calculator, plot the function [latex]f(x)=x^2+k[/latex]. Now change the [latex]k[/latex] value to shift the graph down [latex]4[/latex] units, then up [latex]4[/latex] units.

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.

Remember that the negative sign inside the argument of the vertex form of a parabola (in the parentheses with the variable [latex]x[/latex] ) is part of the formula [latex]f(x)=(x-h)^2 +k[/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.

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.

Using an online graphing calculator, plot the function [latex]f(x)=(x-h)^2[/latex]. Now change the [latex]h[/latex] value to shift the graph [latex]2[/latex] units to the right, then [latex]2[/latex] units to the 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.

Using an online graphing calculator plot the function [latex]f(x)=ax^2[/latex]. Now adjust the [latex]a[/latex] value to create a graph that has been compressed vertically by a factor of [latex]\frac{1}{2}[/latex] and another that has been vertically stretched by a factor of [latex]3[/latex]. What are the equations of the two graphs?