Transformations of Quadratic Functions

The graph of the quadratic function $f(x)=\text{-}{x}^2+2x+1$ is drawn on the coordinate plane. By applying transformations to the parabola that corresponds to $f,$ draw the graph of $g(x)=\text{-}{x}^2+4x.$

A reflection of a function is a transformation that flips a graph over a line called the line of reflection. A reflection in the $x\text{-}$axis is achieved by changing the sign of every output value of the function rule. In other words, the sign of the $y\text{-}$coordinate of every point on the graph of a function should be changed. Consider the quadratic parent function $y=x^2.$ The reflection of the corresponding parabola can be shown on a coordinate plane. A reflection in the $y\text{-}$axis is instead achieved by changing the sign of every input value. However, since $(\text{-}x{)}^2$ is equivalent to $x^2,$ reflecting $y=x^2$ in the $y\text{-}$axis does not change the graph. For this reason, the reflection of the graph of another quadratic function will be shown. This transformation can also be shown on a coordinate plane.

In the coordinate plane, the parabola that corresponds to the quadratic function $y=af(bx)$ can be seen. Observe how the graph is vertically and horizontally stretched and shrunk by changing the values of $a$ and $b.$

A function graph is vertically stretched or shrunk by multiplying the output of a function rule by some constant $a,$ where $a>0.$ This constant must be positive, otherwise a reflection is involved. Consider the quadratic function $y=x^2+1.$ If $a$ is greater than $1,$ the parabola is vertically stretched by a factor of $a.$ Conversely, if $a$ is less than $1,$ the graph is vertically shrunk by a factor of $a.$ If $a=1,$ then there is no stretch nor shrink. Here, the $y\text{-}$coordinates of all points on the graph are multiplied by the factor $a.$ Similarly, a function graph is horizontally stretched or shrunk by multiplying the input of a function rule by some constant $b,$ where $b>0.$ Again, the constant must be positive because if it was negative, a reflection would be required. In this case, if $b$ is greater than $1,$ the graph is horizontally shrunk by a factor of $b.$ Conversely, if $b$ is less than $1,$ the graph is horizontally stretched by a factor of $b.$ If $b=1,$ then there is neither a stretch nor shrink of the graph. Here, the $x\text{-}$coordinates of all points on the graph are multiplied by the factor $\frac{1}{b}.$ The information about vertical and horizontal stretches and shrinks of the graph of a function $f$ can be summed up in a table.

	Vertical	Horizontal
Stretch	$af(x),$ with $a>1$	$f(ax),$ with $0<a<1$
Shrink	$af(x),$ with $0<a<1$	$f(ax),$ with $a>1$

The graph of the quadratic function $y=x^2$ is shown in the coordinate plane. The graph of a horizontal or vertical stretch or shrink is also shown.

In the coordinate plane, the graph of the quadratic function $y=(x-h{)}^2+k$ can be seen. Observe how the graph is horizontally and vertically translated by changing the values of $h$ and $k.$

A translation of a function is a transformation that shifts a graph vertically or horizontally. A vertical translation is achieved by adding some number to every output value of a function rule. Consider the quadratic function $y=x^2.$ If $k$ is a positive number, the translation is performed upwards. Conversely, if $k$ is negative, the translation is performed downwards. If $k=0,$ then there is no translation. This transformation can be shown on a coordinate plane. A horizontal translation is instead achieved by subtracting a number from every input value. In this case, if $h$ is a positive number, the translation is performed to the right. Conversely, if $h$ is negative, the translation is performed to the left. If $h=0,$ then there is no translation. It is worth noting that since $h$ is subtracted from $x,$ if $h$ is positive, then a number is subtracted from $x.$ On the other hand, if $h$ is negative, a number is added to the variable $x.$ The vertical and horizontal translations of the graph of a function $f$ can be summarized in a table.

Translation
Vertical	Horizontal
Upwards $f(x)+k,$ with $k>0$	To the Right $f(x-h),$ with $h>0$
Downwards $f(x)+k,$ with $k<0$	To the Left $f(x-h),$ with $h<0$

The graph of the quadratic function $y=x^2$ and a vertical or horizontal translation are shown in the coordinate plane.