Vertical Transformations Change Outputs, Not Inputs
- Vertical transformations modify the $y$-values (outputs) of a function while leaving the $x$-values (inputs) in the same places.
- Namely, it moves every point of a graph up or down by the same amount.
The Rule For $f(x)+k$
If the original graph is $y=f(x)$, then $$y=f(x)+k$$ is the graph of $f$ translated $k$ units in the $y$-direction.
- If $k>0$, the graph shifts up $k$ units.
- If $k<0$, the graph shifts down $|k|$ units.
Definition
Vertical translation
A transformation that adds a constant $k$ to a function so that every point moves up ($k>0$) or down ($k<0$) by the same number of units.
Point Mapping For Vertical Translations
- If $(x,y)$ lies on $y=f(x)$, then after the translation $y=f(x)+k$ the corresponding point is $(x,y+k)$.
- So the $x$-coordinate stays the same and only the $y$-value changes.
Example
- Suppose $f(2)=3$.
- Then the point $(2,3)$ is on $y=f(x)$.
- On $y=f(x)+4$, the point becomes $(2,7)$.
- On $y=f(x)-5$, the point becomes $(2,-2)$.
Translations Of Quadratics: Vertex Moves Clearly
- Quadratics are especially easy to translate if you use vertex form.
- A quadratic written as $$y=(x-h)^2+k$$ has vertex at $(h,k)$. In particular, the parent function $y=x^2$ has vertex $(0,0)$, so:
- $y=x^2+4$ shifts the vertex to $(0,4)$.
- $y=x^2-5$ shifts the vertex to $(0,-5)$.
- This is why it is often helpful to rewrite a quadratic in vertex form when identifying transformations.
Note
- In many problems you start with an expanded quadratic like $x^2+6x+4$.
- Completing the square rewrites it in vertex form so the translations are obvious: $$x^2+6x+4=(x+3)^2-5$$
- From this, you can read a horizontal translation left $3$ and a vertical translation down $5$.
- Horizontal translation will be covered in the next article.
Worked Examples With Linear And Quadratic Functions
Example
Transforming A Line
- Start with the parent line $y=x$.
- Find the transformations to get $y=2x-9$.
- Multiply by $2$: $y=2f(x)$ is a vertical dilation (stretch) by factor $2$.
- Subtract $9$: $y=2f(x)-9$ is a vertical translation down $9$.
- So $y=2x-9$ is a vertical stretch by $2$, then a downward shift by $9$.
Example
Reading Vertical Changes From Vertex Form
- Write $y=x^2+6x+4$ in vertex form: $$x^2+6x+4=(x+3)^2-5.$$
- Relative to $y=x^2$:
- $(x+3)^2$ indicates a horizontal translation left $3$.
- $-5$ indicates a vertical translation down $5$.
- So the vertex moves from $(0,0)$ to $(-3,-5)$.
Self review
What happens to the point $(1,-2)$ on $y=f(x)$ under the transformation $y=f(x)+5$?
