4.06 Transformations of quadratics

There are specific names given to different types of polynomials functions.  A quadratic function is a polynomial function of degree $2$2, and functions of this form share many of the same properties.

Key characteristics of $y=x^2$y=x2

The most basic type of quadratic function is an equation of the form $y=x^2$y=x2.  When graphed, this relation creates a shape that is called a parabola.  It is the parent function for all other quadratic functions because all other quadratic functions are transformations of this function.  Below is a graph of the function $y=x^2$y=x2.  What key characteristics can you identify from the graph?

By the graph above we can see that the graph of $y=x^2$y=x2 is symmetric across the line $x=0$x=0, or the line of symmetry.  It has a minimum turning point at the point $\left(0,0\right)$(0,0). This turning point value is called the vertex.  In the case of this graph, the vertex is also the same point as both the $x$x- and $y$y-intercepts.

 

Transformations of the parent function $y=x^2$y=x2

Let's explore how a quadratic function changes using the applet below. What happens when we change the sliders for dilation, reflection, vertical translation and horizontal translation? Can you describe what changes these values make to the quadratic function?

These movements are called transformations. Transform means change, and these transformations change the simple quadratic $y=x^2$y=x2 into other quadratics by moving (translating), flipping (reflecting) and making the graph appear more or less steep (dilating).

By using the above applet, step through these instructions:

• Start with the simple quadratic $y=x^2$y=x2
• Dilate the quadratic by factor of $2$2
• Reflect on the $x$x-axis
• Translate vertically by $2$2 and horizontally by $-3$−3 units

How has the graph changed? Can you visualize the changes without using the applet? What is the resulting equation?

 

Equations and transformations of quadratics

Transforming a quadratic will change its equation. Here are some of the most common types of quadratic equations, and what they mean with regards to the transformations that have occurred.

$y=ax^2$y=ax2

If $a<0$a<0 (that is, $a$a is negative) then we have a reflection parallel to the $x$x-axis. It's like the quadratic has been flipped upside down.

 

$y=ax^2$y=ax2

This is a quadratic that has been dilated vertically by a factor of $a$a.

If $\left|a\right|>1$|a|>1 then the graph is steeper than $y=x^2$y=x2.

If $\left|a\right|<1$|a|<1 then the graph is flatter than $y=x^2$y=x2. 

 

$y=ax^2+k$y=ax2+k

In the graph $y=ax^2+k$y=ax2+k, the quadratic has been vertically translated by $k$k units.  

If $k>0$k>0 then the translation is up.

If $k<0$k<0 then the translation is down.

 

$y=\left(x-h\right)^2$y=(x−h)2

The $h$h indicates the horizontal translation.  

If $h>0$h>0, that is the factor in the parentheses is $\left(x-h\right)$(x−h) than we have a horizontal translation of $h$h units right.

If $h<0$h<0, that is the factor in the parentheses is $\left(x-\left(-h\right)\right)=\left(x+h\right)$(x−(−h))=(x+h) than we have a horizontal translations of $h$h units left.  

 

The vertex

The vertex (sometimes called turning point) of a quadratic is the point where the function turns. At this point, the function can have a maximum or minimum value.

It changes from being decreasing to increasing, like in this positive quadratic.

Or changes from being increasing to decreasing, like in this negative quadratic.  

 

Vertex form

The form $y=a\left(x-h\right)^2+k$y=a(x−h)2+k is called vertex form. It's really useful for two reasons:

• It tells us the vertex immediately, and knowing the vertex we can draw a pretty good sketch of any quadratic.
• It explains to us a number of transformations that have occurred to the quadratic from the simple quadratic $y=x^2$y=x2.

 

 

Practice questions

Question 1

Question 2

Question 3