Quadratic Equations

Lesson

Let's explore how a quadratic function changes using the applet below. What happens when we change the sliders for enlargement, 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`=`x`2 into other quadratics by moving (translating), flipping (reflecting) and making the graph appear more or less steep (enlarging).

By using the above applet, step through these instructions:

- Start with the simple quadratic $y=x^2$
`y`=`x`2 - Enlarge 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 visualise the changes without using the applet? What is the resulting equation?

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`=`a``x`2

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.

Shows the reflection of

$y=x^2$`y`=`x`2 to $y=-x^2$`y`=−`x`2

Shows the reflection of

$y=\left(x-1\right)\left(x+2\right)$`y`=(`x`−1)(`x`+2) to $y=-\left(x-1\right)\left(x+2\right)$`y`=−(`x`−1)(`x`+2)

$y=ax^2$`y`=`a``x`2

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

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

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

$y=ax^2+k$`y`=`a``x`2+`k`

In the graph $y=ax^2+k$`y`=`a``x`2+`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 brackets 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 brackets 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 turning point of a quadratic is the point where the function turns. This point is also known as the vertex of the parabola.

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

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

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

- It tells us the turning point immediately, and knowing the turning point 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`=`x`2.

The form $y=a\left(x-h\right)^2+k$`y`=`a`(`x`−`h`)2+`k` shows us:

If $\left|a\right|>1$|`a`|>1 then the quadratic is steeper than $x^2$`x`2.

If $\left|a\right|<1$|`a`|<1 then the quadratic is flatter than $x^2$`x`2.

If $a<0$`a`<0 then the quadratic is a reflection of $ax^2$`a``x`2 on a line parallel to the $x$`x`-axis.

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

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

If $h<0$`h`<0 then the translation is to the left, and $x+h$`x`+`h` appears in the brackets.

If $h>0$`h`>0 then the translation is to the right, and $x-h$`x`−`h` appears in the brackets.

The function $y=-5x^2$`y`=−5`x`2 has what enlargement factor?

Enlargement factor $=$= $\editable{}$

The quadratic $y=x^2$`y`=`x`2 has been transformed into $y=3\left(x-4\right)^2$`y`=3(`x`−4)2. Has there been a reflection?

Yes, about the $x$

`x`-axisAYes, about the $y$

`y`-axisBNo$\text{ }$

CYes, about the $x$

`x`-axisAYes, about the $y$

`y`-axisBNo$\text{ }$

C

The quadratic $y=x^2$`y`=`x`2 has been transformed into $y=-4\left(x+10\right)^2$`y`=−4(`x`+10)2. Identify the vertical translation.

$0$0 units up or down

A$4$4 units up

B$10$10 units down

C$10$10 units up

D$0$0 units up or down

A$4$4 units up

B$10$10 units down

C$10$10 units up

D

The quadratic $y=x^2$`y`=`x`2 has been transformed into the graph to the right.

Loading Graph...

What is the vertical translation?

$4$4 units up

A$4$4 units down

B$3$3 units down

C$3$3 units up

D$4$4 units up

A$4$4 units down

B$3$3 units down

C$3$3 units up

DWhat is the horizontal translation?

$3$3 units right

A$4$4 units left

B$3$3 units left

C$4$4 units right

D$3$3 units right

A$4$4 units left

B$3$3 units left

C$4$4 units right

DHas the curve been reflected about the $x$

`x`-axis?No

AYes

BNo

AYes

BHas this quadratic had a change in enlargement from the standard $y=x^2$

`y`=`x`2 curve?Yes

ANo

BYes

ANo

B