Quadratic Equations

Lesson

We've looked at quadratics and how we can graph them as parabolas. We also looked at groups of different features of these quadratics, including concavity, turning points, intercepts, gradient, as well as their symmetrical shape.

In this chapter, we are going to look more at turning points. A quadratic will have either a minimum or a maximum value at its turning point.

Parabolas that are concave up have a minimum value. This means their $y$`y` value will never be less than a certain value.

Similarly, parabolas that are concave down have a maximum value. This means their $y$`y` value will never be *more* than a certain value.

Remember!

The minimum or maximum value always occurs at the axis of symmetry.

It's easy to see a maximum and minimum values when we look at graphs. However, we can also find these minimum or maximum values algebraically. To do this, we need to find the axis of symmetry

$x=\frac{-b}{2a}$`x`=−`b`2`a`

Remember, the general form of a quadratic is $y=ax^2+bx+c$`y`=`a``x`2+`b``x`+`c`, so we just need to substitute in the relevant values. For example, let's find the equation of the axis of symmetry for $y=2x^2-4x+7$`y`=2`x`2−4`x`+7.

In this example, $a=2$`a`=2 and $b=-4$`b`=−4, so let's substitute them into the formula:

$x$x |
$=$= | $\frac{-\left(-4\right)}{2\times2}$−(−4)2×2 |

$=$= | $\frac{4}{4}$44 | |

$\therefore$∴ $x$x |
$=$= | $1$1 |

- Determine whether the parabola is concave up or down. This will tell us whether we're finding a minimum or maximum value.
- Find the $x$
`x`value of the turning point using the axis of symmetry formula. - Substitute the value from step 2 into the original equation to find the $y$
`y`value.

Consider the equation $y=2x^2$`y`=2`x`2.

Is every value of $y$

`y`positive?No

AYes

BNo

AYes

BThe graph of $y=2x^2$

`y`=2`x`2 is given below.Loading Graph...State the axis of symmetry of the parabola.

What is the minimum value of $y$

`y`?$y$

`y`$=$=$\editable{}$

Consider the curve $y=2\left(x-3\right)^2+3$`y`=2(`x`−3)2+3.

Is the graph of the curve concave up or concave down?

Concave up

AConcave down

BConcave up

AConcave down

BWhat is the axis of symmetry of the parabola?

What is the minimum value of the graph?

$y$

`y`$=$=$\editable{}$At which value of $x$

`x`does the minimum value occur?$x$

`x`$=$= $\editable{}$If the graph was shifted $4$4 units downwards, what would be the new minimum value?

Consider the equation $y=\left(x-12\right)\left(x-6\right)$`y`=(`x`−12)(`x`−6)

State the zeros of the function on the same line, separated by a comma.

Find the axis of symmetry.

The graph of the function is:

Concave up

AConcave down

BConcave up

AConcave down

BThe graph has a:

Maximum value

AMinimum value

BMaximum value

AMinimum value

BDetermine the minimum $y$

`y`value of the function.

Find optimal solutions, using numerical approaches

Investigate relationships between tables, equations and graphs