Maximum and minimum values of quadratics

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, slope, 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. 

 

Maximum and Minimum Values

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.

 

 

Finding the Equation of 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=−b2a​

Remember, the general form of a quadratic is $y=ax^2+bx+c$y=ax2+bx+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=2x2−4x+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

 

Finding the Turning Point

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

 

Examples

Question 1

Question 2

Question 3