Quadratic Relations

Lesson

As we have seen, the vertex (or turning point) of a parabola can be found very easily when we are able to write the function in the form $y=a(x-h)^2+k$`y`=`a`(`x`−`h`)2+`k`: the vertex is $(h,k)$(`h`,`k`).

We have also seen that if we are first given the form $y=ax^2+bx+c$`y`=`a``x`2+`b``x`+`c`, we can complete the square to turn it into the form $y=a(x-h)^2+k$`y`=`a`(`x`−`h`)2+`k`.

But there's another way!

If we want to find the $x$`x` value of the turning point or vertex, we can use the formula $x=-\frac{b}{2a}$`x`=−`b`2`a`.

This is a nice little formula that has been derived from the process of completing the square of the general form $y=ax^2+bx+c$`y`=`a``x`2+`b``x`+`c`.

This is quite a complicated piece of algebra and it leads into the development of the quadratic formula, which we'll look at in detail soon. For now, let's just see how we can use the vertex formula.

Because all parabolas are symmetrical, this formula is helping us find the midpoint between the two $x$`x`-intercepts. That is, $x=-\frac{b}{2a}$`x`=−`b`2`a` is the equation of the axis of symmetry. So we can either use this formula, or if we already have the $x$`x`-intercepts, we can find the half-way point between them.

The steps involved are:

- Look at the equation in the form $y=ax^2+bx+c$
`y`=`a``x`2+`b``x`+`c` - Identify the $a$
`a`and the $b$`b`value - Substitute them into the formula $x=-\frac{b}{2a}$
`x`=−`b`2`a` to find the $x$`x`value of the vertex - To find the $y$
`y`value of the vertex, use the $x$`x`value just found and substitute it into the original equation of the parabola given.

Consider the function $P\left(x\right)=x^2-4x+2$`P`(`x`)=`x`2−4`x`+2.

Find the $x$

`x`-coordinate of the vertex.Find the $y$

`y`-coordinate of the vertex.Graph the function.

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Consider the function $y=x^2+4x+3$`y`=`x`2+4`x`+3.

Determine the equation of the axis of symmetry.

Hence determine the minimum value of $y$

`y`.Hence state the coordinates of the vertex of the curve.

Here is the graph of another quadratic function. State the coordinates of its vertex.

Loading Graph...What is the relationship between the graph of $y=x^2+4x+3$

`y`=`x`2+4`x`+3 and the graph provided?Select all that apply.

They have the same $x$

`x`-intercepts.AThey share the same turning point.

BThey have the same concavity.

CThey have the same $x$

`x`-intercepts.AThey share the same turning point.

BThey have the same concavity.

C

$y=x^2-18x+10$`y`=`x`2−18`x`+10

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

BState the value of $x$

`x`at which the minimum value of the function occurs.State the minimum $y$

`y`value of the function.

Consider the function $P\left(x\right)=x^2+5x+3$`P`(`x`)=`x`2+5`x`+3.

Find the $x$

`x`-coordinate of the vertex.Find the $y$

`y`-coordinate of the vertex.Graph the function.

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Determine the zeros and the maximum or minimum value of a quadratic relation from its graph or from its defining equation