Quadratic Relations

Lesson

The characteristic shape of the graph of a quadratic function is called a parabola. It has a point called a vertex, which could represent either its minimum or its maximum value.

The characteristic shape of the graph of a quadratic function is called a parabola. It has a point called a vertex, which could represent either its minimum or its maximum value.

If you experiment with the applet above, you will find that the parabola has a minimum when the number multiplying the $x^2$`x`2 term is positive and it has a maximum when the coefficient of the $x^2$`x`2 term is negative. Changing this coefficient also has the effect of zooming in or zooming out. It changes the scale of the parabola picture.

You will also notice that when the constant term is changed, the parabola is shifted in the vertical direction without any change of scale.

One of the most basic forms of a quadratic is one where the formula has only an $x^2$`x`2-term and the constant term. The complete version looks like this:

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

where the numbers $a$`a` and $k$`k` are the coefficients that control the shape and position of the parabola relative to the $x$`x`- and $y$`y`-axes.

It is an interesting fact that the graph of every quadratic function is a parabola and all parabolas have exactly the same shape. The apparent differences between them are to do with changes of scale and orientation and with changes in position relative to the origin.

Consider the equation $y=-x^2$`y`=−`x`2

Complete the following table of values.

$x$ `x`$-3$−3 $-2$−2 $-1$−1 $0$0 $1$1 $2$2 $3$3 $y$ `y`$\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Plot the points in the table of values.

Loading Graph...Hence plot the curve.

Loading Graph...Are the $y$

`y`values ever positive?No

AYes

BWhat is the maximum $y$

`y`value?Write down the equation of the axis of symmetry.

Consider the quadratic $y=3x^2-3$`y`=3`x`2−3.

Complete the table.

$x$ `x`$-3$−3 $-2$−2 $-1$−1 $0$0 $1$1 $2$2 $3$3 $y$ `y`$\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Hence graph $y=3x^2-3$

`y`=3`x`2−3.Loading Graph...

Consider the function $y=\left(x-2\right)^2$`y`=(`x`−2)2

Complete the following table of values.

$x$ `x`$0$0 $1$1 $2$2 $3$3 $4$4 $y$ `y`$\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Sketch a graph of the function.

Loading Graph...What is the minimum $y$

`y`value?What $x$

`x`value corresponds to this minimum $y$`y`value?What are the coordinates of the vertex? Give your answer in the form $\left(a,b\right)$(

`a`,`b`).