Middle Years

Lesson

Graphs of quadratic equations of the form $y=Ax^2+Bx+C$`y`=`A``x`2+`B``x`+`C` (where $A$`A`, $B$`B`, and $C$`C` are any number and $A\ne0$`A`≠0) are called parabolas.

Like lines, parabolas will always have a $y$`y`-intercept. This is the point on the graph which touches the $y$`y`-axis. We can find this by setting $x=0$`x`=0 and finding the value of $y$`y`.

Similarly, we can look for $x$`x`-intercepts by setting $y=0$`y`=0 and then solving for $x$`x`. Because this is a quadratic equation, there could be $0$0, $1$1, or $2$2 solutions, and there will be the same number of $x$`x`-intercepts.

Parabolas have an axis of symmetry which is the vertical line $x=-\frac{B}{2A}$`x`=−`B`2`A`. This is also the midpoint of the $x$`x`-intercepts if they exist.

The point on the parabola which intersects the axis of symmetry is called the vertex of the parabola. The $x$`x`-value of the vertex will be the axis of symmetry, and we can find the $y$`y`-value by substituting this $x$`x`-value into the equation.

Finally, parabolas have a concavity. If the vertex is the minimum point on the graph then the parabola is concave up and if the vertex is the maximum point on the graph then the parabola is concave down.

A parabola can be vertically translated by increasing or decreasing the $y$`y`-values by a constant number. So to translate $y=x^2$`y`=`x`2 up by $k$`k` units gives us $y=x^2+k$`y`=`x`2+`k`.

Similarly, a parabola can be horizontally translated by increasing or decreasing the $x$`x`-values by a constant number. However, the $x$`x`-value together with the translation must be squared together. That is, to translate $y=x^2$`y`=`x`2 to the left by $h$`h` units we get $y=\left(x+h\right)^2$`y`=(`x`+`h`)2.

A parabola can be vertically scaled by multiplying every $y$`y`-value by a constant number. So to expand the parabola $y=x^2$`y`=`x`2 by a scale factor of $a$`a` we get $y=ax^2$`y`=`a``x`2. We can compress a parabola by dividing by the scale factor instead.

Finally, we can reflect a parabola about the $x$`x`-axis by taking the negative. So to reflect $y=x^2$`y`=`x`2 about the $x$`x`-axis gives us $y=-x^2$`y`=−`x`2. Notice that reflecting will change the concavity (in this case from concave up to concave down).

Summary

The graph of a quadratic equation of the form $y=Ax^2+Bx+C$`y`=`A``x`2+`B``x`+`C` is a parabola.

Parabolas have a $y$`y`-intercept and can have $0$0, $1$1, or $2$2 $x$`x`-intercepts, depending on the solutions to the quadratic equation.

Parabolas have a vertical axis of symmetry and a vertex which is the point on the graph which intersects the axis of symmetry.

Parabolas are either concave up or concave down, depending on whether the vertex is the minimum or maximum point on the graph.

Parabolas can be transformed in the following ways (starting with the parabola defined by $y=x^2$`y`=`x`2):

- Vertically translated by $k$
`k`units: $y=x^2+k$`y`=`x`2+`k` - Horizontally translated by $h$
`h`units: $y=\left(x-h\right)^2$`y`=(`x`−`h`)2 - Vertically scaled by a scale factor of $a$
`a`: $y=ax^2$`y`=`a``x`2 - Vertically reflected about the $x$
`x`-axis: $y=-x^2$`y`=−`x`2

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

BNo

AYes

BWhat is the maximum $y$

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

Consider the equation $y=3x^2$`y`=3`x`2.

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 $y=3x^2$

`y`=3`x`2.Loading Graph...

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

Complete the following table of values.

$x$ `x`$1$1 $2$2 $3$3 $4$4 $5$5 $y$ `y`$\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Plot the graph.

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`).