4 Quadratics and polynomials

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 $y=3x^2+2$`y`=3`x`2+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 plot the graph of $y=3x^2+2$

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

Consider the parabola $y=-\left(x+2\right)^2-5$`y`=−(`x`+2)2−5.

Which successive transformations turn $y=x^2$

`y`=`x`2 into the equation $y=-\left(x+2\right)^2-5$`y`=−(`x`+2)2−5?Reflection about the $x$

`x`-axis, horizontal shift $2$2 units to the left and vertical shift $5$5 units downAHorizontal shift $2$2 units to the right, vertical shift $5$5 units down, reflection about the $x$

`x`-axisBReflection about the $x$

`x`-axis, vertical shift $2$2 units down and horizontal shift $5$5 units to the leftCVertical shift $2$2 units up, horizontal shift $5$5 units to the left, reflection about the $x$

`x`-axis.DReflection about the $x$

`x`-axis, horizontal shift $2$2 units to the left and vertical shift $5$5 units downAHorizontal shift $2$2 units to the right, vertical shift $5$5 units down, reflection about the $x$

`x`-axisBReflection about the $x$

`x`-axis, vertical shift $2$2 units down and horizontal shift $5$5 units to the leftCVertical shift $2$2 units up, horizontal shift $5$5 units to the left, reflection about the $x$

`x`-axis.DHence state the coordinates of the vertex of the curve.

Vertex $=$=$\left(\editable{},\editable{}\right)$(,)

Plot the graph of the equation.

Loading Graph...What is the axis of symmetry?

Consider the equation $y=\left(x-2\right)^2-16$`y`=(`x`−2)2−16.

Find the $x$

`x`-intercepts. Write all solutions on the same line, separated by a comma.Find the $y$

`y`-intercept.Determine the coordinates of the vertex.

Vertex $=$=$\left(\editable{},\editable{}\right)$(,)

Plot the graph of the equation.

Loading Graph...

Explore the connection between algebraic and graphical representations of relations such as simple quadratic, reciprocal, circle and exponential, using digital technology as appropriate