NSW Y9 Maths Advanced Equations Quadratic Equations

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An equation in which the highest power of the variable is 2 is called a quadratic equation, for example, \(y = 2{x^2} + 5\).

The graph of \(y = {x^2}\) is of parabola's shape and is concave up with turning point or vertex as \((0,0)\).

The graph of \(y = -{x^2}\) is of parabola's shape and is concave down with turning point or vertex as \((0,0)\).

The graph of \(y = a{x^2}\) where \(a\) is a constant (number), the size of \(a\) affects whether the parabola is 'wide' or 'narrow'. For example,

\(y = \frac{{{x^2}}}{2},y = - 4{x^2}\). 

The graph of \(y = a{x^2}+c\) where \(a\) and \(c\) are constants, the effect of \(c\) is to move the parabola up or down from the origin.

Example 1

a. Graph \(y={x^2},y={x^2}+3,y=-{x^2}-2 and y=-{x^2}-5\) on the same set of axes.

b. Compare each parabola with the parabola \(y={x^2}\)

\(y={x^2}+3\) is \(y={x^2}\) shifted up \(3 units\)

\(y=-{x^2}-2\) is \(y={x^2}\) turned upside down and shifted down \(2 units\)

\(y=-{x^2}-5\) is \(y={x^2}\) turned upside down and shifted down \(5 units\)

c. What is the vertex of each parabola and is each parabola concave up or concave down?

\(y={x^2}\) has vertex \((0,0)\) and is concave up.

\(y={x^2}+3\) has vertex \((0,3)\) and is concave up.

\(y=-{x^2}-2\) has vertex \((0,-2)\) and is concave down.

\(y=-{x^2}-5\) has vertex \((0,-5)\) and is concave down.