Quadratics

Parabolas (y=x²)

Vertical Shift y=x²+c

Horizontal Shift y=(x-b)²

Inverting Parabolas y=-x²

Stretching Parabolas y=ax²

Plotting Parabolas

From intercept form y=(x-a)(x-b)

From vertex form y=a(x-b)²+c

Quadratic Patterns

Parabolas (y=x²)

Parabolas are symmetrical
This means if you know the shape of one half you can complete the other half
The turning point on a parabola is called a vertex
Parabolas without a coefficient in front of the x² term increase by 1, 3, 5, 7,…

Vertical Shift y=x²+c

To move a parabola up/down we add/subtract to the end of the equation
E.g. y=x² is a standard equation (shown in red)
E.g. y=x²+2 shifts the vertex up two (shown in blue)
E.g. y=x²-3 shifts the vertex down three (shown in green)

Horizontal Shift y=(x-b)²

To move a parabola right 'b' units we subtract from the x
E.g. y=x² is a standard equation (shown in red)
E.g. y=(x-2)² shifts the vertex right two (shown in blue)
E.g. y=(x+3)² shifts the vertex left three (shown in green)

Note: Be careful as this may be different from what you initially expected.

Inverting Parabolas y=-x²

A negative coefficient in front of the x² term means the parabola is inverted (upside down)

Stretching Parabolas y=ax²

A larger coefficient gives a vertically stretched parabola
E.g. y=5x² (shown in green)
A coefficient between 0 and 1 gives a horizontally stretch parabola
E.g. y=¼x² (shown in blue)

Parabola Transformations - Desmos

Plotting Parabolas

From intercept form y=(x-a)(x-b)

Find the x-intercept/s: By making y zero
Find the y-intercept: By making x zero
Find the axis of symmetry: Halfway between the x-intercepts
Find vertex: substitute x-value for line of symmetry into equation

5. Find additional points: substitute x-values into the equation (if required)

From vertex form y=a(x-b)²+c

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Quadratic Patterns

Proof as to why half the second difference is 'a' (the coefficient of the x²)

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