061parabolas

 Year 9 into 10 Step Up Program


 * Key Features of a Parabola **toc

Parabolas
(pronounced pa - __**rab**__ - ola)

A __**parabola**__ is a graph produced by a quadratic function, like: math . \qquad \qquad y = \big( x-2 \big)^2 \; \textit{ or } \; y = -x^2 - 4x math

{All quadratic functions have an x 2 term when the brackets are expanded}

Turning Points
Parabolas have a __**turning point**__ where the graph changes direction (or turns) from going down to going up.

The red graph { y = (x – 2) 2 } has a ** __minimum__ ** turning point. The minimum point is the lowest point on the graph.

The blue graph { y = –x 2 – 4x } has a** __maximum__ ** turning point. The __**maximum point**__ is the highest point on the graph

You will often be asked to describe the nature of the turning point (ie minimum or maximum) and state the coordinates of the turning point.

Axis of Symmetry
Parabolas are __**symmetrical**__ (the two sides of the graph are a mirror image of each other).

The __**axis of symmetry**__ will be a __**vertical line**__ passing through the turning point.

If the__ x-coordinate __ of the turning point is ** 2 **,
 * the ** equation of the axis of symmetry ** will be ** x = 2 **.
 * ** x = 2 ** .. describes a vertical line where all the points on that line have an x-coordinate of 2

The __**equations of the axes of symmetry**__ in the above example are: math . \qquad x = 2 \; \textit{ and } \; x = -2 math


 * NOTE: ** Because a parabola is symmetrical, the ** turning point ** is always half way between the two ** x-intercepts **.

Y-Intercept
The ** y-intercept ** is the point where the graph crosses the ** y-axis **.

A parabola in this form will always have a y-intercept.

The y-intercept can be found by
 * reading from the graph or
 * substituting **//x = 0//** into the equation.

X-Intercepts
The ** x-intercept ** is the point where the graph crosses the ** x-axis **.

Not all parabolas in this form have x-intercepts.
 * Some have one x-intercept
 * Some have two x-intercepts
 * Some have none.

If the x-intercepts do exist, they can be found by
 * reading from the graph or
 * substituting **//y = 0//** and solving for x.


 * Remember ** the ** turning point ** will always be half way between the two ** x-intercepts ** (if there are two).

** Example 1 **
In the graph shown here: (i) State the nature of the turning point Minimum Point

(ii) State the coordinates of the turning point math \big( 1, \; -4 \big) math

(iii) State the equation of the axis of symmetry math x = 1 math

(iv) State the coordinates of the y-intercept math \big( 0, \; -3 \big) math

(v) State the coordinates of the two x-intercepts math \big(-1, \; 0 \big) \; \textit{ and } \; \big(3, \; 0 \big) math

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