065summary

 Year 9 into 10 Step Up Program  11D


 * Parabolas - Summary (so far) **toc

Parabolas
A parabola:
 * is the graph produced by a quadratic function (with an x 2 term)
 * can be __upright__ (open end up) or __inverted__ (open end down)
 * has a __turning point__ which is either the __minimum__ point or the __maximum__ point
 * has an __axis of symmetry__ which is a vertical line through the turning point.

[[image:062grph1.gif width="270" height="220" align="right"]] Standard Parabola
The standard parabola (y = x 2 )
 * is upright
 * has a minimum turning point at (0, 0)
 * has an axis of symmetry : x = 0
 * has y-intercept at (0, 0)
 * has x-intercept at (0, 0)

Dilations

Equations of the form y = ax 2.
 * produce a __dilation__ in the standard parabola
 * if a > 1, the parabola will be __thinner__
 * if 0 < a < 1, the parabola will be __wider__
 * the turning point does not change (0, 0)
 * the axis of symmetry does not change : x = 0



Reflections
Equations of the form y = –x 2.
 * produce a __reflection__ across the x-axis
 * ie the parabola is __inverted__
 * the shape of the paraobla does not change
 * the turning point does not change (0, 0)
 * the axis of symmetry does not change : x = 0

Vertical Translations


Equations of the form y = x 2 + k
 * produce a __translation up__ by a distance of k
 * ie the parabola __translates__ (shifts) __up__ by k
 * if k < 0, the parabola __translates down__
 * the turning point moves up to (0, k)
 * the shape of the parabola does not change
 * the axis of symmetry does not change : x = 0

Sketching Parabolas
A sketch is a neat drawing of the graph
 * the axes should be drawn and labelled X and Y
 * the scale does __not__ need to be marked in
 * an effort should be made to keep distances equivalent
 * eg 2 should be further than 1.4 but not twice as far
 * the turning point should be labelled with coordinates
 * any x and y intercepts should be labelled with coordinates
 * the equation of the graph should be written next to the graph

Combinations
We can graph equations like y = ax 2 + k by combining the transformations described above
 * the different parts of the equation do a different job
 * the **+ k** translates the graph (shift up or down)
 * the **a** dilates the graph (thinner or wider)
 * negative values of **a** mean the graph is inverted.
 * the turning point will be at (0, k)
 * x-intercepts can be found by solving ax 2 + k = 0

** Example **
Sketch y = 3x 2 – 4.

__**Solution**__


 * a is positive so the graph is upright
 * a > 1 so the graph is thinner
 * c = – 4 so the graph is shifted down by 4
 * minimum turning point at (0, –4)
 * x-intercepts are:
 * solve 3x 2 – 4 = 0 gives
 * x = –1.2, x = 1.2
 * (–1.2, 0) and (1.2, 0)

Go to Top of Page

.