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Year 9 into 10 Step Up Program  Yr 10 6C 

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Full Turning Point Form
y = ±a(x – h)² + k

Each part of the equation does a specific thing to the standard parabola ( y = x²)


 * A negative sign in front of the bracket reflects the parabola across the x-axis.
 * This means the parabola is __inverted__ (upside down).
 * The inverted parabola has the open end facing down.
 * In an inverted parabola, the turning point is a __maximum__ point.


 * If there is no negative sign in front of the bracket, the parabola is __upright__.
 * The upright parabola has the open end facing up.
 * In an upright parabola, the turning point is a __minimum__ point.


 * a > 1 means the parabola is thinner than y = x 2 . (dilated)
 * 0 < a < 1 means the parabola is wider than y = x 2 . (dilated)


 * The value inside the bracket with the x (h) shifts the parabola to the left or right in the __**opposite**__ direction to the sign of h.
 * The x-value of the turning point is h (reverse the sign).


 * The value after the bracket (k) shifts the parabola up or down in the **same** direction as the sign.
 * The y-value of the turning point is k

__Turning point__ is at (h, k)

__Axis of symmetry__ is a vertical line through the turning point : x = h

Find the __y-intercept__ by substituting x = 0 into the rule.

Find the __x-intercepts__ by substituting y = 0 and solving for x

Demonstration
See an interactive demonstration of parabolas in turning point form here.

** Example **
math \text{Sketch } y = -\frac{1}{2} \big( x - 2 \big)^2 + 3 math


 * __Solution__**

Parabola is __inverted__.

Parabola is __wider__

Parabola is shifted right 2 and up 3

Turning point : (2, 3)

Axis of symmetry : x = 2

y-intercept : math \\ .\qquad y = -\frac{1}{2} \big( 0-2 \big)^2 + 3 \\ \\ . \qquad y = 1 \\ \\ . \qquad \text{coordinates : } (0, \; 1) math

x-intercepts : math \\ . \qquad 0 = -\frac{1}{2} \big( x-2 \big)^2 + 3 \\ \\ . \qquad \frac{1}{2} \big( x-2 \big)^2 = 3 \\ \\ . \qquad \big( x-2 \big)^2 = 6 \\ \\ . \qquad x - 2 = \pm 2.449 \quad \{ \textit{square root of 6} \} math

math \\ . \qquad x - 2 = -2.4 \;\; \textit{ and } \;\; x - 2 = 2.4 \\ \\ . \qquad x = -0.4 \quad \textit{ and } \quad x = 4.4 \\ \\ . \qquad \text{coordinates : } (-0.4, \; 0) \;\; \textit{ and } \;\; (4.4, \; 0) math

In VCE Maths Methods, we would write these x-intercepts as: math . \qquad x = 2 \pm \sqrt{6} math
 * NOTE **

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