063sketchax2

 Year 9 into 10 Step Up Program


 * Sketching y = ax 2 **toc

Standard Parabola
We know //y = x 2 // produces a table of values:

When we graph //y = x 2 //, it produces a parabola.

With turning point at (0, 0)

Axis of symmetry : //x = 0//

Y-Intercept (0, 0)

X-Intercept (0, 0)

The question today is:

What happens if we put a number (coefficient) in front of the //x// 2 ?
 * We call this coefficient : a

** Question 1 **
Fill out a table of values for //y = 2x// 2. (note: 2x 2 → __first__ square the x and __then__ times by 2) Then graph on the same axes as //y = x// 2 (use a different colour pen) What is different between the two graphs? What is the same between the two graphs?
 * a = 2

** Question 2 **
Fill out a table of values for //y = 0.5x// 2. Then graph on the same axes as //y = x// 2 (use a different colour pen) What is different between the two graphs? What is the same between the two graphs?
 * a = 0.5

** Solution 1 **
Table of values for //y = 2x// 2. Now sketch on the same axes as //y = x// 2.

The new graph is __**thinner**__ than //y = x// 2. But!

It has the same turning point (0, 0)

Same axis of symmetry

Same x and y intercepts

** Solution 2 **
Table of values for //y = 0.5x// 2. Now sketch on the same axes as //y = x// 2.

The new graph is __**wider**__ than //y = x// 2.

But

It has the same turning point (0, 0)

Same axis of symmetry

Same x and y intercepts

Summary (so far)
When graphing //y = ax// 2.

The value of **a** causes a __**dilation**__ in the parabola.
 * If **a** > 1 : the parabola will be __**thinner**__
 * If 0 < **a** < 1 : the parabola will be __**wider**__

Some things don't change
 * The turning point will be a minimum at (0, 0)
 * The axis of symmetry will be at x = 0
 * The x and y intercepts are at (0, 0)

** Question 3 **
Fill out the table of values for //y = –x// 2. Then graph on the same axes as //y = x// 2. What is different between the two graphs? What is the same between the two graphs?
 * a = //– 1//

**Question 4**
Fill out the table of values for //y = –¼ x// 2. Then graph on the same axes as //y = x// 2. What is different between the two graphs? What is the same between the two graphs?
 * a = //–¼//

** Solution 3 **
Table of values for //y = –x// 2. Now sketch on the same axes as y = x 2.

The graph is **__inverted__**.

It is a **__reflection__** of y = x 2 across the x-axis.

But

It has the same turning point (0, 0)

Same axis of symmetry

Same x and y intercepts

** Solution 4 **
Table of values for //y = –¼ x// 2. Now sketch on the same axes as //y = x// 2. The graph is __**inverted**__. It is a __**reflection**__ of y = x 2 across the x-axis.

AND

It is __**wider**__.

But

It has the same turning point (0, 0)

Same axis of symmetry

Same x and y intercepts.

Summary
When graphing y = ax 2.

The value of **a** causes a __**dilation**__ in the parabola.
 * If **a** > 1 or **a** < –1 : the parabola will be __**thinner**__
 * If –1 < **a** < 1 : the parabola will be __**wider**__.


 * If a < 0 : the parabola will be __**inverted**__. (reflected across the x-axis) and the turning point will be a maximum.
 * If a > 0 : the parabola will be __**upright**__ and the turning point will be a minimum.

Some things don't change
 * The turning point will be at (0, 0)
 * The axis of symmetry will be x = 0
 * The x and y intercepts are at (0, 0)



Compare to y = mx + c
The //**a**// value in y = ax 2 can be compared to the //**m**// value in the equation of a straight line y = mx + c

Increasing //**m**// makes the straight line steeper and increasing //**a**// makes the parabola steeper.

If //**m**// is negative, the straight line slope down and if //**a**// is negative, the parabola slope down.

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