Everything Maths Grade 10

Functions of the formy=ax^2 +q EMA4K

Investigation: The effects ofaandqon a parabola.

Complete the table and plot the following graphs on the same system of axes:

1.y 1 =x^2  2

2.y 2 =x^2  1

3.y 3 =x^2

4.y 4 =x^2 + 1

5.y 5 =x^2 + 2

x  2  1 0 1 2

y 1

y 2

y 3

y 4

y 5

Use your results to deduce the effect ofq.

Complete the table and plot the following graphs on the same system of axes:

1.y 6 = 2 x^2

2.y 7 =x^2

3.y 8 =x^2

4.y 9 = 2x^2

x  2  1 0 1 2

y 6

y 7

y 8

y 9

Use your results to deduce the effect ofa.

The effect ofq

The effect ofqis called a vertical shift because all points are moved the same distance in the same direction
(it slides the entire graph up or down).

• Forq > 0 , the graph off(x)is shifted vertically upwards byqunits. The turning point off(x)is above
  they-axis.


• Forq < 0 , the graph off(x)is shifted vertically downwards byqunits. The turning point off(x)is below
  they-axis.

The effect ofa

The sign ofadetermines the shape of the graph.

• Fora > 0 , the graph off(x)is a “smile” and has a minimum turning point at(0;q). The graph off(x)
  is stretched vertically upwards; asagets larger, the graph gets narrower.
  For 0 < a < 1 , asagets closer to 0, the graph off(x)gets wider.


• Fora < 0 , the graph off(x)is a “frown” and has a maximum turning point at(0;q). The graph off(x)
  is stretched vertically downwards; asagets smaller, the graph gets narrower.
  For 1 < a < 0 , asagets closer to 0, the graph off(x)gets wider.

a> 0 (a positive smile) a< 0 (a negative frown)

160 6.3. Quadratic functions