Cambridge Additional Mathematics

86 Quadratics (Chapter 3)

Example 19 Self Tutor

Use the discriminant to determine the relationship between the graph of each function and thex-axis:

a y=x^2 +3x+4 b y=¡ 2 x^2 +5x+1

a a=1, b=3, c=4

) ¢=b^2 ¡ 4 ac

=9¡4(1)(4)

=¡ 7

b a=¡ 2 , b=5, c=1

) ¢=b^2 ¡ 4 ac

=25¡4(¡2)(1)

=33

Since ¢> 0 , the graph cuts thex-axis

twice.

Since a< 0 , the graph is concave

down.

EXERCISE 3D.4

1 Use the discriminant to determine the relationship between the graph andx-axis for:

a y=x^2 +x¡ 2 b y=x^2 ¡ 4 x+1 c f(x)=¡x^2 ¡ 3

d f(x)=x^2 +7x¡ 2 e y=x^2 +8x+16 f f(x)=¡ 2 x^2 +3x+1

g y=6x^2 +5x¡ 4 h f(x)=¡x^2 +x+6 i y=9x^2 +6x+1

2 Consider the graph of y=2x^2 ¡ 5 x+1.

a Describe the shape of the graph.

b Use the discriminant to show that the graph cuts thex-axis twice.

c Find thex-intercepts, rounding your answers to 2 decimal places.

d State they-intercept.

e Hence, sketch the function.

3 Consider the graph of f(x)=¡x^2 +4x¡ 7.

a Use the discriminant to show that the graph does not cut thex-axis.

b Is the graph positive definite or negative definite?

c Find the vertex andy-intercept.

d Hence, sketch the function.

4 Show that:

a x^2 ¡ 3 x+6> 0 for allx b 4 x¡x^2 ¡ 6 < 0 for allx

c 2 x^2 ¡ 4 x+7 is positive definite d ¡ 2 x^2 +3x¡ 4 is negative definite.

5 Explain why 3 x^2 +kx¡ 1 is never positive definite for any value ofk.

6 Under what conditions is 2 x^2 +kx+2 positive definite?

x x

Since ¢< 0 , the graph does not cut the

x-axis.

Since a> 0 , the graph is concave up.

The graph is positive definite, which

means that it lies entirely above thex-axis.

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_03\086CamAdd_03.cdr Thursday, 3 April 2014 4:45:58 PM BRIAN