Positive and Negative Definite

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Question 1 of 5

1. Question
Which of the following graphs is a negative definite?
- 1.
- 2.
- 3.
- 4.
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Positive Definite

$Δ$ $Delta$ $<$ $<$ $0$ $0$ and $a$ $a$ $>$ $>$ $0$ $0$

A Positive Definite is a function that is always positive for all values of $x$ $x$ . It is also a parabola that is concave up and is above the $x$ $x$ axis

Negative Definite

$Δ$ $Delta$ $<$ $<$ $0$ $0$ and $a$ $a$ $<$ $<$ $0$ $0$

A Negative Definite is a function that is always negative for all values of $x$ $x$ . It is also a parabola that is concave down and is below the $x$ $x$ axis

Check the characteristics of each graph to identify which is a Negative Definite

This parabola is concave up and is above the $x$ $x$ axis

Therefore, it is a Positive Definite

This parabola is concave down and is below the $x$ $x$ axis

Therefore, it is a Negative Definite

Question 2 of 5

Identify whether the function below is a Negative Definite

y = - 2 x^{2} + 2 x - 1

$y=-2x^2+2x-1$

1. Not a Negative Definite
2. Negative Definite

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Negative Definite

Δ

$Delta$

<

$<$

0

$0$ and $a$ $a$

<

$<$

0

$0$

Discriminant Formula

Δ = b^{2} - 4 a c

$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$

A Negative Definite is a function that is always negative for all values of

x

$x$ . It is also a parabola that is concave down and is below the

x

$x$ axis

First, compute for the discriminant

y = - 2 x^{2} + 2 x - 1

$y=-2x^2+2x-1$

$a = - 2$ $a=-2$ $b = 2$ $b=2$ $c = - 1$ $c=-1$

$Δ$ $Delta$	$=$ $=$	${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$	Discriminant Formula
	$=$ $=$	${\color{#9a00c7}{2}}^2-4\color{#00880A}{(-2)}\color{#007DDC}{(-1)}$	Substitute values
	$=$ $=$	$4 - 8$ $4-8$
	$=$ $=$	$- 4$ $-4$

The value is negative, hence

Δ

$Delta$

<

$<$

0

$0$

Next, check the value of

a

$a$

y =

$y=$ $- 2$ $-2$

x^{2} + 2 x - 1

$x^2+2x-1$

$a = - 2$ $a=-2$

The value is negative, hence $a$ $a$

<

$<$

0

$0$

Therefore, we have established that the function is a Negative Definite

Question 3 of 5

Identify whether the function below is a Positive Definite

2 x^{2} - x + 3 > 0

$2x^2-x+3>0$

1. Not a Positive Definite
2. Positive Definite

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Positive Definite

Δ

$Delta$

<

$<$

0

$0$ and $a$ $a$

>

$>$

0

$0$

Discriminant Formula

Δ = b^{2} - 4 a c

$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$

A Positive Definite is a function that is always positive for all values of

x

$x$ . It is also a parabola that is concave up and is above the

x

$x$ axis

First, compute for the discriminant

2 x^{2} - x + 3

$2x^2-x+3$

$a = 2$ $a=2$ $b = - 1$ $b=-1$ $c = 3$ $c=3$

$Δ$ $Delta$	$=$ $=$	${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$	Discriminant Formula
	$=$ $=$	${\color{#9a00c7}{-1}}^2-4\color{#00880A}{(2)}\color{#007DDC}{(3)}$	Substitute values
	$=$ $=$	$1 - 24$ $1-24$
	$=$ $=$	$- 23$ $-23$

The value is negative, hence

Δ

$Delta$

<

$<$

0

$0$

Next, check the value of

a

$a$

y =

$y=$ $2$ $2$

x^{2} - x + 3

$x^2-x+3$

$a = 2$ $a=2$

The value is positive, hence $a$ $a$

>

$>$

0

$0$

Therefore, we have established that the function is a Positive Definite

Question 4 of 5

For which values of

k

$k$ will the function below be a Positive Definite

y = k x^{2} + 2 k x + 9

$y=kx^2+2kx+9$

1. $k$ $k$ $<$ $<$ $- 9$ $-9$ and $k$ $k$ $>$ $>$ $0$ $0$
2. $k$ $k$ $<$ $<$ $0$ $0$ and $k$ $k$ $>$ $>$ $9$ $9$
3. $- 9$ $-9$ $<$ $<$ $k$ $k$ $<$ $<$ $0$ $0$
4. $0$ $0$ $<$ $<$ $k$ $k$ $<$ $<$ $9$ $9$

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Positive Definite

Δ

$Delta$

<

$<$

0

$0$ and $a$ $a$

>

$>$

0

$0$

Discriminant Formula

Δ = b^{2} - 4 a c

$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$

A Positive Definite is a function that is always positive for all values of

x

$x$ . It is also a parabola that is concave up and is above the

x

$x$ axis

First, compute for the discriminant

y = k x^{2} + 2 k x + 9

$y=kx^2+2kx+9$

$a = k$ $a=k$ $b = 2 k$ $b=2k$ $c = 9$ $c=9$

$Δ$ $Delta$	$=$ $=$	${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$	Discriminant Formula
	$=$ $=$	${\color{#9a00c7}{2k}}^2-4\color{#00880A}{(k)}\color{#007DDC}{(9)}$	Substitute values
	$=$ $=$	$4 k^{2} - 36 k$ $4k^2-36k$

Remember that for a function to be a Positive Definite,

Δ

$Delta$

<

$<$

0

$0$

Substitute the

Δ

$Delta$ computed previously, and then solve for

k

$k$

$Δ$ $Delta$	$<$ $<$	$0$ $0$
$4 k^{2} - 36 k$ $4k^2-36k$	$<$ $<$	$0$ $0$
$4 k (k - 9)$ $4k(k-9)$	$<$ $<$	$0$ $0$
$k = 0$ $k=0$		$k = 9$ $k=9$

To determine which region around

k = 0

$k=0$ and

k = 9

$k=9$ would be included, plot these points and make a rough sketch of

4 k^{2} - 36 k

$4k^2-36k$

Replace the

x

$x$ axis with

k

$k$ axis and draw an upward parabola since

4

$4$ is positive

Remember that

Δ

$Delta$ must be negative

Therefore,

0

$0$

<

$<$

k

$k$

<

$<$

9

$9$

0

$0$

<

$<$

k

$k$

<

$<$

9

$9$

Question 5 of 5

For which values of

m

$m$ will the function below be a Positive Definite

m x^{2} + 4 m x + 16

$mx^2+4mx+16$

1. $0$ $0$ $<$ $<$ $m$ $m$ $<$ $<$ $4$ $4$
2. $m$ $m$ $<$ $<$ $- 4$ $-4$ and $m$ $m$ $>$ $>$ $0$ $0$
3. $m$ $m$ $<$ $<$ $0$ $0$ and $m$ $m$ $>$ $>$ $4$ $4$
4. $- 4$ $-4$ $<$ $<$ $k$ $k$ $<$ $<$ $0$ $0$

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Positive Definite

Δ

$Delta$

<

$<$

0

$0$ and $a$ $a$

>

$>$

0

$0$

Discriminant Formula

Δ = b^{2} - 4 a c

$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$

A Positive Definite is a function that is always positive for all values of

x

$x$ . It is also a parabola that is concave up and is above the

x

$x$ axis

First, compute for the discriminant

m x^{2} + 4 m x + 16

$mx^2+4mx+16$

$a = m$ $a=m$ $b = 4 m$ $b=4m$ $c = 16$ $c=16$

$Δ$ $Delta$	$=$ $=$	${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$	Discriminant Formula
	$=$ $=$	${\color{#9a00c7}{4m}}^2-4\color{#00880A}{(m)}\color{#007DDC}{(16)}$	Substitute values
	$=$ $=$	$16 m^{2} - 64 m$ $16m^2-64m$

Remember that for a function to be a Positive Definite,

Δ

$Delta$

<

$<$

0

$0$

Substitute the

Δ

$Delta$ computed previously, and then solve for

m

$m$

$Δ$ $Delta$	$<$ $<$	$0$ $0$
$16 m^{2} - 64 m$ $16m^2-64m$	$<$ $<$	$0$ $0$
$16 m (m - 4)$ $16m(m-4)$	$<$ $<$	$0$ $0$
$m = 0$ $m=0$		$m = 4$ $m=4$

To determine which region around

m = 0

$m=0$ and

m = 4

$m=4$ would be included, plot these points and make a rough sketch of

16 m^{2} - 64 m

$16m^2-64m$

Replace the

x

$x$ axis with

m

$m$ axis and draw an upward parabola since

16

$16$ is positive

Remember that

Δ

$Delta$ must be negative

Therefore,

0

$0$

<

$<$

m

$m$

<

$<$

4

$4$

0

$0$

<

$<$

m

$m$

<

$<$

4

$4$

Positive and Negative Definite

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Quiz summary

Information

1. Question

Hint

Great Work!

Positive Definite

Negative Definite

2. Question

Hint

Correct!

Negative Definite

Discriminant Formula

3. Question

Hint

Keep Going!

Positive Definite

Discriminant Formula

4. Question

Hint

Fantastic!

Positive Definite

Discriminant Formula

5. Question

Hint

Excellent!

Positive Definite

Discriminant Formula

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