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Positive and Negative DefinitePositive and Negative Definite
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Question 1 of 5
1. Question
Which of the following graphs is a negative definite?Hint
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Positive Definite
`Delta``<``0` and `a``>``0`A Positive Definite is a function that is always positive for all values of `x`. It is also a parabola that is concave up and is above the `x` axisNegative Definite
`Delta``<``0` and `a``<``0`A Negative Definite is a function that is always negative for all values of `x`. It is also a parabola that is concave down and is below the `x` axisCheck the characteristics of each graph to identify which is a Negative Definite
This parabola is concave up and is above the `x` axisTherefore, it is a Positive Definite
This parabola is concave down and is below the `x` axisTherefore, it is a Negative Definite
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Question 2 of 5
2. Question
Identify whether the function below is a Negative Definite`y=-2x^2+2x-1`Hint
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Negative Definite
`Delta``<``0` and `a``<``0`Discriminant Formula
$$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$A Negative Definite is a function that is always negative for all values of `x`. It is also a parabola that is concave down and is below the `x` axisFirst, compute for the discriminant`y=-2x^2+2x-1``a=-2` `b=2` `c=-1``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` The value is negative, hence `Delta``<``0`Next, check the value of `a``y=``-2``x^2+2x-1``a=-2`The value is negative, hence `a``<``0`Therefore, we have established that the function is a Negative Definite -
Question 3 of 5
3. Question
Identify whether the function below is a Positive Definite`2x^2-x+3>0`Hint
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Positive Definite
`Delta``<``0` and `a``>``0`Discriminant Formula
$$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$A Positive Definite is a function that is always positive for all values of `x`. It is also a parabola that is concave up and is above the `x` axisFirst, compute for the discriminant`2x^2-x+3``a=2` `b=-1` `c=3``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` `=` `-23` The value is negative, hence `Delta``<``0`Next, check the value of `a``y=``2``x^2-x+3``a=2`The value is positive, hence `a``>``0`Therefore, we have established that the function is a Positive Definite -
Question 4 of 5
4. Question
For which values of `k` will the function below be a Positive Definite`y=kx^2+2kx+9`Hint
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Positive Definite
`Delta``<``0` and `a``>``0`Discriminant Formula
$$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$A Positive Definite is a function that is always positive for all values of `x`. It is also a parabola that is concave up and is above the `x` axisFirst, compute for the discriminant`y=kx^2+2kx+9``a=k` `b=2k` `c=9``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 `=` `4k^2-36k` Remember that for a function to be a Positive Definite, `Delta``<``0`Substitute the `Delta` computed previously, and then solve for `k``Delta` `<` `0` `4k^2-36k` `<` `0` `4k(k-9)` `<` `0` `k=0` `k=9` To determine which region around `k=0` and `k=9` would be included, plot these points and make a rough sketch of `4k^2-36k`Replace the `x` axis with `k` axis and draw an upward parabola since `4` is positive
Remember that `Delta` must be negativeTherefore, `0``<``k``<``9``0``<``k``<``9` -
Question 5 of 5
5. Question
For which values of `m` will the function below be a Positive Definite`mx^2+4mx+16`Hint
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Positive Definite
`Delta``<``0` and `a``>``0`Discriminant Formula
$$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$A Positive Definite is a function that is always positive for all values of `x`. It is also a parabola that is concave up and is above the `x` axisFirst, compute for the discriminant`mx^2+4mx+16``a=m` `b=4m` `c=16``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 `=` `16m^2-64m` Remember that for a function to be a Positive Definite, `Delta``<``0`Substitute the `Delta` computed previously, and then solve for `m``Delta` `<` `0` `16m^2-64m` `<` `0` `16m(m-4)` `<` `0` `m=0` `m=4` To determine which region around `m=0` and `m=4` would be included, plot these points and make a rough sketch of `16m^2-64m`Replace the `x` axis with `m` axis and draw an upward parabola since `16` is positive
Remember that `Delta` must be negativeTherefore, `0``<``m``<``4``0``<``m``<``4`
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- Intro to Quadratic Functions (Parabolas) 1
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