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FromTheMoon [43]
3 years ago
14

Find the quadratic function that passes through the points: (-1,6), (1,4), and (2,9).

Mathematics
1 answer:
Masja [62]3 years ago
8 0

Answer:

The quadratic function that passes through given points is                           y = 2 x² - x + 3  .

Step-by-step explanation:

The given quadratic function as

y = a x² + b x + c

The equation passes through the points ( - 1 , 6 ) , ( 1 , 4 ) and ( 2, 9 )

As The points passes through equation then

At points  ( - 1 , 6 )

6 = a (1)² + b ×( - 1 ) + c

Or, a - b + c = 6           .....A

Again At points  ( 1 , 4 )

4 = a (1)² + b × 1 + c

Or, a + b + c = 4            .......B

<u>Similarly At points  ( 2 , 9 )</u>

9 = a (2)² + b × 2 + c

Or, 4 a +2 b + c = 9        ....,,,C

<u>Now solving equation A and B</u>

(  a - b + c ) + (  a + b + c ) = 6 + 4

Or, a + c = \frac{10}{2}  

I.e a + c = 5           ......D

<u>Similarly Solving equation B and C</u>

( 4 a +2 b + c  ) - 2 × ( a + b + c ) = 9 - 2 × 4

Or, ( 4 a - 2 a + 2 b - 2 b + c - 2 c ) = 9 - 8

Or, ( 2 a - c ) = 1        .....E

<u>Solving D and E</u>

( a + c ) + ( 2 a - c ) = 5 + 1

Or, 3 a = 6

∴  a = \frac{6}{3}

I.e a = 2

<u>Put the value of a in Eq D</u>

So ,  a + c = 5

Or,  c = 5 - a

∴  c = 5 - 2 = 3

I.e  c = 3

<u>Put The value of a and c in Eq A</u>

a - b + c = 6      

Or, b = a + c - 6

Or . b = 2 + 3 - 6

∴ , b = 5 - 6

I.e   b = - 1

Now, <u>Putting the values of a , b , c in the given quadratic equation</u>

I.e y = a x² + b x + c

Or, y = 2 x² + ( - 1 ) x + 3

∴ The quadratic eq is  y = 2 x² - x + 3

Hence The quadratic function that passes through given points is                 y = 2 x² - x + 3  . Answer

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Step-by-step explanation:

(5 - 3)⁴ - 2(7) + 8²

PEMDAS

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adoni [48]

Answer:

  f(x) = {x-3 for x ≤ -1; -3x+14 for x > 5}

Step-by-step explanation:

To write the piecewise function, we can consider the pieces one at a time. For each, we need to define the domain, and the functional relation.

__

<h3>Left Piece</h3>

The domain is the horizontal extent. It is shown as -∞ to -1, with -1 included.

The relation has a slope (rise/run) of +1, and would intersect the y-axis at -3 if it were extended.

The first piece can be written ...

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<h3>Right Piece</h3>

The domain is shown as 5 to ∞, with 5<em> not included</em>.

The relation is shown as having a slope (rise/run) of (-3)/(1) = -3. If extended, it would intersect the point (5, -1), so we can write the point-slope equation as ...

  y -(-1) = -3(x -5)

  y = -3x +15 -1 = -3x +14

The second piece can be written ...

  f(x) = -3x +14 for x > 5

__

<h3>Whole function</h3>

Putting these pieces together, we have ...

  \boxed{f(x)=\begin{cases}x-3&\text{for }x\le-1\\-3x+14&\text{for }5 < x\end{cases}}

_____

<em>Additional comment</em>

Sometimes it is convenient to write inequalities in number-line order (using < or ≤ symbols). This gives a visual indication of where the variable stands in relation to the limit(s). Perhaps a more conventional way to write the domain for the second piece is, <em>x > 5</em>.

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Can you help me with my math?
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Answer:

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