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The table represents a quadratic function f(x) . x & f(x) -4 & 7 -3 & 6 -2 & 7 -1 & 10 0 & 15 1 & 22 2 & 31 If the equation of the function f(x) is written in standard form f(x)=a x^2+b x+c , what is the valu 3 6 16 22

Pergunta

The table represents a quadratic function f(x) .

 x & f(x) 
-4 & 7 
-3 & 6 
-2 & 7 
-1 & 10 
 0 & 15 
 1 & 22 
 2 & 31 


If the equation of the function f(x) is written in standard form f(x)=a x^2+b x+c , what is the valu
3
6
16
22

The table represents a quadratic function f(x) . x & f(x) -4 & 7 -3 & 6 -2 & 7 -1 & 10 0 & 15 1 & 22 2 & 31 If the equation of the function f(x) is written in standard form f(x)=a x^2+b x+c , what is the valu 3 6 16 22

Solução

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DeniseEspecialista · Tutor por 3 anos

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To find the value of \( b \) in the quadratic function \( f(x) = ax^2 + bx + c \), we can use the given points to set up a system of equations. However, since this is a multiple-choice question and we are looking for the coefficient \( b \), we can observe the pattern in the table.<br /><br />The differences between consecutive \( f(x) \) values are:<br />- From \( x = -4 \) to \( x = -3 \): \( 6 - 7 = -1 \)<br />- From \( x = -3 \) to \( x = -2 \): \( 7 - 6 = 1 \)<br />- From \( x = -2 \) to \( x = -1 \): \( 10 - 7 = 3 \)<br />- From \( x = -1 \) to \( x = 0 \): \( 15 - 10 = 5 \)<br />- From \( x = 0 \) to \( x = 1 \): \( 22 - 15 = 7 \)<br />- From \( x = 1 \) to \( x = 2 \): \( 31 - 22 = 9 \)<br /><br />The second differences (differences of the differences) are constant:<br />- From \( x = -4 \) to \( x = -2 \): \( 1 - (-1) = 2 \)<br />- From \( x = -3 \) to \( x = -1 \): \( 3 - 1 = 2 \)<br />- From \( x = -2 \) to \( x = 0 \): \( 5 - 3 = 2 \)<br />- From \( x = -1 \) to \( x = 1 \): \( 7 - 5 = 2 \)<br />- From \( x = 0 \) to \( x = 2 \): \( 9 - 7 = 2 \)<br /><br />Since the second differences are constant, this confirms that the function is quadratic. The first difference at \( x = 0 \) is 7, which corresponds to the linear term's contribution when \( x = 0 \). Therefore, the value of \( b \) is 6.
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