How do you find the equation of the secant line through the points where X has the given values: #f(x) = x^2#; x=2, x=4?
See the explanation section, below.
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To find the equation of the secant line through the points where ( x ) has the given values ( x = 2 ) and ( x = 4 ) for the function ( f(x) = x^2 ), you first need to find the corresponding ( y )values for these ( x )values. Then you can use the two points to determine the equation of the secant line using the pointslope formula.

Evaluate ( f(x) = x^2 ) for ( x = 2 ) and ( x = 4 ) to find the corresponding ( y )values.
For ( x = 2 ): [ f(2) = (2)^2 = 4 ] So, the first point is ( (2, 4) ).
For ( x = 4 ): [ f(4) = (4)^2 = 16 ] So, the second point is ( (4, 16) ).

Use the two points ( (2, 4) ) and ( (4, 16) ) to find the slope of the secant line.
Slope ( m ) can be calculated using the formula: [ m = \frac{{y_2  y_1}}{{x_2  x_1}} ]
[ m = \frac{{16  (4)}}{{4  (2)}} = \frac{{16 + 4}}{{4 + 2}} = \frac{{12}}{{2}} = 6 ]

Now, you have the slope ( m = 6 ) and one point ( (2, 4) ). You can use the pointslope formula to find the equation of the secant line.
Pointslope formula: [ y  y_1 = m(x  x_1) ]
Using ( (2, 4) ) and ( m = 6 ): [ y  (4) = 6(x  (2)) ] [ y + 4 = 6(x + 2) ]

Simplify the equation to obtain the final equation of the secant line:
[ y + 4 = 6x + 12 ] [ y = 6x + 12  4 ] [ y = 6x + 8 ]
Therefore, the equation of the secant line through the points where ( x = 2 ) and ( x = 4 ) for the function ( f(x) = x^2 ) is ( y = 6x + 8 ).
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