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?

Answer 1

See the explanation section, below.

The secant line to the graph of #f(x) = -x^2#; through the points where #x=-2#, and #x=-4# is the line through the points:
#(-2,f(-2))# and #(-4, f(-4))#
Which are the points: #(-2,-4)# and #(-4,-16)#
The slope is #m = ((-16)-(-4))/((-4)-(-2)) = (-12)/-2 =6#
The line has equation #y+4=6(x+2)#
Or #y+16=6(x+4)#
Or #y=6x+8#
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Answer 2

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 point-slope formula.

  1. 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) ).

  2. 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 ]

  3. Now, you have the slope ( m = 6 ) and one point ( (-2, -4) ). You can use the point-slope formula to find the equation of the secant line.

    Point-slope formula: [ y - y_1 = m(x - x_1) ]

    Using ( (-2, -4) ) and ( m = 6 ): [ y - (-4) = 6(x - (-2)) ] [ y + 4 = 6(x + 2) ]

  4. 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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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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