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For the function f(x)=-x^(2)+2x-6, find the slope of the secant line between x=-4 and x=3. Answer:

For the function f(x)=−x2+2x−6 f(x)=-x^{2}+2 x-6 , find the slope of the secant line between x=−4 x=-4 and x=3 x=3 .\newlineAnswer:

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Full solution

Q. For the function f(x)=−x2+2x−6 f(x)=-x^{2}+2 x-6 , find the slope of the secant line between x=−4 x=-4 and x=3 x=3 .\newlineAnswer:
  1. Calculate f(−4)f(-4): To find the slope of the secant line between two points on a function, we use the formula for slope, which is the change in yy divided by the change in xx, or f(x2)−f(x1)x2−x1\frac{f(x_2) - f(x_1)}{x_2 - x_1}. We need to calculate the function values at x=−4x = -4 and x=3x = 3.
  2. Calculate f(3)f(3): First, we calculate f(−4)f(-4). We substitute xx with −4-4 into the function f(x)=−x2+2x−6f(x) = -x^2 + 2x - 6.\newlinef(−4)=−(−4)2+2(−4)−6=−16−8−6=−30f(-4) = -(-4)^2 + 2(-4) - 6 = -16 - 8 - 6 = -30.
  3. Find slope formula: Next, we calculate f(3)f(3). We substitute xx with 33 into the function f(x)=−x2+2x−6f(x) = -x^2 + 2x - 6. f(3)=−(3)2+2(3)−6=−9+6−6=−9f(3) = -(3)^2 + 2(3) - 6 = -9 + 6 - 6 = -9.
  4. Calculate slope: Now we have the function values f(−4)=−30f(-4) = -30 and f(3)=−9f(3) = -9. We can use these to find the slope of the secant line. The slope mm is given by (f(3)−f(−4))/(3−(−4))(f(3) - f(-4)) / (3 - (-4)).
  5. Calculate slope: Now we have the function values f(−4)=−30f(-4) = -30 and f(3)=−9f(3) = -9. We can use these to find the slope of the secant line. The slope mm is given by (f(3)−f(−4))/(3−(−4))(f(3) - f(-4)) / (3 - (-4)).We calculate the slope mm. m=(−9−(−30))/(3−(−4))=(−9+30)/(3+4)=21/7=3m = (-9 - (-30)) / (3 - (-4)) = (-9 + 30) / (3 + 4) = 21 / 7 = 3.

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