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For the function f(x)=2x^(2)+2x+3, find the slope of the secant line between x=-4 and x=1. Answer:

For the function f(x)=2x2+2x+3 f(x)=2 x^{2}+2 x+3 , find the slope of the secant line between x=−4 x=-4 and x=1 x=1 .\newlineAnswer:

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Q. For the function f(x)=2x2+2x+3 f(x)=2 x^{2}+2 x+3 , find the slope of the secant line between x=−4 x=-4 and x=1 x=1 .\newlineAnswer:
  1. Calculate Function Values: To find the slope of the secant line between two points on a curve, we use the formula for the slope of a line through two points, which is (f(x2)−f(x1))/(x2−x1)(f(x_2) - f(x_1)) / (x_2 - x_1), where x1x_1 and x2x_2 are the x-coordinates of the two points.
  2. Substitute x Values: First, we need to find the value of the function f(x)f(x) at x=−4x = -4. We do this by substituting x=−4x = -4 into the function f(x)=2x2+2x+3f(x) = 2x^2 + 2x + 3.f(−4)=2(−4)2+2(−4)+3=2(16)−8+3=32−8+3=27f(-4) = 2(-4)^2 + 2(-4) + 3 = 2(16) - 8 + 3 = 32 - 8 + 3 = 27.
  3. Find Slope of Secant Line: Next, we need to find the value of the function f(x)f(x) at x=1x = 1. We do this by substituting x=1x = 1 into the function f(x)=2x2+2x+3f(x) = 2x^2 + 2x + 3.f(1)=2(1)2+2(1)+3=2(1)+2+3=2+2+3=7f(1) = 2(1)^2 + 2(1) + 3 = 2(1) + 2 + 3 = 2 + 2 + 3 = 7.
  4. Find Slope of Secant Line: Next, we need to find the value of the function f(x)f(x) at x=1x = 1. We do this by substituting x=1x = 1 into the function f(x)=2x2+2x+3f(x) = 2x^2 + 2x + 3.f(1)=2(1)2+2(1)+3=2(1)+2+3=2+2+3=7f(1) = 2(1)^2 + 2(1) + 3 = 2(1) + 2 + 3 = 2 + 2 + 3 = 7.Now we have the function values at both points: f(−4)=27f(-4) = 27 and f(1)=7f(1) = 7. We can use these to find the slope of the secant line. The slope mm is given by (f(1)−f(−4))/(1−(−4))(f(1) - f(-4)) / (1 - (-4)).m=(7−27)/(1−(−4))=(−20)/(5)=−4m = (7 - 27) / (1 - (-4)) = (-20) / (5) = -4.

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