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

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

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Full solution

Q. For the function f(x)=−x2−3x+3 f(x)=-x^{2}-3 x+3 , find the slope of the secant line between x=−4 x=-4 and x=4 x=4 .\newlineAnswer:
  1. Define Slope Formula: 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}. Here, x1=−4x_1 = -4 and x2=4x_2 = 4.
  2. Find f(−4)f(-4): First, we need to find the value of the function at x=−4x = -4, which is f(−4)=−(−4)2−3(−4)+3f(-4) = -(-4)^2 - 3(-4) + 3.
  3. Find f(4)f(4): Calculating f(−4)f(-4) gives us f(−4)=−16+12+3=−1f(-4) = -16 + 12 + 3 = -1.
  4. Calculate Slope: Next, we need to find the value of the function at x=4x = 4, which is f(4)=−(4)2−3(4)+3f(4) = -(4)^2 - 3(4) + 3.
  5. Use Two Points: Calculating f(4)f(4) gives us f(4)=−16−12+3=−25f(4) = -16 - 12 + 3 = -25.
  6. Calculate Secant Slope: Now we have the two points on the function: (−4,−1)(-4, -1) and (4,−25)(4, -25). We can use these to find the slope of the secant line.
  7. Simplify Slope Calculation: The slope of the secant line is (f(4)−f(−4))/(4−(−4))=(−25−(−1))/(4−(−4))(f(4) - f(-4)) / (4 - (-4)) = (-25 - (-1)) / (4 - (-4)).
  8. Simplify Slope Calculation: The slope of the secant line is (f(4)−f(−4))/(4−(−4))=(−25−(−1))/(4−(−4))(f(4) - f(-4)) / (4 - (-4)) = (-25 - (-1)) / (4 - (-4)). Simplifying the slope calculation gives us (−25+1)/(4+4)=−24/8=−3(-25 + 1) / (4 + 4) = -24 / 8 = -3.

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