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For the function f(x)=x^(2)-6, find the slope of the secant line between x=-3 and x=-1. Answer:

For the function f(x)=x2−6 f(x)=x^{2}-6 , find the slope of the secant line between x=−3 x=-3 and x=−1 x=-1 .\newlineAnswer:

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Q. For the function f(x)=x2−6 f(x)=x^{2}-6 , find the slope of the secant line between x=−3 x=-3 and x=−1 x=-1 .\newlineAnswer:
  1. Calculate Function Values: 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 (rise over run). We need to calculate the function values at x=−3x = -3 and x=−1x = -1 first.
  2. Find Slope Formula: Calculate the function value at x=−3x = -3.f(−3)=(−3)2−6=9−6=3f(-3) = (-3)^2 - 6 = 9 - 6 = 3
  3. Calculate Slope: Calculate the function value at x=−1x = -1.f(−1)=(−1)2−6=1−6=−5f(-1) = (-1)^2 - 6 = 1 - 6 = -5
  4. Calculate Slope: Calculate the function value at x=−1x = -1.f(−1)=(−1)2−6=1−6=−5f(-1) = (-1)^2 - 6 = 1 - 6 = -5Now we have two points on the function: (−3,f(−3))=(−3,3)(-3, f(-3)) = (-3, 3) and (−1,f(−1))=(−1,−5)(-1, f(-1)) = (-1, -5). We can use these points to find the slope of the secant line.
  5. Calculate Slope: Calculate the function value at x=−1x = -1.f(−1)=(−1)2−6=1−6=−5f(-1) = (-1)^2 - 6 = 1 - 6 = -5Now we have two points on the function: (−3,f(−3))=(−3,3)(-3, f(-3)) = (-3, 3) and (−1,f(−1))=(−1,−5)(-1, f(-1)) = (-1, -5). We can use these points to find the slope of the secant line.Use the slope formula: slope=y2−y1x2−x1\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}.Let's plug in our values: slope=−5−3−1−(−3)\text{slope} = \frac{-5 - 3}{-1 - (-3)}
  6. Calculate Slope: Calculate the function value at x=−1x = -1.f(−1)=(−1)2−6=1−6=−5f(-1) = (-1)^2 - 6 = 1 - 6 = -5Now we have two points on the function: (−3,f(−3))=(−3,3)(-3, f(-3)) = (-3, 3) and (−1,f(−1))=(−1,−5)(-1, f(-1)) = (-1, -5). We can use these points to find the slope of the secant line.Use the slope formula: slope=y2−y1x2−x1\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}.Let's plug in our values: slope=−5−3−1−(−3)\text{slope} = \frac{-5 - 3}{-1 - (-3)}Simplify the expression: slope=−5−3−1+3=−82=−4\text{slope} = \frac{-5 - 3}{-1 + 3} = \frac{-8}{2} = -4

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