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

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

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Q. For the function f(x)=x2−2 f(x)=x^{2}-2 , find the slope of the secant line between x=1 x=1 and x=4 x=4 .\newlineAnswer:
  1. Slope Formula: To find the slope of the secant line between two points on a function, we use the slope formula: slope=f(x2)−f(x1)x2−x1\text{slope} = \frac{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. Find y-coordinate at x=11: First, we need to find the y-coordinate for the point where x=1x = 1 by plugging it into the function f(x)=x2−2f(x) = x^2 - 2.\newlinef(1)=(1)2−2=1−2=−1f(1) = (1)^2 - 2 = 1 - 2 = -1.
  3. Find y-coordinate at x=4x=4: Next, we need to find the y-coordinate for the point where x=4x = 4 by plugging it into the function f(x)=x2−2f(x) = x^2 - 2.f(4)=(4)2−2=16−2=14f(4) = (4)^2 - 2 = 16 - 2 = 14.
  4. Calculate Slope: Now we have two points: (1,f(1))=(1,−1)(1, f(1)) = (1, -1) and (4,f(4))=(4,14)(4, f(4)) = (4, 14). We can use these points to find the slope of the secant line.slope=f(4)−f(1)4−1=14−(−1)4−1=14+13=153.\text{slope} = \frac{f(4) - f(1)}{4 - 1} = \frac{14 - (-1)}{4 - 1} = \frac{14 + 1}{3} = \frac{15}{3}.
  5. Final Result: Calculating the slope gives us 153=5\frac{15}{3} = 5. So, the slope of the secant line between x=1x = 1 and x=4x = 4 for the function f(x)=x2−2f(x) = x^2 - 2 is 55.

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