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For the function f(x)=x^(2)+3, find the slope of the secant line between x=-5 and x=-3. Answer:

For the function f(x)=x2+3 f(x)=x^{2}+3 , find the slope of the secant line between x=−5 x=-5 and x=−3 x=-3 .\newlineAnswer:

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Q. For the function f(x)=x2+3 f(x)=x^{2}+3 , find the slope of the secant line between x=−5 x=-5 and x=−3 x=-3 .\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 (change in y)/(change in x)(\text{change in } y) / (\text{change in } x), 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=−5x = -5 and x=−3x = -3.
  2. Substitute x Values: First, calculate the function value at x=−5x = -5. We substitute xx with −5-5 into the function f(x)=x2+3f(x) = x^2 + 3.\newlinef(−5)=(−5)2+3=25+3=28f(-5) = (-5)^2 + 3 = 25 + 3 = 28.
  3. Find Slope: Next, calculate the function value at x=−3x = -3. We substitute xx with −3-3 into the function f(x)=x2+3f(x) = x^2 + 3.f(−3)=(−3)2+3=9+3=12f(-3) = (-3)^2 + 3 = 9 + 3 = 12.
  4. Simplify Expression: Now we have the function values at both points: f(−5)=28f(-5) = 28 and f(−3)=12f(-3) = 12. We can use these values to find the slope of the secant line.\newlineSlope = f(−3)−f(−5)−3−(−5)=12−28−3+5\frac{f(-3) - f(-5)}{-3 - (-5)} = \frac{12 - 28}{-3 + 5}.
  5. Simplify Expression: Now we have the function values at both points: f(−5)=28f(-5) = 28 and f(−3)=12f(-3) = 12. We can use these values to find the slope of the secant line. Slope = f(−3)−f(−5)−3−(−5)=12−28−3−5\frac{f(-3) - f(-5)}{-3 - (-5)} = \frac{12 - 28}{-3 - 5}. Simplify the expression to find the slope. Slope = 12−28−3+5=−162=−8\frac{12 - 28}{-3 + 5} = \frac{-16}{2} = -8.

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