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

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

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Full solution

Q. For the function f(x)=x2+2x+4 f(x)=x^{2}+2 x+4 , find the slope of the secant line between x=1 x=1 and x=4 x=4 .\newlineAnswer:
  1. Calculate f(1)f(1): 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)(f(x_2) - f(x_1)) / (x_2 - x_1). We need to calculate the function values at x=1x = 1 and x=4x = 4.
  2. Calculate f(4)f(4): First, calculate f(1)f(1) by substituting x=1x = 1 into the function f(x)=x2+2x+4f(x) = x^2 + 2x + 4.\newlinef(1)=(1)2+2(1)+4=1+2+4=7f(1) = (1)^2 + 2(1) + 4 = 1 + 2 + 4 = 7.
  3. Use slope formula: Next, calculate f(4)f(4) by substituting x=4x = 4 into the function f(x)=x2+2x+4f(x) = x^2 + 2x + 4. \newlinef(4)=(4)2+2(4)+4=16+8+4=28f(4) = (4)^2 + 2(4) + 4 = 16 + 8 + 4 = 28.
  4. Find slope: Now, use the slope formula with f(1)f(1) and f(4)f(4) to find the slope of the secant line.Slope=f(4)−f(1)4−1=28−74−1=213=7.\text{Slope} = \frac{f(4) - f(1)}{4 - 1} = \frac{28 - 7}{4 - 1} = \frac{21}{3} = 7.

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