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For the function f(x)=x^(2)+4x+4, find the slope of the secant line between x=-7 and x=-5. Answer:

For the function f(x)=x2+4x+4 f(x)=x^{2}+4 x+4 , find the slope of the secant line between x=−7 x=-7 and x=−5 x=-5 .\newlineAnswer:

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Full solution

Q. For the function f(x)=x2+4x+4 f(x)=x^{2}+4 x+4 , find the slope of the secant line between x=−7 x=-7 and x=−5 x=-5 .\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 (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}, where x1x_1 and x2x_2 are the x-values of the two points.
  2. Calculate f(−7)f(-7): First, we need to find the yy-values for the xx-values given. Let's start with x=−7x = -7. We plug it into the function f(x)=x2+4x+4f(x) = x^2 + 4x + 4 to get f(−7)f(-7).\newlinef(−7)=(−7)2+4(−7)+4=49−28+4=25f(-7) = (-7)^2 + 4(-7) + 4 = 49 - 28 + 4 = 25.
  3. Calculate f(−5)f(-5): Next, we find the y-value for x=−5x = -5. We plug it into the function f(x)=x2+4x+4f(x) = x^2 + 4x + 4 to get f(−5)f(-5).\newlinef(−5)=(−5)2+4(−5)+4=25−20+4=9f(-5) = (-5)^2 + 4(-5) + 4 = 25 - 20 + 4 = 9.
  4. Find Slope of Secant Line: Now we have the two points: (−7,25)(-7, 25) and (−5,9)(-5, 9). We can use these to find the slope of the secant line.\newlineSlope = (f(−5)−f(−7))/(−5−(−7))=(9−25)/(−5+7)=(−16)/2=−8(f(-5) - f(-7)) / (-5 - (-7)) = (9 - 25) / (-5 + 7) = (-16) / 2 = -8.

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