Chapter Closure
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Function:& y=a(x-(-3))^2+(-12) Vertex:& (-3,-12) This simplifies to y=a(x+3)^2-12. Let's show this transformation.
Finally, we want to stretch the parabola by a factor of 0.25 and make it open downward. If a=0.25 in the graphing form, the parabola will stretch by a factor of 0.25. However, to make it open downward, the value of a must also be negative. With this information, we can write the function. Function:& y= -0.25(x+3)^2-12 Vertex:& (-3,-12) Stretch Factor:& -0.25 Let's show this final transformation.
General Equation:& y= a(x-h)^3+k Locator Point:& (h,k) Stretch Factor:& a We have been given both the function's stretch factor and locator point. Let's substitute these into the general equation. Function:& y= 2(x- (-6))^3+1 Locator Point:& (-6,1) Stretch Factor:& 2 The function simplifies to y=2(x+6)^3+1.
General Equation:& y=a(1/x-h)+k [0.8em] Locator Point:& (h,k) [0.8em] Horizontal Asymptote:& y=k [0.8em] Vertical Asymptote:& x=h To end up with a hyperbola with the desired asymptotes, we have to substitute k=-6 and h=2 in the general equation. General Equation:& y=1/x-2+(-6) [0.8em] Locator Point:& (2,-6) [0.8em] Horizontal Asymptote:& y=-6 [0.8em] Vertical Asymptote:& x=2 The function simplifies to y= 1x-2-6