MATH SOLVE

4 months ago

Q:
# graph the following function and identify any shifts, stretches, and symmetry and x and y intercepts:y = f(x) = (x - 3)^2 + 2

Accepted Solution

A:

Shifts:Shifted three units right and two units up to match the blue parabolaStretches:It doesn't have any stretchesSymmetry:[tex]x=3[/tex]x and y intercepts:x intercpets: It has no x-interceptsy-intercepts: (0, 11)Explanation:The pattern of the quadratic function is:[tex]f(x)=x^2[/tex]Whose graph is shown below as the red parabola.So here we need to identify some characteristics of:[tex]y = f(x) = (x - 3)^2 + 2[/tex]Whose graph is shown below as the blue parabola.As you can see, the blue parabola is a transformation of the red parabola. The rule is as follows:The red parabola has been shifted three units right and two units up to match the blue parabola.This is so because, for any function:[tex]y=f(x)[/tex]We have the following transformations:[tex]y=h(x)=f(x+c)+k \\ \\ \\ \bullet \ c>0 \ Shift \ f(x) \ c \ units \ to \ the \ left \\ \\ \bullet \ c<0 \ Shift \ f(x) \ c \ units \ to \ the \ right \\ \\ \bullet \ k>0 \ Shift \ f(x) \ k \ units \ up \\ \\ \bullet \ k<0 \ Shift \ f(x) \ k \ units \ down[/tex]On the other hand, the blue graph has neither stretches nor x intercepts. Finally, its axes of symmetry is [tex]x=3[/tex]. The y-intercept is (0, 11) is indicated in the figure.Learn more:Shifting graphs: