Take notice of where the closed and open circles are.
y=-(x-2)^2, & x < 2
x+2, & x ≥ 2
Practice makes perfect
To write the piecewise function for the given graph, we need to find the equation of the line representing each piece and then restrict the domain accordingly.
The First Piece
Let's take a look at the first piece of graph.
As we can see, our piece is half of a parabola that has a vertex (0,2) and passes through the point (0,-4). To write the rule of this function, let's first recall the vertex form of a quadratic function.
y= a(x- h)^2+kIn this form, ( h,k) is the vertex of the parabola. Since we know that the vertex of our function is ( 2, ), we have that h= 2 and k= . We can use these values to partially write our equation.
y= a(x- 2)^2+ ⇒ y=a(x-2)^2
Finally, to find the value of a, we will use the fact that the function passes through (0,-4). We can substitute 0 for x and 0 for y in the above equation and solve for a.
Now that we know that a= -1, we can complete the equation of the first piece of our function.
y= -1(x- 2)^2
The Second Piece
Now let's take a look at the second piece.
This line has a slope of 1 and the y-intercept is 2. We can write the equation for this piece in slope-intercept form.
y=1x+ 2 ⇔ y=x+2
Combining the Pieces
We can add the equations of these lines to the piecewise function notation.
y(x)=
-(x-2)^2
x+2
Finally, we need to determine the domain for each equation. The "jump" occurs at x=2, so this will be where the domains are divided. We cannot have an overlap in our domains, so we need to take notice of where the closed and open circles are located.
The open circle at the end of the first piece tells us that its domain does not include 2. The closed circle at the beginning of the second piece tells us that its domain includes 2.
y=
-(x-2)^2, & x< 2
x+2, & x≥2
