This is a parabola, which is the graph of a quadratic function. We can write the equation of quadratic functions using graphing form.
Graphing Form:& y=a(x-h)^2+k
Vertex:& (h,k)
When written in graphing form, the vertex has the coordinates (h,k). Our function has its vertex at (2,3), so h=2 and k=3.
Function:& y=a(x-2)^2+3
Vertex:& (2,3)
We can find a by substituting a point on the graph for x and y and solve the resulting equation. The graph seems to pass through (1,4), so let's use that.
We found that a=1, we can add this to our equation.
y=1 (x-2)^2+3
⇕
y=(x-2)^2+3
b Now let's consider the second graph.
This is a cubic function. Cubic functions can be written using the following equation where h is how much the graph is translated horizontally and k is how much it is translated vertically when compared to the parent function, f(x)=x^3.
y=(x-h)^3+k
Let's draw the graph of the parent function together with our function. We will also mark the horizontal and vertical distance between the graphs.
We can see that, when compared to the parent function's graph, our graph is translated 2 units to the right and 3 units up. Thus, h=2 and k=3. Using this, we can write an equation for the graph.
y=(x-2)^3+3
c Finally, let's look at the third graph.
This is a quadratic function in which the parabola is opening downward. Since we know the parabola's vertex, we can write its equation in graphing form.
Graphing Form:& y=a(x-h)^2+k
Vertex:& (h,k)
Our function has the vertex in (- 6,0). In the equation we find the vertex as the constants h and k. Therefore, we will have h=- 6 and k= .
Function:& y=a(x-(- 6))^2+
Vertex:& (-6, )
This simplifies to y=a(x+6)^2. Let's now find a by substituting a point on the graph for x and y in the equation. Let's use (- 5,- 2).
