When asked to compare one graph to another, we usually need to look at our transformations. Let's start this one with looking at the graphs.
From the graph, we can see that the red curve is narrower, reflected across the x -axis and translated up compared to the blue curve. We can get the same information by looking ath the algebraic tools of transformations.
Transformations of f(x)
Vertical Translations
Translation up k units, k>0
y=f(x)+ k
Translation down k units, k<0
y=f(x)+ k
Vertical Stretch or Compression
Vertical stretch, a>1
y= af(x)
Vertical compression, 0< a< 1
y= af(x)
Reflections
In the x-axis
y=- f(x)
In the y-axis
y=f(- x)
From our function, h = -16t^2+ 14, we can see that 14 represents a vertical shift 14 units up. From -16, we can also conclude that the function is vertically stretched by a factor of 16. Finally, the negative on the quadratic term means the function is reflected across the x-axis.
