A quadratic function y = f(x) is plotted on a graph and the vertex of the resulting parabola is (4, 3) . What is the vertex of function defined as g(x) = - f(x) + 5

Answer 1

The generic way to write an equation for a quadratic function is:

f(x) = a(x - h)^2 + k

The h term indicates how much the graph is shifted left or right; along the x axis. The k term indicates how much the graph is shifted up and down; along the y axis. The a term indicates how much the graph is stretched or how steep or flat the curve is.

Therefore, in this expression the vertex is a (h,k).

Assuming we don't need to worry about the value of a we can plug in the vertex given to the equation for f(x):

f(x) = (x-4)^2 + 3

The problem gives us:

g(x) = -f(x) + 5

g(x) = -(x-4)^2 -3 +5

g(x) = -(x-4)^2 + 2

In this can a = -1, h = 4, and k = 2.

Because the value of x or h is unchanged the vertex is only being moved along the y-axis. Meaning the vertex moves to (4,2).

Answer 2

Notice that g(x) = -f(x) + 5 means "Take the function f(x), flip it up-down and then shift it up 5".

Here I refer to the flip as "up-down" because textbooks can be confusing with their wording here. You could say you're "reflecting the function in the y-axis" or "reflecting the function over the x-axis" and these mean the same thing.

Notice that in our translation there is no right or let shift, and no left-right flip. It reflects up-down and shifts up. So g(x) is directly above or below f(x) for every value of x.

We know that f(4) = 3 is the vertex on the curve of the quadratic function f(x). Thus g(x) is still a parabola whose vertex is at g(4).

g(4) = -f(4) + 5 = -3 + 5 = 2