SOLUTION: f(x)=1/4(x-5)^2+6 find the vertex line of symmetry and the max and min value of f(x) graph the function

Question 263706: f(x)=1/4(x-5)^2+6 find the vertex line of symmetry and the max and min value of f(x) graph the function
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
When you learn about various kinds of graphs (lines, parabolas, circles, etc.) you learn how to recognize which graph an equation will have and you learn a form of equation for that graph in which you can "read" the critical information for the graph directly from the equation.

Lines do not have squared terms. This is a parabola because it has and no or xy terms. Unfortunately there is a variety of forms of the equation for a parabola that we can "read". First there is a form for vertically oriented parabolas (where the "bowl" opens upward or downward) and there is a form for horizontally oriented (where the "bowl" opens to the right or left). And on top of this the forms seem to vary from one book to another.

Your equation is a vertically oriented one (because the x is the squared variable, not the y). The form we can "read" for vertically oriented parabolas may be one of the following (or something equivalent to one of these):

The "a" and the "p" from these might be different letters. And there may be a different coefficient to the "p". The most important part of these is the "h". And you can see that there is consistency to where the "h" is found. The "h" is the x coordinate of the vertex. With this we can find everything we need for your problem.

Your equation has a 5 for "h" so the x coordinate of your vertex is 5. We can use this to find the y coordinate of the vertex:

So the vertex is (5, 6). (You might notice that the "k" in the equation was the y coordinate of the vertex.) Since this is a vertically oriented parabola, the axis of symmetry is the vertical line through the vertex: x = 5.

And finally since this is a vertically oriented parabola the vertex is either the maximum value for the parabola (if it opens downward) or the minimum value (if it opens upward). We can determine upward or downward a couple of ways:

"Read" the equation. If the "a" or "p" (or whatever letter is used) is positive then the parabola opens upward. (A negative "a" or "p" means the parabola opens downward.)
Find another point. Choose another x value and find its y value. If this 2nd y value is greater than the y of the vertex, the vertex is a minimum (i.e. the parabola opens upward) and if the 2nd y value is less that the y of the vertex then the vertex is a maximum (i.e. the parabola opens downward).

Either of these methods tell you that your parabola opens upward so the vertex is the minimum value for the function.
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