Find two or three people to work with at the boards, and
work through the fol lowing problems .
Take turns writing, everyone should get a chance to write for some of the
problems. Also, make
sure everyone in your group understands the solution. Simply getting the answer
is not the point
of these worksheets, rather discuss the solution to make sure you and your other
group members
understand it.
Quadratic Functions and Optimization
1. Sketch each of the following parabolas on the same set of axes. Make sure
their positions relative
to each other are accurate.
2. Sketch graphs of each of the following functions, and
specify the vertex, axis of symmetry, max-
imum or minimum values, and x- and y-intercepts.
3. Find a quadratic function such that the axis of
symmetry is the line x = 1, the y- intercept is 1,
and there is only one x-intercept. (It might help to sketch a graph of the
parabola first.)
4. For which value of c will the minimum value of the function f(x) = x2 + 2x +
c be 
?
5. (a) Complete the square to show that the coordinates of the vertex of the
parabola y = ax2+bx+c
are (-b/2a,-D/4a) where D = b2- 4ac.
(b) Compute the average value of the two x-intercepts of the graph of y = ax2 + bx + c. (As sume
b2 - 4ac > 0.) How does your answer relate to part (a) and what does this mean?
6. Find two numbers whos difference is 100 and whose
product is a minimum. (Hint, first find a
function of one variable for the product of the two numbers, then find the minimum
of that function.)
7. Find the are of the largest rectangle that can be inscribed in a right
traingle with legs of lengths
3 cm and 4 cm if two sides of the rectangle lie along the legs. (You might want
to use simular
triangles).
8. Find the dimensions of a rectangle with perimiter P whose area is a maximum.
Your answer
will be in terms of P . What is the significance of your answer ?
9. (a) A point (x, y) lies on the line y = 2x+1. Ex press
the distance between the point (x, y) and
the point (5, 1) as a function of x.
(b) Find the coordinates of the point on the line y = 2x + 1 which is closest to
the point (5, 1).
(You can consider just the quadratic that is inside the square root for this
problem . Why?)
