Math 475 Homework 9a
Due April 22rd, 2011
1. Derive our class solution for x2 + Bx + C = 0, then explain how you can solve ANY quadratic equation
using this special case and derive the Quadratic Formula.
2. Axis of Symmetry, Four Ways
Let’s carry out four approaches to deducing the formula for the axis of symmetry of a quadratic. For
these approaches, do not use the formula; we are justifying it!
(a) For each of the following quadratic equations, (i) find the x-intercepts, (i) find the axis of symmetry
halfway between from the x-intercepts; and (iii) put it in standard form and explain the pattern that
relates the coefficients to the axis of symmetry.
y = (x − 4)(x − 2)
y = (x + 7)(x − 2)
y = (x + 5)(x + 1)
y = (x − j)(x − k)
i. What is the relationship between the axis of symmetry of y = (x − 4)(x − 2), y = 3(x − 4)(x − 2)
and y = 700(x − 4)(x − 2)? Explain your reasoning.
ii. Describe the relationship between the axis of symmetry of y = Ax2 + Bx + C and y = x2 +
(B/A)x + (C/A). Then guess a formula for the axis of symmetry of y = Ax2 + Bx + C.
(b) Consider y = x2 − 6x + 10. Lets agree that if we can find two x-values which give the same y value,
then the axis of symmetry lies halfway between the two.
i. Find the y-intercept. Then find another x value whose y-value equals the y-intercept. Use this
information to calculate the axis of symmetry.
ii. Repeat the calculations in (2(b)i) for the general quadratic y = Ax2 + Bx + C and find a formula
for the axis of symmetry.
(c) Explain how you can deduce the axis of symmetry of a quadratic from the quadratic formula.
(d) Explain how you can deduce the axis of symmetry of a quadratic from its derivative.
3. Which of the following could be the equation for the parabola in the figure? Only one answer is correct.
Give a mathematical reason why you chose or rejected each answer.
(a) y = x2 − 6x + 7
(b) y = −x2 − 6x − 11
(c) y = −x2 + 6x
(d) y = −x2 + 6x − 8
(e) None of the above are possible.
4. Finding a root of a polynomial is equivalent to finding a linear factor. We can show this as follows. Let p
be a polynomial. Prove that p(a) = 0 ⇐⇒ p(x) = (x − a)r(x) for some other polynomial r such that the
degree of r is less than the degree of p.
(Hint. The reverse direction is easy. The forward direction can be done using induction on the degree of
p. If p has degree n + 1 and has leading coefficient A, then you can show that p(x) − (x − a)(Axn ) has
at most degree n. Now apply the inductive hypothesis.)