Quadratic Functions and Their Graphs
A quadratic function is one of the form f(x) = ax2 +bx+c,
where a ≠ 0. You should be able (easily and quickly!) to do
any of the following:
Solve a quadratic equation ax2 + bx + c = 0 for x by one
of the following methods:
- factorization of ax2 + bx + c into a product of the form (Ax - B)(Cx
- D),
giving solutions x = -B/2A and
x = -D/C,
- using the quadratic formula,
which you
should immediately and permanently memorize,
since you will be using it a lot,
- completing the square.
Graph any quadratic function f(x) = ax2 + bx + c,
finding
and labeling at least 5 points on the graph, including
- its y-intercept (0, f(0)), i.e. (0, c),
- its x-intercepts, if any, which are given by the solutions of the associated
equation ax2 + bx + c = f(x) = 0,
- its vertex, (-b/2a, f(-b/2a)).
It should not take you more than a couple of minutes to
accomplish all this for a given function f. It is considered a
routine background skill.
You should know, without actually graphing, that the graph
of f(x) = ax2 + bx + c has the same shape and direction as
the graph of y = ax2, that it is a parabola opening up if a > 0 and down if a <
0, that it is symmetric with respect to its
axis, the line x = -b/2a.
EXERCISES:
1. Solve the following equations. [Note that while one can
always use the quadratic formula, if the relevant quadratic
happens to factor over the integers, then factoring is generally quicker and
less likely to produce arithmetic simplification
errors. Whenever you solve by factoring, it is a good habit to mentally multiply
the product back out to make certain
you have not made a mistake.]
(a) 3x2 - 4x = 0
(b) 2x + 5x2 = 0
(c) x2 - 4 = 0
(d) 3x2 - 5 = 0
(e) x2 + 4
(f) 4x2 + 9 = 0
(g) x2 + 2x - 24 = 0
(h) 12x2 + 19x + 5 = 0
(i) 24 - 2x - x2 = 0
(j) x2 + 2x - 20 = 0
(k) 12x2 + 19x + 30 = 0
(l) 6x2 + x + 5 = 0
(m) -11x2 + x - 1 = 0
(n) 9 + 6x + x2 = 0
(o) 4x2 - 44x + 121 = 0
2. Solve for x. (Except in a)-c), you will want to rewrite
the equation in an equivalent form ax2 +bx+c = 0, and proceed
as you did in the exercises of part 1 above.
(a) x2 = 4
(b) 3x2 = 5
(c) 6x2 = -5
(d) 6x2 = -2x2
(e) 5x2 = 2x (careful!)
(f) x2 + 2x = 24
(g) x2 + 12x + 3 = 10x + 21
(h) x2 + 2x = 24
(i) 3x2 + 2x = 2x2 + 20
(j) 4x2 + 19/3x = -10
(k) x + 5 = 6x2
(l) x = 1 + 11x2
(m) (x - 5)2 = 12
(n) (x + 2)(x - 3) = 5
(o) -x(2x + 10) = x - 1
3. Graph the following functions, labeling the coordinates of at least 5 points on the graph, including all x- and y-intercepts and the vertex. [You already found the x-intercepts of many of these functions in the exercises in part 1 above.]
(a) f(x) = 3x2 - 4x
(b) f(x) = x2 - 4
(c) f(x) = 4x2 + 9
(d) f(x) = -4x2 - 121
(e) f(x) = x2 + 2x - 24
(f) f(x) = x2 + 2x - 20
(g) f(x) = 6x2 + x + 5
(h) f(x) = x2 + 6x + 9
(i) f(x) = -4x2 + 44x - 121
(j) f(x) = (x - 5)2 - 12
(k) f(x) = -x(x + 2) + 3x2 - 4x(3x + 1)