Worksheet 32
Graphing quadratic equations.
The graph of y = ax2 + bx +c will be either “U” shaped or a “U” turned upside down.
Two things you have to know:
K If a>0 then the “U” will open up. If a<0, then the “U” will open down.
K The x-coordinate of the vertex is
.
We then find the y-coordinate of the vertex by putting the x-coordinate in the equation.
The exercises (and the test question) will ask for the equation of the axis of symmetry. That is
the equation of a vertical line x = k, where k is the x-coordinate of the vertex.
Also, you will be asked for the y-intercept. That is the point (0, b) where you get b by putting 0
in for x in the equation.
You may be asked for the x-intercepts. This amounts to solving the equation ax2 + bx +c = 0.
There generally are two answers to this, but there might be only one or none. (On a test, I would
imagine that this would have to be a not hard to solve equation, say, an easy factorization.)
You may be asked to give the domain and range of the equation. The domain will be all x. The
range will start with the y-coordinate of the vertex and will include it and all numbers greater if
the graph opens up or if the graph opens down, the range will be the y-coordinate of the vertex
and all smaller numbers.
1. Does y = x2 +3x -4 open up or down?
Answer: Since the coefficient of x2 is 1 and 1 is positive, it opens up.
2. Does y = -3x2 +11x +500 open up or down?
3. Sketch a graph of
y = x2 + 2x - 35
a. The coefficient of x2 is 1. The graph opens up.
b. The x-coefficient of the vertex is -2/2 = -1.
c. The y-coefficient of the vertex is (-1)2 +2(-1) -35 = -36.
(What is the minimum value of the graph? Does it have a maximum value?)
d. The axis of symmetry is x = -1.
e. The y-intercept is (0,-35).
f. To find the x-intercepts we solve x2 + 2x - 35=0. (I picked an easy one.)
(x - 5)(x + 7) = 0
x = 5 and -7
The x-intercepts are (5, 0) and (-7, 0)
4. Sketch a graph of y = x2 -x - 12