Question 1 :
Describe in your own words how to graph a parabola. How do you know if the parabola
opens upward or downward? What is the vertex of a parabola and how do you find it?
Demonstrate with an example. What is a mathematical model and give examples when
using a quadratic function?
The equation of a parabola has the form y = ax2 + bx + c.
You can determine the graph opens upward or downward by noting the value of
the coefficient a. If a is a positive value, then the graph opens upward. If a is a
negative value, then the graph opens downward.
The vertex of the parabola is the point where the graph has its maximum or
minimum value. Using the coefficients of the equation, the x-coordinate of the
vertex has the value x = -b / 2*a. Once the x-coordinate of the vertex is known,
the value of the corresponding y-coordinate can be determined by evaluating the
equation of the parabola at x = -b/2*a.
Once the vertex is known, you can select a few values of x on either side of the
vertex and evaluate the equation at those values. This will give a set of points,
(x, y), that are on the graph of the parabola. You can then plot those points and
sketch the curve that passes through them.
For example, y = 2x2 + 4x + 5:
The x-coordinate of the vertex is x = -b / 2*a = -4 / 2*2 = -4 / 4 = -1
The y-coordinate of the vertex is y = 2(-1)2 + 4(-1) + 5 = 2 – 4 + 5 = 3
The vertex is (-1, 3)
The value of “a” is positive, so the graph opens upward.
Now select three values on either side of x = -1, and evaluate the equation at each
value:
x value
x = -4:
x = -3:
x = -2:
y value
2(-4)2 + 4(-4) + 5 = 32 – 16 + 5 = 21
2(-3)2 + 4(-3) + 5 = 18 – 12 + 5 = 11
2(-2)2 + 4(-2) + 5 = 8 – 8 + 5 = 5
coordinates
(-4, 21)
(-3, 11)
(-2, 5)
x = 0:
x = 1:
x = 2:
2(0)2 + 4(0) + 5 = 0 + 0 + 5 = 5
2(1)2 + 4(1) + 5 = 2 + 4 + 5 = 11
2(2)2 + 4(2) + 5 = 8 + 8 + 5 = 21
(0, 5)
(1, 11)
(2, 21)
The graph looks like this:
A mathematical model is an equation, or a set of equations, that describes the
behavior of an object or system. One example is the path of a object that is
launched into the air. When air resistance is ignored, the height of the object
is modeled by the equation h = h0 + v0t – (½)gt2, where h is the height of the
object at time t, h0 is the beginning height, v0 is the initial velocity in the
vertical direction, and g is the acceleration due to gravity.
Question 2:
Write a monomial and a polynomial using x as the variable. Find their product.
Monomial: 4x
Polynomial: x2 + 4x + 4
Product: (4x)(x2 + 4x + 4) = 4x3 + 16x2 + 16x
Question 3:
Write two binomials of the form (a√b + c√f) and (a√b - c√f). Then find their product.
(2
)(
) (
)(
) (
)(
) (
)(
) (
)(
3 + 4 5 2 3 − 4 5 = 2 3 2 3 + 4 5 2 3 + 2 3 −4 5 + 4 5 −4 5
= 4 (3) + 8 15 − 8 15 −16 ( 5)
= 12 − 80
= −68
)