Tiffany N. KeysŐ Assignment 2:
Look at that Parabola!
A quadratic function has the form y = ax2 + bx +c, where a is not equal to zero. The U-shaped graph of a quadratic function is called a parabola. The graphs of all quadratic functions are related to the graphs y = x2 and y = - x2
y = x2 y = - x2
These graphs have the following characteristics:
¤ The origin is the lowest point on the graph of y = x2 and the highest point of the graph of y = -x2
¤ The lowest point or the highest point on the graph of a quadratic function is called the vertex.
¤ The axis of symmetry for the graph of a quadratic function is the vertical line through the vertex. The graphs above are both symmetric about the axis of symmetry or, in these cases, the y – axis.
INVESTIGATION: Construct graphs for the parabola y = ax2 + bx +c for different values of a, b, and c.
First, letŐs begin with substituting in values for a:
Y = 3x2 y = 2x2 y = x2
Y = -3x2 y = -2x2 y = -x2
á The parabola always passes through the origin.
á The coordinates of the vertex of the parabola do not change in each equation as different values for a are substituted in.
á The concavity of the parabola is the aspect of the graph changes in each equation. As a increases, the concavity of the parabola decreases. If a is negative, then it is reflected across the x-axis.
Next, letŐs observe what happens when different values for b are substituted in:
Y = x2+3x y = x2+2x y = x2 + x
Y = x2- 3x y = x2- 2x y = x2 - x
á As above, the parabola always passes through the origin.
á However, the coordinates of the vertex of the parabola are the aspect of the graph that changes in each equation. As b increases, the vertex shifts across the x-axis. If b is negative, then it reflected across the y-axis.
á The concavity of each parabola does not change in each equation as different values of b are substituted in.
Lastly, letŐs notice the difference in the graph when values for c are substituted in:
Y = x2+ x + 3 y = x2 + x + 2 y = x2 + x + 1
Y = x2+ x - 3 y = x2 + x - 2 y = x2 + x - 1
á The parabola does not pass through the same point as c changes.