Shifts of a Parabola
Let's dive into an important concept—parabolas and how they move across the coordinate plane. Understanding this well will significantly improve your grasp of quadratic equations, especially for your GRE prep.
1. General Equation of a ParabolaA parabola is represented by the general form:
\(y = a(x − h)^2 + k\)
or
\(x = a(y − k)^2 + h\)
Where:
• (h,k) is the vertex of the parabola.
• a affects the direction and width of the parabola.
2. Movements of ParabolasTo understand how parabolas move, let’s break down the transformations:
a. Vertical Shift:
•
Upward: When we add a constant to the equation, it shifts the parabola upward. For
example:
\(y = x^2 + 2\)
This shifts the graph up by 2 units, and the vertex is at (0, 2).
•
Downward: Subtracting a constant shifts the parabola downward. For instance:
\(y = x^2 – 4\)
This shifts the graph down by 4 units, with the vertex at (0, −4).
b. Horizontal Shift:
•
Left Shift: Adding a value inside the squared term shifts the parabola left. For example:
\(y = (x + 6)^2\)
This shifts the parabola 6 units to the left, and the vertex is at (-6, 0).
•
Right Shift: Subtracting a value moves the parabola to the right. For example:
\(y = (x - 8)^2\)
This shifts the graph 8 units to the right, and the vertex is at (8, 0).
c. Combining Shifts:
We can combine both horizontal and vertical shifts. For example:
\(y = (x + 6)^2 – 4\)
This shifts the parabola 6 units to the left and 4 units down, giving the vertex at (-6, -4).
3. Direction of Opening:
• If a > 0, the parabola opens upwards (like the green parabolas).
• If a < 0, it opens downwards.
4. Graph Interpretation:
The image also provides a table showing how the points change as we plot the parabola y = x^2.
This basic parabola (centered at (0, 0)) shifts and changes based on the equations we explored.