Finding a parabola typically means determining its equation and sketching its graph. Here's a step-by-step guide on how to achieve this, focusing on finding key features and using them to define the parabola:
1. Identifying the Form of the Equation
Parabolas can be represented in a few different forms:
- Standard Form (Vertex Form):
y = a(x - h)^2 + korx = a(y - k)^2 + h. This form directly reveals the vertex of the parabola as (h, k). - General Form:
y = ax^2 + bx + corx = ay^2 + by + c. While not as immediately informative as vertex form, you can convert to vertex form.
The choice of which equation to use depends on the information given. If you know the vertex, vertex form is ideal. If you know three points on the parabola, general form is often easier to work with.
2. Determining the Orientation
The variable that's squared dictates the axis of symmetry:
- If 'x' is squared (e.g.,
y = ax^2 + bx + c), the parabola opens either upwards (if a > 0) or downwards (if a < 0). The axis of symmetry is a vertical line. - If 'y' is squared (e.g.,
x = ay^2 + by + c), the parabola opens either to the right (if a > 0) or to the left (if a < 0). The axis of symmetry is a horizontal line.
3. Finding Key Features
To sketch a parabola, you'll need to determine key features like the vertex, axis of symmetry, intercepts, and any additional points.
3.1. Vertex
- Vertex Form: The vertex (h, k) is directly given in the equation
y = a(x - h)^2 + korx = a(y - k)^2 + h. - General Form: To find the vertex (h, k) of
y = ax^2 + bx + c:h = -b / 2a(x-coordinate of the vertex)k = f(h)(y-coordinate of the vertex – substitute 'h' back into the equation)
- For
x = ay^2 + by + c:k = -b / 2a(y-coordinate of the vertex)h = f(k)(x-coordinate of the vertex - substitute 'k' back into the equation)
3.2. Axis of Symmetry
- The axis of symmetry is a vertical line
x = hif the parabola opens up or down. - The axis of symmetry is a horizontal line
y = kif the parabola opens left or right.
3.3. Intercepts
- Y-intercept: Set
x = 0in the equation and solve fory. - X-intercept(s): Set
y = 0in the equation and solve forx. You might need to use the quadratic formula:x = (-b ± √(b^2 - 4ac)) / 2a- If
b^2 - 4ac > 0, there are two distinct x-intercepts. - If
b^2 - 4ac = 0, there is one x-intercept (the vertex touches the x-axis). - If
b^2 - 4ac < 0, there are no real x-intercepts.
- If
- When the parabola opens to the side, the intercepts are reversed. Set
y = 0to find the x-intercept, andx = 0to find the y-intercept.
3.4. Additional Points
Calculate a few additional points on either side of the vertex to get a more accurate graph. Choose x-values (or y-values if the parabola opens left/right) near the vertex and plug them into the equation to find the corresponding y-values (or x-values). Symmetry can help you find points easily.
4. Graphing the Parabola
- Plot the vertex, intercepts, and any additional points you've calculated.
- Draw a smooth curve through the points, ensuring the parabola is symmetrical about the axis of symmetry.
- Extend the parabola beyond the plotted points, indicating that it continues indefinitely.
Example
Let's say we have the equation y = x^2 - 4x + 3.
- Form: General Form (
y = ax^2 + bx + c) - Orientation: Opens upwards (a = 1, which is positive)
- Vertex:
h = -b / 2a = -(-4) / (2 * 1) = 2k = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1- Vertex: (2, -1)
- Axis of Symmetry:
x = 2 - Y-intercept: Set
x = 0:y = (0)^2 - 4(0) + 3 = 3. Y-intercept: (0, 3) - X-intercepts: Set
y = 0:0 = x^2 - 4x + 3. Factor:0 = (x - 3)(x - 1). X-intercepts: x = 1 and x = 3, giving us points (1, 0) and (3, 0). - Additional Points: Let's find the point where x = 4:
y = (4)^2 - 4(4) + 3 = 16 - 16 + 3 = 3. Point: (4, 3).
Now plot these points and draw the parabola.
In summary, finding a parabola involves determining its form, orientation, key features (vertex, axis of symmetry, intercepts), plotting these points, and drawing a smooth curve through them.
