To graph a quadratic intercept, you identify the x-intercepts from the intercept form of the quadratic equation, plot those points, then find and plot the vertex to complete the parabola.
Here's a detailed breakdown:
Understanding Intercept Form
A quadratic equation in intercept form looks like this:
- y = a(x - p)(x - q)
- Where p and q are the x-intercepts (the points where the parabola crosses the x-axis).
- The value a determines the parabola's direction (upward if positive, downward if negative) and its "stretch."
Steps to Graph a Quadratic from Intercept Form:
-
Identify the x-intercepts: From the equation y = a(x - p)(x - q), the x-intercepts are directly p and q. Remember to consider the sign within the parentheses. For example, if the equation is y = (x - 2)(x + 3), the x-intercepts are at x = 2 and x = -3.
- Example: In the equation y = 2(x - 1)(x + 4), the x-intercepts are at x=1 and x=-4.
-
Plot the x-intercepts: Mark these points on the x-axis of your graph.
-
Find the Axis of Symmetry: The axis of symmetry is a vertical line that passes through the midpoint of the x-intercepts. You can find it by averaging the x-intercepts:
- Axis of Symmetry (x) = (p + q)/2
- Example from our equation y = 2(x - 1)(x + 4) the axis of symmetry would be (1 + -4)/2= -1.5
-
Find the Vertex: The vertex is the point where the parabola changes direction.
- The x-coordinate of the vertex is the same as the axis of symmetry.
- Substitute the x-coordinate of the axis of symmetry (the x-coordinate of the vertex) into the original intercept form equation to solve for the y-coordinate of the vertex.
- Example: from our equation y = 2(x - 1)(x + 4) the x-coordinate of the vertex is -1.5, plug that into the equation: y = 2(-1.5 - 1)(-1.5 + 4) so y= 2(-2.5)(2.5) thus y = -12.5 so the vertex is (-1.5, -12.5).
-
Plot the Vertex: Plot the vertex on the graph.
-
Find an additional point if needed for accuracy.
-
Sketch the Parabola: Draw a smooth curve that passes through the x-intercepts and the vertex. Make sure the parabola opens upward if 'a' is positive and downward if 'a' is negative. As shown in the reference, the parabola will be slightly off the graph if needed, but you can still connect the dots.
Visual Representation of the Process
| Step | Description |
|---|---|
| 1. Identify x-intercepts | Find p and q from the equation y=a(x-p)(x-q). |
| 2. Plot x-intercepts | Mark p and q on the x-axis. |
| 3. Find Axis of Symmetry | Calculate x = (p + q)/2. |
| 4. Find the Vertex | Substitute the axis of symmetry's x value into equation and solve for y. |
| 5. Plot the Vertex | Mark the vertex (x, y) on the graph. |
| 6. Find additional points | If needed for more accuracy, substitute other x values into the equation. |
| 7. Sketch the Parabola | Draw the smooth curve. |
