To graph a quadratic sequence, which is represented by a quadratic function (parabola), follow these steps:
Steps to Graph a Quadratic Sequence
Here's a detailed breakdown of how to graph a quadratic sequence, referencing the provided steps:
- Determine the Parabola's Orientation:
- Check the coefficient of the $x^2$ term, typically denoted as 'a' in the general form $f(x) = ax^2 + bx + c$.
- If a > 0, the parabola opens upwards, resembling a U shape.
- If a < 0, the parabola opens downwards, resembling an inverted U shape.
- Find the Axis of Symmetry:
- The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves.
- Its equation is given by: x = h, where h = -b / 2a. This formula helps you find the x-coordinate of the vertex.
- Locate the Vertex:
- The vertex is the turning point of the parabola, either the minimum point (if the parabola opens upwards) or the maximum point (if the parabola opens downwards).
- The vertex coordinates are (h, k), where h is the x-coordinate (found in step 2) and k is the y-coordinate, calculated by substituting h back into the quadratic equation: k = f(h).
- Identify the Y-intercept:
- The y-intercept is the point where the parabola crosses the y-axis.
- It's found by setting x = 0 in the quadratic equation, so the y-intercept is f(0). This is usually just the 'c' value in the standard form.
- Determine the X-intercepts (if they exist):
- The x-intercepts are the points where the parabola crosses the x-axis.
- They are found by setting f(x) = 0 and solving the resulting quadratic equation for x. You can solve this using factoring, the quadratic formula, or other appropriate techniques.
- For example, using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
- Graph the Parabola:
- Plot the vertex, the y-intercept, and the x-intercepts (if any).
- Draw a smooth curve through the plotted points, ensuring that the parabola maintains its shape (U or inverted U) based on step 1.
- Use the axis of symmetry as a guide to ensure symmetry when drawing your curve. You can find additional points by substituting other x-values into the function, for example, the value to the left and right of the vertex to get the graph correct
Example
Let’s say you have the quadratic function: $f(x) = x^2 - 4x + 3$
- Step 1: a = 1, since a is positive, the parabola opens upward.
- Step 2: h = -(-4) / (2 * 1) = 4 / 2 = 2. Therefore, the axis of symmetry is x = 2.
- Step 3: To find the k value, substitute x = 2 into the function: k = 2^2 - 4*2 + 3 = 4 - 8 + 3 = -1. The vertex is at (2, -1).
- Step 4: To find the y-intercept, set x = 0: f(0) = 0^2 - 4*0 + 3 = 3. So the y-intercept is at (0, 3).
- Step 5: To find the x-intercepts, set f(x) = 0: x^2 - 4x + 3 = 0. This factors to (x - 3)(x - 1) = 0. So the x-intercepts are at (1, 0) and (3, 0).
- Step 6: Plot these key points, and draw the U-shaped curve of the parabola through them.
By following these steps, you can accurately graph any quadratic sequence represented by a quadratic function.
