Lesson Objectives

- Learn how to sketch the graph of a parabola

## How to Sketch the Graph of a Parabola

In this lesson, we will learn how to sketch the graph of a parabola. In the last lesson, we learned how to find the vertex form of a parabola. Recall the two forms:

Example #1: Graph each parabola. $$f(x)=2x^2 + 4x - 2$$ First, we will plot the vertex. To find the vertex, we write the function in vertex form: $$f(x)=2(x + 1)^2 - 4$$ Our vertex occurs at the point (-1, -4): To get additional points, we will use the step pattern. This pattern is extremely simple. We take the value of a and multiply by a pattern: $$1, 3, 5,...$$ In other words, we multiply a, which is 2 here, by 1, then by 3, then by 5, and so on and so forth. This would give us values of: $$2, 6, 10$$ From the vertex, we would move one unit right and then 2 units up. This gives us a point at: $$(-1 + 1, -4 + 2)$$ $$(0, -2)$$ Next, we would use the next number in the step pattern. We would move one unit right and then 6 units up. This gives us a point at: $$(0 + 1, -2 + 6)$$ $$(1, 4)$$ At this point, we can just reflect the points across the axis of symmetry (x = -1) to get additional points. $$(-2, -2)$$ $$(-3, 4)$$ Let's plot these points and then sketch our parabola:

### Standard Form of a Parabola

$$f(x)=ax^2 + bx + c$$### Vertex Form of a Parabola

$$f(x)=a(x - h)^2 + k$$ From the vertex form, we immediately know the vertex of the parabola. This occurs at the point (h, k).### Graphing a Parabola Using the Step Pattern

There are many ways to graph a parabola, however, the easiest way to graph most parabolas is using the step pattern. Let's see an example.Example #1: Graph each parabola. $$f(x)=2x^2 + 4x - 2$$ First, we will plot the vertex. To find the vertex, we write the function in vertex form: $$f(x)=2(x + 1)^2 - 4$$ Our vertex occurs at the point (-1, -4): To get additional points, we will use the step pattern. This pattern is extremely simple. We take the value of a and multiply by a pattern: $$1, 3, 5,...$$ In other words, we multiply a, which is 2 here, by 1, then by 3, then by 5, and so on and so forth. This would give us values of: $$2, 6, 10$$ From the vertex, we would move one unit right and then 2 units up. This gives us a point at: $$(-1 + 1, -4 + 2)$$ $$(0, -2)$$ Next, we would use the next number in the step pattern. We would move one unit right and then 6 units up. This gives us a point at: $$(0 + 1, -2 + 6)$$ $$(1, 4)$$ At this point, we can just reflect the points across the axis of symmetry (x = -1) to get additional points. $$(-2, -2)$$ $$(-3, 4)$$ Let's plot these points and then sketch our parabola:

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