And factors, as we remember from Primary Year, are whole numbers that divide other whole numbers. Better yet, if I can factorize something, I can write it as the product of two or more factors. So 4 and 8 are factors of If I tell you that h multiplied by k equals to 0 , the null factor law says that either h or k must be 0. In order to better understand, let us deconstruct the standard form of a quadratic function:.
There are two types of intercepts: x-int and y-int. An x-int occurs when the graph cuts the x -axis, which can be found by letting y become 0 and solving for x. When we apply this idea to our newly formed function 2 , we get:. Now we apply the null factor law. But which one? Either of these could be 0 , so now we have:. Simple algebra tells us that x can be either p or q. The vertex is [latex] h,k. If you want to convert a quadratic in vertex form to one in standard form, simply multiply out the square and combine like terms.
For example, the quadratic. It is more difficult to convert from standard form to vertex form. Then we square that number. We then complete the square within the parentheses. The axis of symmetry for a parabola is given by:. More specifically, it is the point where the parabola intercepts the y-axis.
Privacy Policy. Skip to main content. Quadratic Functions and Factoring. Search for:. Graphs of Quadratic Functions. The graph contains three points and a parabola that goes through all three.
The corresponding function is shown in the text box below the graph. If you drag any of the points, then the function and parabola are updated. See the section on manipulating graphs. The functions in parts a and b of Exercise 1 are examples of quadratic functions in standard form. If a is positive, the graph opens upward, and if a is negative, then it opens downward. Any quadratic function can be rewritten in standard form by completing the square. See the section on solving equations algebraically to review completing the square.
The steps that we use in this section for completing the square will look a little different, because our chief goal here is not solving an equation. Similarly, if it has already started opening up it will not turn around and start opening down all of a sudden. The dashed line with each of these parabolas is called the axis of symmetry. Every parabola has an axis of symmetry and, as the graph shows, the graph to either side of the axis of symmetry is a mirror image of the other side.
This means that if we know a point on one side of the parabola we will also know a point on the other side based on the axis of symmetry. We will see how to find this point once we get into some examples.
We should probably do a quick review of intercepts before going much farther. We also saw a graph in the section where we introduced intercepts where an intercept just touched the axis without actually crossing it. Finding intercepts is a fairly simple process. So, we will need to solve the equation,. There is a basic process we can always use to get a pretty good sketch of a parabola.
Here it is. Now, there are two forms of the parabola that we will be looking at. This first form will make graphing parabolas very easy.
Unfortunately, most parabolas are not in this form. The second form is the more common form and will require slightly and only slightly more work to sketch the graph of the parabola. There are two pieces of information about the parabola that we can instantly get from this function.
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