10.6 Graphing Quadratic Equations—Vertex and Intercept Method
One useful strategy that is used to get a quick sketch of a quadratic equation is to identify 3 key points of the quadratic: its vertex and the two intercept points. From these 3 points, it’s possible to sketch out a rough graph of what the quadratic graph looks like.
The intercepts are where the quadratic equation crosses the
The vertex is found by using the quadratic equation where the discriminant equals zero, which gives us the
The vertex then takes the form of
What is new here is finding the vertex, so consider the following examples.
Example 10.6.1
Find the vertex of
For this equation,
This means that the
We now use this
The vertex is at
The
For this problem, the quadratic factors to
Trying to sketch this curve will be somewhat challenging if there is to be any semblance of accuracy.
When this happens, it is quite easy to fill in some of the places where there may have been coordinates by using a data table.
For this graph, choose values from
First, find the value of
Put this value in the table and then carry on to complete all of it.
2 | 9 |
1 | 0 |
0 | −7 |
−1 | −12 |
−2 | −15 |
−3 | −16 |
−4 | −15 |
−5 | −12 |
−6 | −7 |
−7 | 0 |
−8 | 9 |
Placing all of these coordinates on the graph will generate a graph showing increased detail, as shown below. All that remains is to draw a curve that connects the points on the graph. The level of detail required to draw the curve only depends on the unique characteristics of the curve itself.
Remember:
For the quadratic equation
The following questions will ask you to sketch the quadratic function using the vertex and the x-intercepts and then later to draw a data table to find the coordinates of data points from which to draw a curve.
Both approaches are quite valuable, the difference is only in the details, which if required can use both techniques to general a curve in increased detail.
Example 10.6.2
Find the vertex of
In the equation,
This means that the
We now use this
The vertex is at
The
For this problem, the quadratic factors to
Trying to sketch this curve will be somewhat challenging if there is to be any semblance of accuracy.
When this happens, it is quite easy to fill in some of the places where there may have been coordinates by using a data table.
For this graph, choose values for
Put this value in the table and then carry on to complete all of it.
5 | 7 |
4 | 0 |
3 | −5 |
2 | −8 |
1 | −9 |
0 | −8 |
−1 | −5 |
−2 | 0 |
−3 | 7 |
Placing all of these coordinates on the graph will generate a graph showing increased detail as shown below. All that remains is to draw a curve that connects the points on the graph. The level of detail you require to draw the curve only depends on the unique characteristics of the curve itself.
Remember:
For the quadratic equation
The following questions will ask you to sketch the quadratic function using the vertex and the
Both approaches are quite valuable. The difference is only in the detail. If required, you can use both techniques to generate a curve in increased detail.
Questions
Find the vertex and intercepts of the following quadratics. Use this information to graph the quadratic.
First, find the line of symmetry for each of the following equations. Then, construct a data table for each equation. Use this table to graph the equation.