Construct graphs for the parabola

for different values of a, b, and c. (a, b, c can be any rational numbers).

How can we draw a graph of a quadratic function?

Explore the way to draw the graph as following steps.

< Step 1>

What does "a" mean?

Consider a graph of .

As you can see, the shape of the graph is parabola. If a>0, the parabola opens upward and the graph appears in the upper part(1st and 2nd quadrants) of a coordinate plane. On the other hand, if a<0, then the parabola opens downward and the graph appears in the lower part(3rd and 4th quadrants) of a coordinate plane. Moreover, the larger the value | a |, the narrower the width of a graph. In this graph, the vertex is (0, 0) and the maximum or minimum value is 0.

<Step 2>

Consider a graph of a function . Observe below file, and then what happens as the 'c' changes?

 Here, the blue one is and the red one is .

  

As 'c' increases, the graphs are going up along y-axis; that is, the graph of function is to translate by 'c' along y-axis. Thus, the vertex is (0, c) and the minimum or maximum is c.  

<Step 3>

Explore a graph of a function .

 Here, the blue one is , and he red one is .

As 'p' changes, the graph is moving along x-axis; that is the graph of the function is to translate by 'p' along x-axis. Thus, the vertex is (p, 0) and minimum or maximum is 0.

<Step 4>

Explore a graph of a function .

By <step 2> and <step 3>, its graph is to translate by 'p' along x-axis first, and then to shift by 'q' vertically.

For example, draw the graph of the function .

 As you can see, we can get the graph through the following process.

First, draw the graph of , and then shift the graph by -3 along x-axis, which is the graph of . Finally, translate  by 1 along y-axis, then we can get the graph of . 

In the form ,  the vertex is (p, q) and minimum or maximum is q.

<Step 5>

Finally, consider a graph of a function .

To use previous steps, we need to transform the equation to of form using the method of completing the square.

Therefore, the graph of is to translate by along x-axis and by along y-axis. 

Now, let's explore using following GSP file. To download the file, Click Here!

If you want to change the value 'a', 'b', and 'c', drag the points 'a' , 'b', or 'c'. Also, you can press action buttons.

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