To begin an exploration of any series of graphs you must first start by looking at a base graph. For this exploration, our base graph will be y = 2x2 + 3x -4. This exploration will move the base graph around the coordinate plane. The purpose of this exercise is to show students how a base graph can be modified from its original form by using linear transformations. A linear transformation relocates the graph on the coordinate plane, but does not change its size or shape. Two types of linear transformation are reflections and translations.
Let's first start our exploration by graphing the equation y = 2x2 + 3x -4 on the coordinate plane.
Now, students need to notice the major points on the graph. For example, where is the vertex, and where does the graph cross the x and y-axis. After these questions are answered, educators need to ask students what would happen if we replaced x by x-4. The new equation would look like the following y = 2(x-4)2 + 3(x-4) -4:
The predictions were correct. Changing the equation to x - 4 moves the
graph 4 units to the right. Now, we can draw some conclusions. First, replacing
the x in the quadratic equations with x - s, with move the graph to the
right s units. The opposite is true for equation x + s. This moves the graph
s units the left.
Next, lets try to manipulate the equation to move the graph into the second quadrant. Ask students what they feel would accomplish this task. First, lets look at the base graph (the graph in red). The vertex lies in the third quadrant and we need to move it to the second quadrant and keep the original size/shape. If students don't have any idea, start off manipulating the "4" variable in the base graph. Let's first try to graph y = 2x2 + 3x -1.
By replacing the "-4" with a "-1", the graph moved
up. However, the graph is not completely in the 2nd quadrant.
Let's try "+1".
Now, we need to move the graph over to the left. Remind the students
of the earlier exercise. Below is the graph of y = 2(x+2)2 +
The graph is completely located in the 2nd quadrant. Recap with the students the step they took to derive the above graph. First, they manipulated one variable at a time to see the effects the variable has on the equation. Then, try different values of the variable to move the graph accordingly. When one task is complete, then begin manipulating the other variables until the final graph is drawn. Moving the base graph around the coordinate plane is called a translation.
Change the base equation to produce a graph concave down that shares the same vertex as the original base equation. I would remind students to plot this on the same set of axes, because it will be easier to tell if they share the same vertex. The base equation is
y = 2x2 + 3x -4. Let's try to manipulate the coefficient "2". We will replace "2" with the following values : 1, -1, -2
Looking at the graphs plotted, we can see that a negative sign moves
the graph into a concave down position, but they still do not share the
same vertex. The graph of y = -2x2 + 3x
-4 is the closest. The only point it shares with the base graph is
the y intercept. How would we graph 2 equations? The vertex of the original
equation is (-.75, -5.125). Remember this is the point we want to share.
Let's try taking a new approach to the problem. We know to flip the graph
over the x-axis we need to y = -f(x). This will place the vertex of the
new graph at (-.75, 5.125). Now we need to move the graph down 15.25 units.
The equation of the new graph is y = -(2x2+3x-4)-10.25
The two equations share a common vertex , but the original is concave up while the new equation is concave down. Flipping the graph over some line is called a reflection.
Using programs in the classroom utilizes the time to explore the unknown, instead of using your time to plot points on chalkboards or overheads.
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