To help your students understand vertical and horizontal shifts in graphs they need to start thinking in terms of x/y-intercepts, not x/y values. We will label this the method the horizontal slide vs the vertical slide.
Horizontal vs Vertical Slides of Function Graphs
Here is the graph of f(x) = x^2
Here is the graph of x^2 with a Vertical shift of 2 units (f(x) = x^2 + 2)
Here is a graph of x^2 with a horizontal shift of 2 units f(x) = (x-2)^2
Students' Trouble In Understanding
Most students tend to understand vertical shifts. It seems intuitive to them that adding 2 to x^2 will shift the graph 2 units in the positive direction. However, students tend not to understand the horizontal shifts. It seems backwards to them. The reason for this is that students are concentrating on what is being done to the variables as opposed to the x/y-intercepts. The task of this activity is not mastery but to shift the students' focus to what's happening with the intercepts instead of the what is being done to the variable.
What You Will Need
- graph paper
- multiple color markers
- Activity Sheet
Step 1: On blank (x,y) coordinate graphing paper have students plot the following graph by generating random points.
f(x) = x^2
Step 2: Have students analyze the graph and determine the x-intercept and the y-intercept.
Step 3: Ask them what would you need to do to the graph of x^2 to change the y-intercept.
Step 4: Have them redraw the graph of x^2 anywhere else they want on the y-axis as long at it doesn't shift to the left or right.
Step 5: Ask the question how many units did your graph shift upward or downward?
Step 6: Have them contemplate what their new function will look like. Will it be x^2 plus 2, minus 2, multiplied by 2, divided by 2, etc.
Step 7: Show them what the new function will look like
f(x) = x^2 + 2
Step 8: Have them determine what the function would like if their graph what shifted up 2 more units. What would it look like if it was shifted down 5 units?
Step 9: Have them draw the two new graphs and write the new functions beside them.
Step 10: Ask them what changed in the graph, what remained the same.
Repeat the process with Horizontal shifts, having them concentrate on the x-intercept as opposed to what is being done to the variable x