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

__Steps:__

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

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