# Absolute Value Functions

Absolute value functions are explored, using an applet, by comparing the graphs of f(x) and h(x) = |f(x)|.

Function f(x) used is a quadratic function of the form

f(x) = ax2 + bx + c

The exploration is carried out by changing the parameters a, b and c included in f(x) above.

## Interactive Tutorial

 a = 1 -10+10 b = 0 -10+10 c = 1 -10+10
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1. click on the button above "draw" to start.
2. Use the sliders to set parameter a to zero, parameter b to zero and parameter c to a positive value; f(x)is a constant function. Compare the graph of f(x) in blue and that of h(x) = |f(x)| in red. Change c to a negative value and compare the graphs again. Use the definition of the absolute value functions to explain how can the graph of |f(x)| be obtained from the graph of f(x).
3. Keep the value of a equal to zero, select non zero values for b to obtain a linear function . How can the graph of h(x) be obtained from that of f(x)?
Hint: use the definition of the absolute value functions and reflection of a graph on the x-axis.
4. Set b and c to zero and select a positive value for a to obtain a quadratic function . Why are the two graphs the same? (Hint: use the definition of the absolute value functions).
5. Set b and c to zero and select a negative value for a to obtain a quadratic function . Why are the two graphs reflection of each other? (Hint: use the definition of the absolute value functions and reflection of a graph on the x-axis).
6. Keep the values of a and b as in 5 above and change gradually c from zero to some positive values. How can the graph of h(x) be obtained from that of f(x)?
7. Select different values for a, b and c and explore.

## Exercises

### Sketch the following functions and absolute value functions

1. f(x) = x - 1 and h(x) = |f(x)|
2. f(x) = x2 - 4 and h(x) = |f(x)|
3. f(x) = -x and h(x) = |f(x)|