Question

answer the following questions in complete sentences. make sure that you are giving thoughtful responses.
for all of the functions, the one you are manipulating will be red and the \parent function will be blue.
parent functions:
\t\t\t\t\t\t\tabsolute value functions:
\tperform the following actions and answer accordingly based on what you observe.
\t- move the slider for \a\.
\t - what happens to the graph when the slider for \a\ moves to the right?
\t
\t - what happens to the graph when the slider for \a\ moves to the left?
\t
\t - what happens to the graph when a is less than 1, but greater than 0?
\t
\t - what happens to the graph when a is less than 0?
\t
\t - based on the information from your answers above, what could you conclude about the value \a\?
\t

• move the slider for \h\.
• relative to the \parent graph\ what happens to your graph when you move the slider to the right?

• relative to the \parent graph,\ what happens to your graph when you move the slider to the left?

• based on your answers above, a change in the \h\ value results in what kind of shift to the function?

move the slider for \k\

• relative to the \parent graph,\ what happens to your graph when you move the slider to the right?

Explanation:

Step1: Analyze "a" in Absolute Value Function

The general form of an absolute value function is \( y = a|x - h| + k \). When moving the slider for "a" (vertical stretch/compression and reflection):

• Moving "a" right (increasing a > 0): As \( a \) increases (slider right, assuming \( a \) is positive and increasing), the graph vertically stretches (becomes narrower) because a larger \( |a| \) makes the function values grow faster. For example, if \( a = 2 \) vs \( a = 1 \), \( y = 2|x| \) is narrower than \( y = |x| \).
• Moving "a" left (decreasing a > 0): As \( a \) decreases towards 0 (slider left, \( a > 0 \)), the graph vertically compresses (becomes wider) because smaller \( |a| \) makes function values grow slower. For \( a = 0.5 \), \( y = 0.5|x| \) is wider than \( y = |x| \).
• \( 0 < a < 1 \): The graph vertically compresses (wider than the parent \( y = |x| \)) since \( |a| < 1 \), reducing the steepness of the V - shape.
• \( a < 0 \): The graph reflects over the x - axis (opens downward) and also has a vertical stretch or compression based on \( |a| \). For example, \( y=-2|x| \) opens downward and is narrower than \( y = |x| \), \( y=-0.5|x| \) opens downward and is wider.
• Conclusion about "a": The value of \( a \) affects the vertical stretch (if \( |a|>1 \)), compression (if \( 0 < |a| < 1 \)) of the absolute value graph, and a negative \( a \) reflects the graph over the x - axis.

Step2: Analyze "h" in Absolute Value Function

The term \( (x - h) \) in \( y = a|x - h| + k \) is related to horizontal shifts.

• Moving "h" right: If we increase \( h \) (slider right), the graph of \( y = a|x - h|+k \) shifts horizontally to the right. For example, \( y = |x - 2| \) is the graph of \( y = |x| \) shifted 2 units to the right.
• Moving "h" left: If we decrease \( h \) (slider left), the graph shifts horizontally to the left. For example, \( y = |x + 2|=|x-(-2)| \) is the graph of \( y = |x| \) shifted 2 units to the left.
• Conclusion about "h": A change in \( h \) results in a horizontal shift of the function. The direction of the shift is opposite to the sign of \( h \) in the formula (right shift when \( h \) is positive, left shift when \( h \) is negative in \( (x - h) \)).

Step3: Analyze "k" in Absolute Value Function (Partial, as per visible question)

The term \( k \) in \( y = a|x - h| + k \) is related to vertical shifts. When moving the slider for "k" to the right (assuming \( k \) is increasing), the graph of the absolute value function shifts vertically upward. For example, \( y = |x|+2 \) is the graph of \( y = |x| \) shifted 2 units up. If we were to move it left (decreasing \( k \)), it would shift downward.

Answer:

s (for each sub - question):

1. When the slider for "a" moves to the right (assuming \( a>0 \) and increasing), the graph of the absolute value function vertically stretches (becomes narrower).
2. When the slider for "a" moves to the left (assuming \( a>0 \) and decreasing towards 0), the graph vertically compresses (becomes wider).
3. When \( a \) is less than 1 but greater than 0, the graph of the absolute value function vertically compresses (is wider than the parent absolute value graph).
4. When \( a \) is less than 0, the graph of the absolute value function reflects over the x - axis (opens downward) and has a vertical stretch or compression depending on \( |a| \).
5. The value of "a" determines the vertical stretch (if \( |a|>1 \)) or compression (if \( 0 < |a| < 1 \)) of the absolute value graph, and a negative "a" reflects the graph over the x - axis.
6. When the slider for "h" moves to the right, the graph of the absolute value function shifts horizontally to the right relative to the parent graph.
7. When the slider for "h" moves to the left, the graph of the absolute value function shifts horizontally to the left relative to the parent graph.
8. A change in the "h" value results in a horizontal shift of the absolute value function.
9. When the slider for "k" moves to the right, the graph of the absolute value function shifts vertically upward relative to the parent graph.